# Isotropic distributions in the p-norm

Carlos A Pinzón

► To cite this version:

Carlos A Pinzón. Isotropic distributions in the p-norm: Uniform directionality in multi-dimensional p-normed spaces.. 2024. hal-04567089

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# Isotropic distributions in the p-norm

Carlos Pinzón*

May 2024

## Abstract

This article explores the concept of isotropy (uniform directionality) in different *p*-norms. This corresponds to distributions in *n* dimensions whose density function can be written as a function of the *p*-norm almost everywhere. First, we remark that sampling uniformly from the surface of a *p*-ball in *d* dimensions is not equivalent to sampling from its volume and projecting to the surface unless *p* is 1, 2 or ∞. As we argue, the volume-based procedure is the correct definition for sampling directions uniformly at random for arbitrary *p*-norms. Second, we prove that only some specific distributions, that we call independent-isotropic, have the property that the joint distribution of i.i.d. coordinates is isotropic. Finally, we generalize the independent-isotropic family with two additional parameters, we show that two different subfamilies can be differentiated (light and heavy-tailed) and we expose multiple connections (geometric, analytical and statistical) that exist between them.

## 1 Introduction

The gaussian distribution has the distinctive property that if X and Y are independent and gaussian, then the vector $V = (X, Y)$ has a 2-dimensional distribution that is radially symmetric (it is the same in all directions), so its density function $f: \mathbb{R}^2 \to \mathbb{R}_{\ge 0}$ can be written as a function of the norm $f(x, y) = g(\|(x, y)\|)$. Except for scaled versions of the gaussian, no other distribution has this property. For instance, as shown in Figure 1, neither the exponential nor the uniform distribution are radially symmetric.

The shapes produced by the Laplace and uniform distributions (Figure 1) are not arbitrary. They are precisely the shapes of the surfaces of the 2D unit balls in different *p*-norms, and, as with the Gaussian distribution and its exclusive relationship with the circle, the Laplace and the uniform are uniquely related with the rhombus and the square.

In this paper, we study this phenomenon in more detail for different *p*-norms and dimensionalities. We obtain a family of *independent-isotropic* distributions that

<footnote>*Independent researcher. caph1993@gmail.com</footnote>
# Isotropic distributions in the p-norm

Carlos Pinzón*

May 2024

## Abstract

This article explores the concept of isotropy (uniform directionality) in different *p*-norms. This corresponds to distributions in *n* dimensions whose density function can be written as a function of the *p*-norm almost everywhere. First, we remark that sampling uniformly from the surface of a *p*-ball in *d* dimensions is not equivalent to sampling from its volume and projecting to the surface unless *p* is 1, 2 or ∞. As we argue, the volume-based procedure is the correct definition for sampling directions uniformly at random for arbitrary *p*-norms. Second, we prove that only some specific distributions, that we call independent-isotropic, have the property that the joint distribution of i.i.d. coordinates is isotropic. Finally, we generalize the independent-isotropic family with two additional parameters, we show that two different subfamilies can be differentiated (light and heavy-tailed) and we expose multiple connections (geometric, analytical and statistical) that exist between them.

## 1 Introduction

The gaussian distribution has the distinctive property that if X and Y are independent and gaussian, then the vector $V = (X, Y)$ has a 2-dimensional distribution that is radially symmetric (it is the same in all directions), so its density function $f: \mathbb{R}^2 \to \mathbb{R}_{\ge 0}$ can be written as a function of the norm $f(x, y) = g(\|(x, y)\|)$. Except for scaled versions of the gaussian, no other distribution has this property. For instance, as shown in Figure 1, neither the exponential nor the uniform distribution are radially symmetric.

The shapes produced by the Laplace and uniform distributions (Figure 1) are not arbitrary. They are precisely the shapes of the surfaces of the 2D unit balls in different *p*-norms, and, as with the Gaussian distribution and its exclusive relationship with the circle, the Laplace and the uniform are uniquely related with the rhombus and the square.

In this paper, we study this phenomenon in more detail for different *p*-norms and dimensionalities. We obtain a family of *independent-isotropic* distributions that

<footnote>*Independent researcher. caph1993@gmail.com</footnote>
<figure><img src="image_1.png" /><figcaption>Figure 1: Three distributions in 2D consisting of i.i.d. coordinates and the geometric shape of their decay patterns. These shapes are 2D spheres in different $p$-norms: $p = 1, 2, \infty$ respectively.</figcaption></figure>

have the property that the joint distribution of i.i.d. coordinates is isotropic, and we show that this family is unique. This distribution matches the Gaussian for $p = 2$, the Laplace for $p = 1$, and the uniform in $(-1, 1)$ for $p = \infty$, which explains the patterns in Figure 1.

Distributions like the Gaussian and the Laplace are light-tailed, and they have heavy-tail analogues, such as the Cauchy and the Lomax distributions. These distributions have power-law decays, and they are needed in financial and insurance applications where the objective is to analyze the magnitude and chance of large risks, represented by the tails. The geometric interconnection between these light and heavy-tailed distributions is also explored in this paper by introducing two parameters that generalize the independent-isotropic distribution.

# 2 Preliminaries

This section presents the notation used in the paper.

Vectors are represented as any of $\mathbf{x} = x_{1..d} = (x_1, ..., x_d)$. We use uppercase letters for random variables, lowercase for their realizations, and $\text{Uni}$, $\text{Gauss}$, $\text{Gamma}$, etc. to denote distributions. Densities can be represented as $p_X(x)$ and joint densities as $p_{X_1,X_2}(x_1, x_2)$.

## 2.1 $p$-norms

A **norm** on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb{R}_{\ge 0}$ that satisfies (i) $\|\mathbf{x}\| = 0$ if and only if $\mathbf{x} = \mathbf{0}$, (ii) scalar multiplication ($\|a\mathbf{x}\| = |a|\|\mathbf{x}\|$), and (iii) the triangle inequality for norms ($\|\mathbf{x} + \mathbf{y}\| \le \|\mathbf{x}\| + \|\mathbf{y}\|$).

For $p \in (0, \infty]$ and $d \in \{1, 2, ...\}$, we define $\|\mathbf{x}\|_p \stackrel{\text{def}}{=} (\sum_{i=1}^d |x_i|^p)^{1/p}$ and $\|\mathbf{x}\|_{\infty} = \lim_{p \to \infty} \|\mathbf{x}\|_p = \max_{i=1}^d |x_i|$. This function is a norm for $p \in [1, \infty]$ and is called the **$p$-norm**. For $p \in (0, 1)$, the correct term is **seminorm** because the triangle inequality fails to hold.
<figure><img src="image_2.png" /><figcaption>Figure 1: Three distributions in 2D consisting of i.i.d. coordinates and the geometric shape of their decay patterns. These shapes are 2D spheres in different $p$-norms: $p = 1, 2, \infty$ respectively.</figcaption></figure>

have the property that the joint distribution of i.i.d. coordinates is isotropic, and we show that this family is unique. This distribution matches the Gaussian for $p = 2$, the Laplace for $p = 1$, and the uniform in $(-1, 1)$ for $p = \infty$, which explains the patterns in Figure 1.

Distributions like the Gaussian and the Laplace are light-tailed, and they have heavy-tail analogues, such as the Cauchy and the Lomax distributions. These distributions have power-law decays, and they are needed in financial and insurance applications where the objective is to analyze the magnitude and chance of large risks, represented by the tails. The geometric interconnection between these light and heavy-tailed distributions is also explored in this paper by introducing two parameters that generalize the independent-isotropic distribution.

# 2 Preliminaries

This section presents the notation used in the paper.

Vectors are represented as any of $\mathbf{x} = x_{1..d} = (x_1, ..., x_d)$. We use uppercase letters for random variables, lowercase for their realizations, and Uni, Gauss, Gamma, etc. to denote distributions. Densities can be represented as $p_X(x)$ and joint densities as $p_{X_1,X_2}(x_1, x_2)$.

## 2.1 $p$-norms

A **norm** on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb{R}_{\ge 0}$ that satisfies (i) $\|\mathbf{x}\| = 0$ if and only if $\mathbf{x} = \mathbf{0}$, (ii) scalar multiplication ($\|a\mathbf{x}\| = |a|\|\mathbf{x}\|$), and (iii) the triangle inequality for norms ($\|\mathbf{x} + \mathbf{y}\| \le \|\mathbf{x}\| + \|\mathbf{y}\|$).

For $p \in (0, \infty]$ and $d \in \{1, 2, ...\}$, we define $\|\mathbf{x}\|_p \stackrel{\text{def}}{=} (\sum_{i=1}^d |x_i|^p)^{1/p}$ and $\|\mathbf{x}\|_{\infty} = \lim_{p \to \infty} \|\mathbf{x}\|_p = \max_{i=1}^d |x_i|$. This function is a norm for $p \in [1, \infty]$ and is called the **$p$-norm**. For $p \in (0, 1)$, the correct term is **seminorm** because the triangle inequality fails to hold.
## 2.2 p-spheres, p-balls and p-simplexes

For fixed $p \in (0, \infty]$ and $d \in \{1, 2, \dots\}$, **unit p-sphere** is the set $\text{Sph}_p^d \subseteq \mathbb{R}^d$ of points with $p$-norm equal to 1 and the **unit p-ball** is the set $\text{Ball}_p^d \subseteq \mathbb{R}^d$ of points with $p$-norm at most 1. In geometric terms, the unit sphere is the convex hull of the unit sphere and the unit sphere is the surface (or topological frontier) of the unit ball. The dimensionalities of $\text{Ball}_p^d$ and $\text{Sph}_p^d$ are $d$ and $d-1$ respectively, both in a Hausdorff and a packing-measure sense.

The **unit simplex**, $\Delta_p^d \stackrel{\text{def}}{=} \text{Sph}_p^d \cap (\mathbb{R}_{\ge 0}^d)$ is the restriction of the hypersphere to vectors with non-negative entries.

## 2.3 Continuous distributions

The Lebesgue decomposition theorem [18] states that any real distribution can be decomposed as a mixture of an absolutely continuous distribution, a discrete distribution and a singular-continuous distribution. A random variable $X$ is singular-continuous if its domain $S \subseteq \mathbb{R}$ has measure zero (like a discrete) yet it does not have point masses (like a continuous). These domains can be built using fractals [8] with Hausdorff dimension lying between 0 (discrete) and 1 (continuous), like the Cantor set.

## 2.4 Notation and manipulation of distributions

We use the notation $\text{Dist}$ to denote an arbitrary distribution over some given domain $D$. In particular, the domain $\mathcal{R}$ is used to denote simultaneously either $\mathbb{R}$ (signed) or $\mathbb{R}_{\ge 0}$ (unsigned). This is because due to symmetry, it is equivalent to discuss isotropy in signed or unsigned domains. For some visualizations and analogies, signed is preferred, while for most of the mathematical purposes unsigned is simpler to manipulate.

We allow multiplication by scalars and usual usual arbitrary operations to be carried on distributions, e.g. $b \text{Dist}$ to denote the distribution of $bX$ for $X \sim \text{Dist}$, and when multiple distributions appear in a single formula, it is assumed that they are independent, e.g. $\text{Dist} + \text{Dist}$ denotes the distribution of $X + Y$ for $X \sim \text{Dist}$ and $Y \sim \text{Dist}$. Also, $\pm \text{Dist}$ denotes the distribution of $X$ or $-X$ with probability $1/2$ each, where $X \sim \text{Dist}$.

Finally, the joint distribution of $n$ i.i.d. random variables $X_i \sim \text{Dist}$ is represented as $[\text{Dist}]_{1 \dots n}$, so $(X, Y) \sim [\text{Dist}]_{1 \dots 2}$ denotes $X, Y \sim \text{Dist}$ and $X \perp Y$. This is used instead of $\text{Dist}^n$ to avoid confusion with $X^n$ for $X \sim \text{Dist}$.

## 2.5 Uniform distributions

We use the notation $\text{Uni}$ to denote the uniform distribution in $[0, 1)$, and $\text{Uni}(D)$ to denote the uniform distribution over a set $D$ that we restrict to be any of $(-1, 1)^d$, $\text{Ball}_p^d$, $\text{Sph}_p^d$, $\Delta_p^d$, or a finite set.
## 2.2 p-spheres, p-balls and p-simplexes

For fixed $p \in (0, \infty]$ and $d \in \{1, 2, \dots\}$, **unit p-sphere** is the set $\text{Sph}_p^d \subseteq \mathbb{R}^d$ of points with $p$-norm equal to 1 and the **unit p-ball** is the set $\text{Ball}_p^d \subseteq \mathbb{R}^d$ of points with $p$-norm at most 1. In geometric terms, the unit sphere is the convex hull of the unit sphere and the unit sphere is the surface (or topological frontier) of the unit ball. The dimensionalities of $\text{Ball}_p^d$ and $\text{Sph}_p^d$ are $d$ and $d-1$ respectively, both in a Hausdorff and a packing-measure sense.

The **unit simplex**, $\Delta_p^d \stackrel{\text{def}}{=} \text{Sph}_p^d \cap (\mathbb{R}_{\ge 0}^d)$ is the restriction of the hypersphere to vectors with non-negative entries.

## 2.3 Continuous distributions

The Lebesgue decomposition theorem [18] states that any real distribution can be decomposed as a mixture of an absolutely continuous distribution, a discrete distribution and a singular-continuous distribution. A random variable $X$ is singular-continuous if its domain $S \subseteq \mathbb{R}$ has measure zero (like a discrete) yet it does not have point masses (like a continuous). These domains can be built using fractals [8] with Hausdorff dimension lying between 0 (discrete) and 1 (continuous), like the Cantor set.

## 2.4 Notation and manipulation of distributions

We use the notation $\text{Dist}$ to denote an arbitrary distribution over some given domain $D$. In particular, the domain $\mathcal{R}$ is used to denote simultaneously either $\mathbb{R}$ (signed) or $\mathbb{R}_{\ge 0}$ (unsigned). This is because due to symmetry, it is equivalent to discuss isotropy in signed or unsigned domains. For some visualizations and analogies, signed is preferred, while for most of the mathematical purposes unsigned is simpler to manipulate.

We allow multiplication by scalars and usual usual arbitrary operations to be carried on distributions, e.g. $b \text{Dist}$ to denote the distribution of $bX$ for $X \sim \text{Dist}$, and when multiple distributions appear in a single formula, it is assumed that they are independent, e.g. $\text{Dist} + \text{Dist}$ denotes the distribution of $X + Y$ for $X \sim \text{Dist}$ and $Y \sim \text{Dist}$. Also, $\pm \text{Dist}$ denotes the distribution of $X$ or $-X$ with probability $1/2$ each, where $X \sim \text{Dist}$.

Finally, the joint distribution of $n$ i.i.d. random variables $X_i \sim \text{Dist}$ is represented as $[\text{Dist}]_{1 \dots n}$, so $(X, Y) \sim [\text{Dist}]_{1 \dots 2}$ denotes $X, Y \sim \text{Dist}$ and $X \perp Y$. This is used instead of $\text{Dist}^n$ to avoid confusion with $X^n$ for $X \sim \text{Dist}$.

## 2.5 Uniform distributions

We use the notation $\text{Uni}$ to denote the uniform distribution in $[0, 1)$, and $\text{Uni}(D)$ to denote the uniform distribution over a set $D$ that we restrict to be any of $(-1, 1)^d$, $\text{Ball}_p^d$, $\text{Sph}_p^d$, $\Delta_p^d$, or a finite set.
In these sets, the (Hausdorff) dimensionality of the domain is clear, and the corresponding Hausdorff measure is positive and finite, so the uniform distribution is well defined. And it is in this sense that it is uniform. For instance, Uni(Ball³) is a distribution over the 3D ball and is uniform in a volumetric sense, while Uni(Sph³) is uniform over its border in a superficial sense.

The most important of these domains is Uni(Ballₚ) in this paper, and an algorithm for sampling from this distribution is to let X ~ Uni((-1, 1)⁴) and accept X only if ||X||ₚ ≤ 1. This works for any combination of d and p, though the acceptance rate is small for small p or large d.

Alternatively, the random variates for Uni(Ball₂) can be defined as R·Uni(Sph²), where R ~ Uni². Notice that R ~ Uni, because small radii represent short circumferences, so they must have smaller probability of being generated than large radii.

## 2.6 Squigonometric functions

The squigonometric [23]<fnref n="1" /> values of πₚ and πₚarc are defined so that the area enclosed by the p-simplex in 2D is πₚ/4 and its p-arc-length (using the p-norm to measure distances) is πₚarc/2. Figure 2 shows the graph of the area-based πₚ and p-arc-length-based πₚarc values of π, for different values of p.

The area-based squigonometric functions sinₚ(t) and cosₚ(t) (shown in Figure 2) are defined, for p < ∞, as the unique solution to the coupled initial value problem

$$C'(t) = -S(t)^{p-1}; \quad S'(t) = C(t)^{p-1}; \quad C(0) = 1; \quad S(0) = 0, \quad (1)$$

and the p-arc-length-based functions sinₚarc(t) and cosₚarc(t) as the unique solution to

$$C'(t) = -S(t)^{p-1}(C(t)^{p(p-1)} + S(t)^{p(p-1)})^{-1/p}; \\ S'(t) = C(t)^{p-1}(C(t)^{p(p-1)} + S(t)^{p(p-1)})^{-1/p}; \quad C(0) = 1; \quad S(0) = 0.$$

These functions generalize sine and cosine for different p-norms, and they coincide if and only if p = 2, in which case, they also coincide with the usual sine and cosine functions. In general sinₚ and cosₚ have period 2πₚ and sinₚarc and cosₚarc have period πₚarc. All of them oscillate between -1 and 1.

In this paper, we will focus mostly on the area-based squigonometry. For p = ∞, we define

$$\sin_{\infty}(t) := \min(1, t); \quad \cos_{\infty}(t) := \min(1, 2-t)$$

for t ∈ [-1, 3], and we extend this to t ∈ ℝ with the functional equations sin_∞(t + 2) = cos_∞(t), sin_∞(-t) = sin_∞(t) and sin_∞(t + 8) = sin_∞(t).

<footnote n="1">¹The words “squigonometric” for trigonometry in the p-norm, and “squirrel” for circle in the p-norm (Sphₚ) come from the visual appearance of Sphₚ between a circle (p = 2) and a square (p = ∞).</footnote>
In these sets, the (Hausdorff) dimensionality of the domain is clear, and the corresponding Hausdorff measure is positive and finite, so the uniform distribution is well defined. And it is in this sense that it is uniform. For instance, Uni(Ball³) is a distribution over the 3D ball and is uniform in a volumetric sense, while Uni(Sph³) is uniform over its border in a superficial sense.

The most important of these domains is Uni(Ballₚ) in this paper, and an algorithm for sampling from this distribution is to let X ~ Uni((-1, 1)⁴) and accept X only if ||X||ₚ ≤ 1. This works for any combination of d and p, though the acceptance rate is small for small p or large d.

Alternatively, the random variates for Uni(Ball₂) can be defined as R·Uni(Sph²), where R ~ Uni². Notice that R ~ Uni, because small radii represent short circumferences, so they must have smaller probability of being generated than large radii.

## 2.6 Squigonometric functions

The squigonometric [23]<fnref n="1" /> values of πₚ and πₚarc are defined so that the area enclosed by the p-simplex in 2D is πₚ/4 and its p-arc-length (using the p-norm to measure distances) is πₚarc/2. Figure 2 shows the graph of the area-based πₚ and p-arc-length-based πₚarc values of π, for different values of p.

The area-based squigonometric functions sinₚ(t) and cosₚ(t) (shown in Figure 2) are defined, for p < ∞, as the unique solution to the coupled initial value problem

$$C'(t) = -S(t)^{p-1}; \quad S'(t) = C(t)^{p-1}; \quad C(0) = 1; \quad S(0) = 0, \quad (1)$$

and the p-arc-length-based functions sinₚarc(t) and cosₚarc(t) as the unique solution to

$$C'(t) = -S(t)^{p-1}(C(t)^{p(p-1)} + S(t)^{p(p-1)})^{-1/p}; \\ S'(t) = C(t)^{p-1}(C(t)^{p(p-1)} + S(t)^{p(p-1)})^{-1/p}; \quad C(0) = 1; \quad S(0) = 0.$$

These functions generalize sine and cosine for different p-norms, and they coincide if and only if p = 2, in which case, they also coincide with the usual sine and cosine functions. In general sinₚ and cosₚ have period 2πₚ and sinₚarc and cosₚarc have period πₚarc. All of them oscillate between -1 and 1.

In this paper, we will focus mostly on the area-based squigonometry. For p = ∞, we define

$$\sin_{\infty}(t) := \min(1, t); \quad \cos_{\infty}(t) := \min(1, 2-t)$$

for t ∈ [-1, 3], and we extend this to t ∈ ℝ with the functional equations sin_∞(t + 2) = cos_∞(t), sin_∞(-t) = sin_∞(t) and sin_∞(t + 8) = sin_∞(t).

<footnote n="1">¹The words “squigonometric” for trigonometry in the p-norm, and “squirrel” for circle in the p-norm (Sphₚ) come from the visual appearance of Sphₚ between a circle (p = 2) and a square (p = ∞).</footnote>
For $x, y \in [0, 1]$, the inverses of p-sine and p-cosine are

$$ \arccos_p(x) = \int_x^1 \frac{1}{(1-u^p)^{\frac{p-1}{p}}} du; \quad \arcsin_p(y) = \int_y^1 \frac{1}{(1-u^p)^{\frac{p-1}{p}}} du; $$

$$ \arccos_{\infty}(x) := \min(1, 2-x); \quad \arcsin_{\infty}(y) := \min(1, y). $$

It holds that

$$ \pi_p = 2 \arcsin_p(1) = 2 \arccos_p(0) = (2/p)B(1/p, 1/p) = \frac{2}{p} \frac{\Gamma(1/p)^2}{\Gamma(2/p)}; \quad \pi_{\infty} = 4. $$

<figure><img src="image_2.png" /><figcaption>Figure 2: Squigonometric functions (left) and areas and perimeters (arc-length) of the unit simplex in different p-norms (right).</figcaption></figure>

The curves $(\sin_p(t), \cos_p(t))$ with $t$ in $[0, \pi_p/4]$, and $(\sin_p^{\text{arc}}(t), \cos_p^{\text{arc}}(t))$ with $t$ in $[0, \pi_p^{\text{arc}}/2]$, are both equal to $\Delta_p^d$. However, the rate at which the curve is drawn is different when varying $t$ uniformly in its respective interval.

Consider the top left plot in Figure 3. Let $A(t)$ be the area of the sector that is enclosed by $\Delta_p^d$ and $(0, 0)$, and which lies below the line that crosses $(0, 0)$ and $(\cos_p(t), \sin_p(t))$, and let $L(t)$ be the p-arc-length of the curve segment of $\Delta_p^d$ from $(1, 0)$ to $(\cos_p^{\text{arc}}(t), \sin_p^{\text{arc}}(t))$.

With the first (area-based) parametrization, $A(t)$ is proportional to $t$ ([23], pages 59-60):

$$ A(t) = \frac{t}{2} \qquad (2) $$

With the second (p-arc-length-based), $L(t)$ is proportional to $t$.

# 3 Isotropy in the p-norm

## 3.1 Isotropy

In a nutshell, isotropy means uniformity in all directions. For example, in physics, electronics and telecommunications, an isotropic antenna is one that
For $x, y \in [0, 1]$, the inverses of p-sine and p-cosine are

$$ \arccos_p(x) = \int_x^1 \frac{1}{(1-u^p)^{\frac{p-1}{p}}} du; \quad \arcsin_p(y) = \int_y^1 \frac{1}{(1-u^p)^{\frac{p-1}{p}}} du; $$

$$ \arccos_{\infty}(x) := \min(1, 2-x); \quad \arcsin_{\infty}(y) := \min(1, y). $$

It holds that

$$ \pi_p = 2 \arcsin_p(1) = 2 \arccos_p(0) = (2/p)B(1/p, 1/p) = \frac{2}{p} \frac{\Gamma(1/p)^2}{\Gamma(2/p)}; \quad \pi_{\infty} = 4. $$

<figure><img src="image_3.png" /><figcaption>Figure 2: Squigonometric functions (left) and areas and perimeters (arc-length) of the unit simplex in different p-norms (right).</figcaption></figure>

The curves $(\sin_p(t), \cos_p(t))$ with $t$ in $[0, \pi_p/4]$, and $(\sin_p^{\text{arc}}(t), \cos_p^{\text{arc}}(t))$ with $t$ in $[0, \pi_p^{\text{arc}}/2]$, are both equal to $\Delta_p^d$. However, the rate at which the curve is drawn is different when varying $t$ uniformly in its respective interval.

Consider the top left plot in Figure 3. Let $A(t)$ be the area of the sector that is enclosed by $\Delta_p^d$ and $(0, 0)$, and which lies below the line that crosses $(0, 0)$ and $(\cos_p(t), \sin_p(t))$, and let $L(t)$ be the p-arc-length of the curve segment of $\Delta_p^d$ from $(1, 0)$ to $(\cos_p^{\text{arc}}(t), \sin_p^{\text{arc}}(t))$.

With the first (area-based) parametrization, $A(t)$ is proportional to $t$ ([23], pages 59-60):

$$ A(t) = \frac{t}{2} \qquad (2) $$

With the second (p-arc-length-based), $L(t)$ is proportional to $t$.

# 3 Isotropy in the p-norm

## 3.1 Isotropy

In a nutshell, isotropy means uniformity in all directions. For example, in physics, electronics and telecommunications, an isotropic antenna is one that
radiates electromagnetic waves uniformly in all directions. A directional antenna, in contrast, radiates more power in some directions than in others, like a parabolic or a Yagi-Uda antenna.

The isotropic antenna has the property that its 3D radiation pattern $f(X, Y, Z)$ in void space consists of concentric spheres centered at the origin (where the antenna is), and the density of power per unit of area at any point $(x, y, z)$ depends exclusively on the straight line distance $||(x, y, z)||$. More precisely, the power density at $(x, y, z)$ is inversely proportional to the surface area of the sphere that contains $(x, y, z)$. This holds because, first, in void space, the power radiated by any antenna propagates without losses in all directions, hence the net power received over any hypothetical convex surface containing the antenna is constant; and second, for an isotropic antenna, the power is distributed over any centered sphere uniformly.

Similarly, in the context of probability theory, we may define continuous isotropic distributions as having the property that its density function can be written as a function of the norm. Since density functions are only unique up to null-sets, the functional relationship should not be imposed everywhere, but only almost everywhere.

**Definition 1.** Let $\mathcal{R} = \mathbb{R}$ or $\mathcal{R} = [0, \infty)$. A $d$-dimensional continuous distribution with density function $f: \mathcal{R}^d \to [0, \infty)$ (unique up to null sets), is said to be p-isotropic (default $p = 2$) if there is a function $g$ such that

$$f(\boldsymbol{x}) = g(\|\boldsymbol{x}\|_p) \text{ for almost all } \boldsymbol{x} \in \mathcal{R}^d. \qquad (3)$$

The signed and unsigned definitions of isotropy in Definition 2 are related to one another: $[\pm X_1, ..., \pm X_d]$ follows a $p$-isotropic signed distribution if and only if $[|X_1|, ..., |X_d|]$ follows a $p$-isotropic unsigned distribution. Basically, the unsigned case is a restriction of the signed one to a fraction of $1/2^d$. A discussion about other types of domain restrictions, in particular non-continuous, is left at the end of this paper.

## 3.2 Decomposition into radius and direction

Definition 1 is just one way of expressing isotropy, based on measure theory. Two alternative, but equivalent definitions for $p = 2$ are:

1. A distribution for $X \in \mathbb{R}^d$ is isotropic if it is invariant to rotations, i.e. $X$ is indistinguishable in distribution from $U X$ for any orthogonal matrix $U \in \mathbb{R}^{d \times d}$.

2. A distribution Dist over $\mathbb{R}^d$ is isotropic if there are two independent random variables $R \in [0, \infty)$ and $S \sim \text{Uni}(\text{Sph}^d)$ (radius and direction) such that $R \cdot S \sim \text{Dist}$.

The first is based on linear algebra, and it can not be generalized to non-euclidean norms. The euclidean norm is the only one that can be written as an
radiates electromagnetic waves uniformly in all directions. A directional antenna, in contrast, radiates more power in some directions than in others, like a parabolic or a Yagi-Uda antenna.

The isotropic antenna has the property that its 3D radiation pattern $f(X, Y, Z)$ in void space consists of concentric spheres centered at the origin (where the antenna is), and the density of power per unit of area at any point $(x, y, z)$ depends exclusively on the straight line distance $||(x, y, z)||$. More precisely, the power density at $(x, y, z)$ is inversely proportional to the surface area of the sphere that contains $(x, y, z)$. This holds because, first, in void space, the power radiated by any antenna propagates without losses in all directions, hence the net power received over any hypothetical convex surface containing the antenna is constant; and second, for an isotropic antenna, the power is distributed over any centered sphere uniformly.

Similarly, in the context of probability theory, we may define continuous isotropic distributions as having the property that its density function can be written as a function of the norm. Since density functions are only unique up to null-sets, the functional relationship should not be imposed everywhere, but only almost everywhere.

**Definition 1.** Let $\mathcal{R} = \mathbb{R}$ or $\mathcal{R} = [0, \infty)$. A $d$-dimensional continuous distribution with density function $f: \mathcal{R}^d \to [0, \infty)$ (unique up to null sets), is said to be p-isotropic (default $p = 2$) if there is a function $g$ such that

$$f(\boldsymbol{x}) = g(\|\boldsymbol{x}\|_p) \text{ for almost all } \boldsymbol{x} \in \mathcal{R}^d. \qquad (3)$$

The signed and unsigned definitions of isotropy in Definition 2 are related to one another: $[\pm X_1, \dots, \pm X_d]$ follows a $p$-isotropic signed distribution if and only if $[|X_1|, \dots, |X_d|]$ follows a $p$-isotropic unsigned distribution. Basically, the unsigned case is a restriction of the signed one to a fraction of $1/2^d$. A discussion about other types of domain restrictions, in particular non-continuous, is left at the end of this paper.

## 3.2 Decomposition into radius and direction

Definition 1 is just one way of expressing isotropy, based on measure theory. Two alternative, but equivalent definitions for $p = 2$ are:

1. A distribution for $X \in \mathbb{R}^d$ is isotropic if it is invariant to rotations, i.e. $X$ is indistinguishable in distribution from $U X$ for any orthogonal matrix $U \in \mathbb{R}^{d \times d}$.

2. A distribution Dist over $\mathbb{R}^d$ is isotropic if there are two independent random variables $R \in [0, \infty)$ and $S \sim \text{Uni}(\text{Sph}^d)$ (radius and direction) such that $R \cdot S \sim \text{Dist}$.

The first is based on linear algebra, and it can not be generalized to non-euclidean norms. The euclidean norm is the only one that can be written as an
inner product, and the only with a rotation group and in which rotation is an isometry. For instance, in Figure 1, the circle is the only one that is invariant under rotations, while for the 1-sphere and $\infty$-sphere, {$U \mathbf{x} : \mathbf{x} \in \text{Sph}_p^2$} $\neq$ $\text{Sph}_p^2$ for all $U$ except 4 particular cases: rotations by multiples of 90°.

But the second can be generalized to p-norms simply by replacing the distribution of the direction vector $S \sim \text{Uni}(\text{Sph}^d)$, or equivalently $S \sim \|\text{Uni}(\text{Ball}^d)\|$. We will use $\|\text{Uni}(\text{Ball}_p^d)\|_p$ (Definition 2) instead of the perhaps more natural choice $\text{Uni}(\text{Sph}_p^d)$ in anticipation of Proposition 2: it is the former that generalizes well to arbitrary p-norms.

**Definition 2.** The signed and unsigned uniform direction distributions, denoted $\text{UniDir}(\text{Sph}_p^d)$ and $\text{UniDir}(\Delta_p^d)$, are $\|\text{Uni}(\text{Ball}_p^d)\|_p$ and $\|\text{Uni}(\text{Ball}_p^d \cap \mathbb{R}_{\ge 0}^d)\|_p$ respectively.

**Proposition 1.** $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$ if and only if $S_1 = \cos_p(T)$ and $S_2 = \sin_p(T)$ where $T \sim \text{Uni}([0, \pi_p/2])$.

*Proof.* (⇒) By Definition 2, if $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$, then $S_1 \le s_1$ is proportional to the area $A(t)$ of the sector in Equation 2, where $s_1 = \cos_p(t)$. Since the total area is $\pi_p/4$, this is $P(S_1 \le s_1) = \frac{A(t)}{\pi_p/4} = \frac{2t}{\pi_p}$.

Let $J$ be the Jacobian of the transformation $s_1 = \cos_p(t)$. We have

$$p_T(t) = \frac{p_{S_1}(s_1)}{J} = \frac{\frac{d}{ds_1}\left(\frac{2t}{\pi_p}\right)}{\left|\frac{dt}{\pi_p}\right|} = \frac{2}{\pi_p},$$

so $T \sim \text{Uni}([0, \pi_p/2])$.

($\Leftarrow$) Let $R \sim p_R(r) = 2r$, for $r \in (0, 1)$, and let $(X_1, X_2) \stackrel{\text{def}}{=} (R \cos_p(t), R \sin_p(t))$.

The Jacobian of the parametrization $(x_1, x_2) = (r \cos_p(t), r \sin_p(t))$ is

$$J = \left| \frac{\partial(x, y)}{\partial(r, t)} \right| = \begin{vmatrix} \cos_p(t) & -r \sin_p(t) \\ \sin_p(t) & r \cos_p(t) \end{vmatrix} = r(\cos_p(t)^p + \sin_p(t)^p) = r. \quad (4)$$

Hence $p_{X_1, X_2}(x_1, x_2) = p_{R, T}(r, t)/J = 2r^{2/\pi_p}/r = \frac{4}{\pi_p}$. This is constant, so $\mathbf{X} = (X_1, X_2)$ is uniformly distributed in $\text{Ball}_p^2 \cap \mathbb{R}_{\ge 0}^2$. and $\mathbf{X}/\|\mathbf{X}\|_p = (S_1, S_2)$. By Definition 2, it follows that $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$.

□

**Proposition 2.** A continuous distribution Dist over $\mathbb{R}_{\ge 0}^2$ is p-isotropic if and only if there are two independent random variables $R \in [0, \infty)$ and $S \sim \text{UniDir}(\Delta_p^2)$ such that $R \cdot S \sim \text{Dist}$.

By symmetry, the same holds over $\mathbb{R}^2$ with $S \sim \text{UniDir}(\text{Sph}_p^2)$.
inner product, and the only with a rotation group and in which rotation is an isometry. For instance, in Figure 1, the circle is the only one that is invariant under rotations, while for the 1-sphere and $\infty$-sphere, {$U \mathbf{x} : \mathbf{x} \in \text{Sph}_p^2$} $\neq$ $\text{Sph}_p^2$ for all $U$ except 4 particular cases: rotations by multiples of 90°.

But the second can be generalized to p-norms simply by replacing the distribution of the direction vector $S \sim \text{Uni}(\text{Sph}^d)$, or equivalently $S \sim \|\text{Uni}(\text{Ball}^d)\|$. We will use $\|\text{Uni}(\text{Ball}_p^d)\|_p$ (Definition 2) instead of the perhaps more natural choice $\text{Uni}(\text{Sph}_p^d)$ in anticipation of Proposition 2: it is the former that generalizes well to arbitrary p-norms.

**Definition 2.** The signed and unsigned uniform direction distributions, denoted $\text{UniDir}(\text{Sph}_p^d)$ and $\text{UniDir}(\Delta_p^d)$, are $\|\text{Uni}(\text{Ball}_p^d)\|_p$ and $\|\text{Uni}(\text{Ball}_p^d \cap \mathbb{R}_{\ge 0}^d)\|_p$ respectively.

**Proposition 1.** $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$ if and only if $S_1 = \cos_p(T)$ and $S_2 = \sin_p(T)$ where $T \sim \text{Uni}([0, \pi_p/2])$.

*Proof.* (⇒) By Definition 2, if $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$, then $S_1 \le s_1$ is proportional to the area $A(t)$ of the sector in Equation 2, where $s_1 = \cos_p(t)$. Since the total area is $\pi_p/4$, this is $P(S_1 \le s_1) = \frac{A(t)}{\pi_p/4} = \frac{2t}{\pi_p}$.

Let $J$ be the Jacobian of the transformation $s_1 = \cos_p(t)$. We have

$$p_T(t) = \frac{p_{S_1}(s_1)}{J} = \frac{\frac{d}{ds_1}\left(\frac{2t}{\pi_p}\right)}{\left(\frac{d}{ds_1}\left(\frac{\pi_p}{2t}\right)\right)} = \frac{2}{\pi_p},$$

so $T \sim \text{Uni}([0, \pi_p/2])$.

($\Leftarrow$) Let $R \sim p_R(r) = 2r$, for $r \in (0, 1)$, and let $(X_1, X_2) \stackrel{\text{def}}{=} (R \cos_p(t), R \sin_p(t))$.

The Jacobian of the parametrization $(x_1, x_2) = (r \cos_p(t), r \sin_p(t))$ is

$$J = \left| \frac{\partial(x, y)}{\partial(r, t)} \right| = \begin{vmatrix} \cos_p(t) & -r \sin_p(t) \\ \sin_p(t) & r \cos_p(t) \end{vmatrix} = r(\cos_p(t)^p + \sin_p(t)^p) = r. \quad (4)$$

Hence $p_{X_1, X_2}(x_1, x_2) = p_{R, T}(r, t)/J = 2r^{2/\pi_p}/r = \frac{4}{\pi_p}$. This is constant, so $\mathbf{X} = (X_1, X_2)$ is uniformly distributed in $\text{Ball}_p^2 \cap \mathbb{R}_{\ge 0}^2$. and $\mathbf{X}/\|\mathbf{X}\|_p = (S_1, S_2)$. By Definition 2, it follows that $(S_1, S_2) \sim \text{UniDir}(\Delta_p^2)$.

□

**Proposition 2.** A continuous distribution Dist over $\mathbb{R}_{\ge 0}^2$ is p-isotropic if and only if there are two independent random variables $R \in [0, \infty)$ and $S \sim \text{UniDir}(\Delta_p^2)$ such that $R \cdot S \sim \text{Dist}$.

By symmetry, the same holds over $\mathbb{R}^2$ with $S \sim \text{UniDir}(\text{Sph}_p^2)$.
Proof. Recall the transformation and the Jacobian of Equation (4).

(⇒) For any arbitrary non-degenerate set $A \subseteq D^2$, we have

$$\int_A p_{R,T}(r,t) dr dt = \int_A p_{X,Y}(x,y) |J| dx dy = \int_A g(r) r dr dt,$$

so for almost every $r, t$, we have $p_{R,T}(r,t) = g(r)r$. This is a function of $r$ only, so $R \perp T$ and $R \perp S$ and $T$ must be uniformly distributed, which implies $S \sim \text{UniDir}(\Delta_p^d)$ from Proposition 1.

(⇔) For any arbitrary non-degenerate set $A \subseteq D^2$, we have

$$\int_A p_{X,Y}(x,y) dx dy = \int_A p_{R,T}(r,t) |J|^{-1} dr dt = \int_A p_R(r) p_T(t) r^{-1} dr dt,$$

so for almost every $x, y$, we have $p_{X,Y}(x,y) = g(r) \frac{2}{\pi} r^{-1}$, a function of $r$. $\square$

Notice that Proposition 2 can serve as the definition of $p$-isotropy, and it has the advantage of expressing $p$-isotropy without direct dependence on measure theory or linear algebra. However, it depends on Proposition 1 which is restricted to 2D distributions.

## 3.3 Direction is uniform in area, not in perimeter

In this section we show that

$$\|\text{Uni}(\text{Ball}_p^d)\|_p \neq \text{Uni}(\text{Sph}_p^d) \text{ for some values of } p \text{ and } d,$$

and we discuss the semantic implications of this fact.

In the euclidean sphere in 3D, a natural way of defining “a direction chosen uniformly at random” is by choosing a random point on the sphere surface, uniformly in the sense of superficial area: Uni(Sph³). An indirect, but equivalent method is to choose a random point inside the sphere volume, uniformly in the volumetric sense, and projecting it from the origin to the surface: $||\text{Ball}^3||$. Proposition 2 indicates that in order to generalize this procedure to arbitrary $p$-norms, the volumetric method should be taken as the definition, and the superficial one as an alternative method that works for some values of $p$.

Analogously, in a 2D circle, the definition of random direction should be based on the area and the alternative method on the perimeter, which can be either the perimeter or the usual euclidean perimeter. Interestingly, the perimeter methods coincide with the area-based methods for $p \in \{1, 2, \infty\}$ but only for these values of $p$. The small plots at the right side of Figure 3 show this phenomenon. For the large plots on the left side, the one at the top shows the relationship between cumulative area and cumulative perimeter as we move the red $\times$ marker, and the one at the bottom shows a method for varying the area uniformly using results from the following sections. As we move the red $\times$ marker through
Proof. Recall the transformation and the Jacobian of Equation (4).

(⇒) For any arbitrary non-degenerate set $A \subseteq D^2$, we have

$$ \int_A p_{R,T}(r,t) dr dt = \int_A p_{X,Y}(x,y) |J| dx dy = \int_A g(r) r dr dt, $$

so for almost every $r, t$, we have $p_{R,T}(r,t) = g(r)r$. This is a function of $r$ only, so $R \perp T$ and $R \perp S$ and $T$ must be uniformly distributed, which implies $S \sim \text{UniDir}(\Delta_p^d)$ from Proposition 1.

(⇔) For any arbitrary non-degenerate set $A \subseteq D^2$, we have

$$ \int_A p_{X,Y}(x,y) dx dy = \int_A p_{R,T}(r,t) |J|^{-1} dr dt = \int_A p_R(r) p_T(t) r^{-1} dr dt, $$

so for almost every $x, y$, we have $p_{X,Y}(x,y) = g(r) \frac{2}{\pi} r^{-1}$, a function of $r$. $\square$

Notice that Proposition 2 can serve as the definition of $p$-isotropy, and it has the advantage of expressing $p$-isotropy without direct dependence on measure theory or linear algebra. However, it depends on Proposition 1 which is restricted to 2D distributions.

## 3.3 Direction is uniform in area, not in perimeter

In this section we show that

$$ \|\text{Uni}(\text{Ball}_p^d)\|_p \neq \text{Uni}(\text{Sph}_p^d) \text{ for some values of } p \text{ and } d, $$

and we discuss the semantic implications of this fact.

In the euclidean sphere in 3D, a natural way of defining “a direction chosen uniformly at random” is by choosing a random point on the sphere surface, uniformly in the sense of superficial area: Uni(Sph³). An indirect, but equivalent method is to choose a random point inside the sphere volume, uniformly in the volumetric sense, and projecting it from the origin to the surface: $||\text{Ball}^3||$. Proposition 2 indicates that in order to generalize this procedure to arbitrary $p$-norms, the volumetric method should be taken as the definition, and the superficial one as an alternative method that works for some values of $p$.

Analogously, in a 2D circle, the definition of random direction should be based on the area and the alternative method on the perimeter, which can be either the perimeter or the usual euclidean perimeter. Interestingly, the perimeter methods coincide with the area-based methods for $p \in \{1, 2, \infty\}$ but only for these values of $p$. The small plots at the right side of Figure 3 show this phenomenon. For the large plots on the left side, the one at the top shows the relationship between cumulative area and cumulative perimeter as we move the red $\times$ marker, and the one at the bottom shows a method for varying the area uniformly using results from the following sections. As we move the red $\times$ marker through
<figure><img src="image_3.png" /><figcaption>Figure 3: Left: Construction of cumulative area (blue shade) and cumulative perimeter (black) for $p = 5$ using a uniform distribution (blue) in the y axis of the bottom plot. Right: Difference between relative perimeters (cumulative divided by total) and relative area for different values of $p$. The x-axis is the relative area.</figcaption></figure>

the indicated points, the area increases uniformly, but the perimeters do not. Though this is difficult to observe in the constructive plot, the right plot shows the discrepancy clearly for $p = 5$.

$p = 1, 2, \infty$ are the classic examples of p-norms because they have simple geometric interpretations. The fact we just presented is another reason to consider them as the main special cases of the p-norm. This is presented formally in Proposition 3.

**Proposition 3.** Let $A(t)/A(1)$ and $L_p(t)/L_p(1)$ denote the relative area and relative $p$-perimeter (measured in the $p$-norm) of the unit simplex in $2D$ parametrized with $x = \cos_p(t)$ and $y = \sin_p(t)$ varying $t \in (0, \pi_p/4)$.

These two functions are equal if and only if $p \in \{1, 2, \infty\}$. Moreover, these functions are also equal to $L_2(t)/L_2(1)$ if and only if $p \in \{1, 2, \infty\}$.

*Proof.* Let $q > 0$ be the norm used for measuring the relative perimeter. Since the relative functions are already equal at 0 and 1, taking values 0 and 1 re-
<figure><img src="image_4.png" /><figcaption>Figure 3: Left: Construction of cumulative area (blue shade) and cumulative perimeter (black) for $p = 5$ using a uniform distribution (blue) in the y axis of the bottom plot. Right: Difference between relative perimeters (cumulative divided by total) and relative area for different values of $p$. The x-axis is the relative area.</figcaption></figure>

the indicated points, the area increases uniformly, but the perimeters do not. Though this is difficult to observe in the constructive plot, the right plot shows the discrepancy clearly for $p = 5$.

$p = 1, 2, \infty$ are the classic examples of p-norms because they have simple geometric interpretations. The fact we just presented is another reason to consider them as the main special cases of the p-norm. This is presented formally in Proposition 3.

**Proposition 3.** Let $A(t)/A(1)$ and $L_p(t)/L_p(1)$ denote the relative area and relative $p$-perimeter (measured in the $p$-norm) of the unit simplex in $2D$ parametrized with $x = \cos_p(t)$ and $y = \sin_p(t)$ varying $t \in (0, \pi_p/4)$.

These two functions are equal if and only if $p \in \{1, 2, \infty\}$. Moreover, these functions are also equal to $L_2(t)/L_2(1)$ if and only if $p \in \{1, 2, \infty\}$.

*Proof.* Let $q > 0$ be the norm used for measuring the relative perimeter. Since the relative functions are already equal at 0 and 1, taking values 0 and 1 re-
respectively, it suffices to show that $A'(t) \propto L'_q(t)$. From Equation (2), we have $A'(t) = 1/2$, so the problem reduces to show that $L'_q(t)$ is a constant.

(Case $p < \infty$.) We have $\frac{dL_q(t)}{dt} = \frac{dL_q(t)}{dx} \frac{dx}{dt}$. The first term is the derivative of the q-arc-length at $x$, which is $-(1 + |\frac{dy}{dx}|^q)^{1/q}$. Provided that $x^p + y^p = 1$, this becomes $-(1 + (x/y)^{q(p-1)})^{1/q}$. The second term is $-y^{p-1}$ from Equation (1). Their product is $(y^{q(p-1)} + x^{q(p-1)})^{1/p}$. There are two ways of forcing this to be constant for all $x$: either making the exponents 0, or using the identity $y^p + x^p = 1$. The first method yields the unique solution $p = 1$, which is valid for every value of $q$. The second method yields the unique solution $q = \frac{p}{p-1}$, for which $q = p$ if and only if $q = p = 2$.

(Case $p = \infty$.) From the definition of $\sin_\infty$ and $\cos_\infty$, we have $x = \min(1, 2-t)$ and $y = \min(1, t)$. For $t \le 1$, $L'_q(t) = \frac{dL_q(t)}{dy} \frac{dy}{dt} = (1 + |\frac{dx}{dy}|^q)^{1/q} = 1$. For $t > 1$, $L'_q(t) = \frac{dL_q(t)}{dx} \frac{dx}{dt} = -(1 + |\frac{dx}{dy}|^q)^{1/q}(-1) = 1$. $\square$

# 4 Lid-isotropy in the p-norm

<figure><img src="image_4.png" /><figcaption>Figure 4: Three 2D isotropic distributions (for $p = 2$) and the radii that contain 25%, 50% and 75% of the mass. Only the gaussian has independent coordinates, in the other two, the coordinates are dependent.</figcaption></figure>

## 4.1 Dependent versus independent coordinates

Definition 2 decomposes isotropic distributions into independent norm and uniform direction. It is easy to derive arbitrary isotropic distributions with this principle by choosing different distributions for $R$. For example, Figure 4 shows three isotropic distributions in 2D for which the function $g$ in Equation (3) is proportional to an exponential, a gaussian and a uniform density. Notice, perhaps counter-intuitively, that these densities are not the same as the density of the radius $R$. The radial density is actually proportional to $r g(r)$, compensating the increase in the length of the circumferences as the radius grows, so that the overall density is governed by $g$. Had the number of dimensions been $d$ and the nD density given by $g(\|x\|)$, then the density of $R$ would have been $r^{d-1} g(r)$.
respectively, it suffices to show that $A'(t) \propto L'_q(t)$. From Equation (2), we have $A'(t) = 1/2$, so the problem reduces to show that $L'_q(t)$ is a constant.

(Case $p < \infty$.) We have $\frac{dL_q(t)}{dt} = \frac{dL_q(t)}{dx} \frac{dx}{dt}$. The first term is the derivative of the q-arc-length at $x$, which is $-(1 + |\frac{dy}{dx}|^q)^{1/q}$. Provided that $x^p + y^p = 1$, this becomes $-(1 + (x/y)^{q(p-1)})^{1/q}$. The second term is $-y^{p-1}$ from Equation (1). Their product is $(y^{q(p-1)} + x^{q(p-1)})^{1/p}$. There are two ways of forcing this to be constant for all $x$: either making the exponents 0, or using the identity $y^p + x^p = 1$. The first method yields the unique solution $p = 1$, which is valid for every value of $q$. The second method yields the unique solution $q = \frac{p}{p-1}$, for which $q = p$ if and only if $q = p = 2$.

(Case $p = \infty$.) From the definition of $\sin_\infty$ and $\cos_\infty$, we have $x = \min(1, 2-t)$ and $y = \min(1, t)$. For $t \le 1$, $L'_q(t) = \frac{dL_q(t)}{dy} \frac{dy}{dt} = (1 + |\frac{dx}{dy}|^q)^{1/q} = 1$. For $t > 1$, $L'_q(t) = \frac{dL_q(t)}{dx} \frac{dx}{dt} = -(1 + |\frac{dx}{dy}|^q)^{1/q}(-1) = 1$. $\square$

# 4 Lid-isotropy in the p-norm

<figure><img src="image_5.png" /><figcaption>Figure 4: Three 2D isotropic distributions (for $p = 2$) and the radii that contain 25%, 50% and 75% of the mass. Only the gaussian has independent coordinates, in the other two, the coordinates are dependent.</figcaption></figure>

## 4.1 Dependent versus independent coordinates

Definition 2 decomposes isotropic distributions into independent norm and uniform direction. It is easy to derive arbitrary isotropic distributions with this principle by choosing different distributions for $R$. For example, Figure 4 shows three isotropic distributions in 2D for which the function $g$ in Equation (3) is proportional to an exponential, a gaussian and a uniform density. Notice, perhaps counter-intuitively, that these densities are not the same as the density of the radius $R$. The radial density is actually proportional to $r g(r)$, compensating the increase in the length of the circumferences as the radius grows, so that the overall density is governed by $g$. Had the number of dimensions been $d$ and the nD density given by $g(\|x\|)$, then the density of $R$ would have been $r^{d-1} g(r)$.
These 3 isotropic distributions are very used in physics, specially with $p \in \{2, \infty\}$ and $d \in \{2, 3\}$, as well as in other areas of science. They are used, for instance in the initialization and noise production for attacks to neural networks ($p = 2, d \gg 1$), and in noise-injection for privacy applications including the laplace mechanism ($p = 1, d = 1$), metric-privacy ($d \in \{2, 3\}$) [2] and differentially private stochastic gradient descent ($d \gg 1$) in neural network training.

However, even though the three distributions are isotropic, only the gaussian has independent coordinates. In the other two, and visually evident for the uniform, the coordinates are dependent.

## 4.2 Circularity of the gaussian and extension to other p-norms

Let us present an intuitive proof of why the gaussian distribution is the only whose product distribution is circular. An elegant video presentation of this proof can be found in [19]. The proof uses an assumption, which simplifies it significantly. This assumption holds indeed, but this is shown only later in this text (Proposition 10) using existing results about the gamma distribution. The objective of this section and the introduction of this assumption is to present a simplified reasoning about the 2D gaussian, and more generally, the distributions that are isotropic with independent marginals.

**Assumption 1.** If $f_d: \mathbb{R}^d \to [0, \infty)$ is a density function that is isotropic and is the joint of independent components, then there is a way choosing the marginal density $f$ and the function $g$ in Equation (3) so that $f$ is differentiable at 0 and

$$f_d(\mathbf{x}) = \prod_{i=1}^{d} f(x_i) = g(\|\mathbf{x}\|_p) \text{ for every (not just almost every) } \mathbf{x} \in \mathbb{R}^d. \quad (5)$$

Note. The assumption that the functional equation is valid everywhere resembles a problem posed by Erdős about Cauchy's functional equation<fnref n="2" />, which was solved in [11].

**Proposition 4.** Under Assumption 1, $[\text{Dist}]_{1..2}$ is isotropic if and only if Dist is a gaussian distribution centered at zero.

*Proof.* ($\Leftarrow$) The gaussian density function is $f(x) = c \cdot \exp(-x^2/\sigma)$ with $c = (\sqrt{2\pi}\sigma)^{-1}$ and satisfies $f(x)f(y) = c \exp(-x^2/\sigma)c \exp(-y^2/\sigma)$. This can be grouped as $c^2 \exp(-(x^2 + y^2)/\sigma) = c^2 \exp(-\|(x, y)\|^2/\sigma)$, a function of the radius, so it is circular.

($\Rightarrow$) Let $f$ be a density function over $\mathbb{R}$ for which there is a function $g: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ satisfying the circularity condition $f(x)f(y) = g(\sqrt{x^2 + y^2})$. In particular, $f(x)f(0) = g(|x|)$ for all $x \in \mathbb{R}$, so $f$ is symmetric and $g(x) = f(x)/f(0)$ for all $x > 0$. Thus, it suffices to find $f$ such that $f(x)f(y) = f(\sqrt{x^2 + y^2})/f(0)$ for all

<footnote n="2">²See [12], Section 1.7.8, for details.</footnote>
These 3 isotropic distributions are very used in physics, specially with $p \in \{2, \infty\}$ and $d \in \{2, 3\}$, as well as in other areas of science. They are used, for instance in the initialization and noise production for attacks to neural networks ($p = 2, d \gg 1$), and in noise-injection for privacy applications including the laplace mechanism ($p = 1, d = 1$), metric-privacy ($d \in \{2, 3\}$) [2] and differentially private stochastic gradient descent ($d \gg 1$) in neural network training.

However, even though the three distributions are isotropic, only the gaussian has independent coordinates. In the other two, and visually evident for the uniform, the coordinates are dependent.

## 4.2 Circularity of the gaussian and extension to other p-norms

Let us present an intuitive proof of why the gaussian distribution is the only whose product distribution is circular. An elegant video presentation of this proof can be found in [19]. The proof uses an assumption, which simplifies it significantly. This assumption holds indeed, but this is shown only later in this text (Proposition 10) using existing results about the gamma distribution. The objective of this section and the introduction of this assumption is to present a simplified reasoning about the 2D gaussian, and more generally, the distributions that are isotropic with independent marginals.

**Assumption 1.** If $f_d: \mathbb{R}^d \to [0, \infty)$ is a density function that is isotropic and is the joint of independent components, then there is a way choosing the marginal density $f$ and the function $g$ in Equation (3) so that $f$ is differentiable at 0 and

$$f_d(\mathbf{x}) = \prod_{i=1}^{d} f(x_i) = g(\|\mathbf{x}\|_p) \text{ for every (not just almost every) } \mathbf{x} \in \mathbb{R}^d. \quad (5)$$

Note. The assumption that the functional equation is valid everywhere resembles a problem posed by Erdős about Cauchy's functional equation<fnref n="2" />, which was solved in [11].

**Proposition 4.** Under Assumption 1, $[\text{Dist}]_{1..2}$ is isotropic if and only if Dist is a gaussian distribution centered at zero.

*Proof.* ($\Leftarrow$) The gaussian density function is $f(x) = c \cdot \exp(-x^2/\sigma)$ with $c = (\sqrt{2\pi}\sigma)^{-1}$ and satisfies $f(x)f(y) = c \exp(-x^2/\sigma)c \exp(-y^2/\sigma)$. This can be grouped as $c^2 \exp(-(x^2 + y^2)/\sigma) = c^2 \exp(-\|(x, y)\|^2/\sigma)$, a function of the radius, so it is circular.

($\Rightarrow$) Let $f$ be a density function over $\mathbb{R}$ for which there is a function $g: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ satisfying the circularity condition $f(x)f(y) = g(\sqrt{x^2 + y^2})$. In particular, $f(x)f(0) = g(|x|)$ for all $x \in \mathbb{R}$, so $f$ is symmetric and $g(x) = f(x)/f(0)$ for all $x > 0$. Thus, it suffices to find $f$ such that $f(x)f(y) = f(\sqrt{x^2 + y^2})/f(0)$ for all

<footnote n="2">²See [12], Section 1.7.8, for details.</footnote>
reals $x$ and $y$. Letting $h(x) \stackrel{\text{def}}{=} f(\sqrt{|x|})$, this condition becomes the exponential identity $h(x)h(y) = h(x+y)$ for all $x, y \in \mathbb{R}$, whose solutions are $h(x) = 0$ and $h(x) = a \exp(b \cdot x)$ (see<fnref n="3" />). So, $f(x) = f(|x|) = a \exp(b \cdot x^2)$, and for $f$ to be a density function, it must hold that $b < 0$ and $a = |b|/\sqrt{2\pi}$. □

Intuitively, the same proof for the circularity of independent gaussians in 2D can be easily extended to $d$ dimensions because $\exp(-kx_1^2) \cdots \exp(-kx_d^2) = \exp(-k(x_1^2 + \cdots + x_d^2))$. So, the gaussian is circular in 2D, spherical in 3D, etc. It is precisely this circularity in 2D, or isotropy in nD, that explains the appearance of the number $\pi$ in the density function of the gaussian distribution, a beautiful connection and celebrated fact in mathematics for which at least 11 different proofs have been found [7].

Moreover, the proof can also be easily extended to non-euclidean p-norms with $p < \infty$. The joint of two i.i.d. random variables is circular in the p-norm if and only if the marginal density can be expressed as $f(x) = c \cdot \exp(-k|x|^p)$, because $f(x)f(y) = c^2 \exp(-k(|x|^p + |y|^p)) = c \cdot \exp(-k \|(x, y)\|_p^p)$. And, again, the same proof can be extended to nD.

These results are proven in Appendix 7 for completeness. With Assumption 1, the proofs are self contained. In Section 4.3, we prove these results more ver- bosely and without Assumption 1.

## 4.3 Characterization of p-iid-isotropy

We proceed to characterize the $p$-isotropic distributions with independent coor- dinates, starting in 2D. For this, we will need the following characterization of the gamma distribution.

**Proposition 5.** (Characterization of the gamma distribution) Let $X$ and $Y$ be two non-constant i.i.d. random variables on $[0, \infty)$. $X+Y$ and $X/Y$ are independent if and only if $X$ and $Y$ have gamma distributions with common scale parameter.

*Proof.* Proven in [14], page 319. The proof was strengthened in [16] for $X, Y \in (-\infty, \infty)$, in which case, either $(X, Y)$ or $(-X, -Y)$ are gamma distributed. Yet another proof is mentioned in [12], result 17.42, and relies on [3]. □

Recall that the gamma distribution $b \cdot \text{Gamma}(d)$ has density

$$f(x) = (1/D) x^{d-1} \exp(-x/b) \qquad (6)$$

for $x \ge 0$ and with integration constant $D = b^d \Gamma(d)$.

We may now characterize $p$-isotropy in 2D for finite and infinite $p$.

<footnote n="3">³Ex. 4.39 of [9], provided $h$ is differentiable at zero.</footnote>
reals x and y. Letting $h(x) := f(\sqrt{|x|})$, this condition becomes the exponential identity $h(x)h(y) = h(x+y)$ for all $x, y \in \mathbb{R}$, whose solutions are $h(x) = 0$ and $h(x) = a \exp(b \cdot x)$ (see<fnref n="3" />). So, $f(x) = f(|x|) = a \exp(b \cdot x^2)$, and for $f$ to be a density function, it must hold that $b < 0$ and $a = |b|/\sqrt{2\pi}$. □

Intuitively, the same proof for the circularity of independent gaussians in 2D can be easily extended to $d$ dimensions because $\exp(-kx_1^2) \cdots \exp(-kx_d^2) = \exp(-k(x_1^2 + \cdots + x_d^2))$. So, the gaussian is circular in 2D, spherical in 3D, etc. It is precisely this circularity in 2D, or isotropy in nD, that explains the appearance of the number $\pi$ in the density function of the gaussian distribution, a beautiful connection and celebrated fact in mathematics for which at least 11 different proofs have been found [7].

Moreover, the proof can also be easily extended to non-euclidean p-norms with $p < \infty$. The joint of two i.i.d. random variables is circular in the p-norm if and only if the marginal density can be expressed as $f(x) = c \cdot \exp(-k|x|^p)$, because $f(x)f(y) = c^2 \exp(-k(|x|^p + |y|^p)) = c \cdot \exp(-k \|(x, y)\|_p^p)$. And, again, the same proof can be extended to nD.

These results are proven in Appendix 7 for completeness. With Assumption 1, the proofs are self contained. In Section 4.3, we prove these results more ver- bosely and without Assumption 1.

## 4.3 Characterization of p-iid-isotropy

We proceed to characterize the $p$-isotropic distributions with independent coor- dinates, starting in 2D. For this, we will need the following characterization of the gamma distribution.

**Proposition 5.** (Characterization of the gamma distribution) Let $X$ and $Y$ be two non-constant i.i.d. random variables on $[0, \infty)$. $X+Y$ and $X/Y$ are independent if and only if $X$ and $Y$ have gamma distributions with common scale parameter.

*Proof.* Proven in [14], page 319. The proof was strengthened in [16] for $X, Y \in (-\infty, \infty)$, in which case, either $(X, Y)$ or $(-X, -Y)$ are gamma distributed. Yet another proof is mentioned in [12], result 17.42, and relies on [3]. □

Recall that the gamma distribution $b \cdot \text{Gamma}(d)$ has density

$$f(x) = (1/D) x^{d-1} \exp(-x/b) \qquad (6)$$

for $x \ge 0$ and with integration constant $D = b^d \Gamma(d)$.

We may now characterize $p$-isotropy in 2D for finite and infinite $p$.

<footnote n="3">³Ex. 4.39 of [9], provided $h$ is differentiable at zero.</footnote>
**Proposition 6.** Let $p \in [1, \infty)$ and let $X, Y$ be i.i.d. with density $f$ over $[0, \infty)$. If $(\pm X, Y \pm)$ is p-isotropic, then $f$ is given by $f(x) \propto \exp(-|x|^p/k)$ for some $k > 0$.

*Proof.* Since $(\pm X, \pm Y)$ is p-isotropic, then $(X, Y)$ can be written as a product $R \cdot S_{1..2}$ of independent variables $R = (X^p + Y^p)^{1/p}$ and $(S_1, S_2) \stackrel{\text{def}}{=} (\frac{X}{R}, \frac{Y}{R})$, where $S \sim \text{UniDir}(\Delta_p^2)$. The independence $R^p \perp S_1^p$ implies that $X^p + Y^p \perp X^p/R^p = 1/(1+Y^p/X^p)$, so, also $Y^p + X^p \perp Y^p/X^p$. Hence, by Proposition 5, $X^p$ and $Y^p$ must be gamma distributed with the same scale. Applying the transformation $x \mapsto x^{1/p}$ to a gamma distribution with parameters $\alpha, \beta$, it follows that $X$ has density given by

$$f(x) = \frac{\beta^\alpha p}{\Gamma(\alpha)} x^{p\alpha-1} e^{-\beta x^p}.$$

It remains to prove that $\alpha = 1/p$ using the fact that, according to Equation (3), the product $f(x)f(y)$ can be written as a function of $\|(x, y)\|_p$ almost everywhere. $f(x)f(y)$ is proportional to the product $(xy)^{p\alpha-1}e^{-b(x^p+y^p)}$ and the exponential factor is already a function of $\|(x, y)\|_p$ everywhere. We will show that the power factor $h(x, y) = (xy)^{p\alpha-1}$ can not, not even almost everywhere, unless $\alpha = 1/p$. From the disintegration theorem, there are null-sets $N, N_r$, so that for every $r \in [0, \infty) \setminus N$ and every $x \in [0, r] \setminus N_r$, $h(x, (r^p - x^p)^{1/p})$ is constant. But this constant, call it $c_r$, equals also the limit

$$c_r = \lim_{x \to 0; x \notin N_r} (x(r^p - x^p)^{1/p})^{p\alpha-1} = \lim_{x \to 0; x \notin N_r} (xr)^{p\alpha-1} = \begin{cases} 0 & \text{if } p\alpha - 1 > 0 \\ 1 & \text{if } p\alpha - 1 = 0 \\ \infty & \text{if } p\alpha - 1 < 0. \end{cases}$$

The case $c_r = 1$ is the only plausible one, because if $c_r = \infty$, then $f(x)f(y)$ is infinite almost everywhere, and if $c_r = 0$, then $f(x)f(y)$ is zero almost everywhere. So $\alpha = 1/p$. $\square$

**Proposition 7.** Let $[X, Y]$ be iid continuous random variables in $[0, \infty)$. If $[\pm X, \pm Y]$ is $\infty$-isotropic, then $X, Y \sim b$ Uni for some $b > 0$.

*Proof.* Let $f: [0, \infty) \to [0, \infty)$ be the density function of $X$ and $Y$, let $F$ be their cumulative density function, let $g$ be the function with $f(x)f(y) = g(\max(x, y))$ and let $G$ be the cumulative integral of $g$. For any $0 \le t \le a < b$ with $F(b) - F(a) > 0$ we have

$$\int_0^t \int_a^b f(x)f(y)dydx = \int_0^t \int_a^b g(\max(x, y))dydx$$

$$\int_0^t f(x) \int_a^b f(y)dydx = \int_0^t \int_a^b g(y)dydx$$

$$F(t) = t \frac{G(b) - G(a)}{F(b) - F(a)}$$
**Proposition 6.** Let $p \in [1, \infty)$ and let $X, Y$ be i.i.d. with density $f$ over $[0, \infty)$. If $(\pm X, \pm Y)$ is p-isotropic, then $f$ is given by $f(x) \propto \exp(-|x|^p/k)$ for some $k > 0$.

*Proof.* Since $(\pm X, \pm Y)$ is p-isotropic, then $(X, Y)$ can be written as a product $R \cdot S_{1..2}$ of independent variables $R = (X^p + Y^p)^{1/p}$ and $(S_1, S_2) \stackrel{\text{def}}{=} (\frac{X}{R}, \frac{Y}{R})$, where $S \sim \text{UniDir}(\Delta_p^2)$. The independence $R^p \perp S_1^p$ implies that $X^p + Y^p \perp X^p/R^p = 1/(1+Y^p/X^p)$, so, also $Y^p + X^p \perp Y^p/X^p$. Hence, by Proposition 5, $X^p$ and $Y^p$ must be gamma distributed with the same scale. Applying the transformation $x \mapsto x^{1/p}$ to a gamma distribution with parameters $\alpha, \beta$, it follows that $X$ has density given by

$$f(x) = \frac{\beta^\alpha p}{\Gamma(\alpha)} x^{p\alpha-1} e^{-\beta x^p}.$$

It remains to prove that $\alpha = 1/p$ using the fact that, according to Equation (3), the product $f(x)f(y)$ can be written as a function of $\|(x, y)\|_p$ almost everywhere. $f(x)f(y)$ is proportional to the product $(xy)^{p\alpha-1}e^{-b(x^p+y^p)}$ and the exponential factor is already a function of $\|(x, y)\|_p$ everywhere. We will show that the power factor $h(x, y) = (xy)^{p\alpha-1}$ can not, not even almost everywhere, unless $\alpha = 1/p$. From the disintegration theorem, there are null-sets $N, N_r$, so that for every $r \in [0, \infty) \setminus N$ and every $x \in [0, r] \setminus N_r$, $h(x, (r^p - x^p)^{1/p})$ is constant. But this constant, call it $c_r$, equals also the limit

$$c_r = \lim_{x \to 0; x \notin N_r} (x(r^p - x^p)^{1/p})^{p\alpha-1} = \lim_{x \to 0; x \notin N_r} (xr)^{p\alpha-1} = \begin{cases} 0 & \text{if } p\alpha - 1 > 0 \\ 1 & \text{if } p\alpha - 1 = 0 \\ \infty & \text{if } p\alpha - 1 < 0. \end{cases}$$

The case $c_r = 1$ is the only plausible one, because if $c_r = \infty$, then $f(x)f(y)$ is infinite almost everywhere, and if $c_r = 0$, then $f(x)f(y)$ is zero almost everywhere. So $\alpha = 1/p$. $\square$

**Proposition 7.** Let $[X, Y]$ be iid continuous random variables in $[0, \infty)$. If $[\pm X, \pm Y]$ is $\infty$-isotropic, then $X, Y \sim b$ Uni for some $b > 0$.

*Proof.* Let $f: [0, \infty) \to [0, \infty)$ be the density function of $X$ and $Y$, let $F$ be their cumulative density function, let $g$ be the function with $f(x)f(y) = g(\max(x, y))$ and let $G$ be the cumulative integral of $g$. For any $0 \le t \le a < b$ with $F(b) - F(a) > 0$ we have

$$\int_0^t \int_a^b f(x)f(y)dydx = \int_0^t \int_a^b g(\max(x, y))dydx$$

$$\int_0^t f(x) \int_a^b f(y) dydx = \int_0^t \int_a^b g(y) dydx$$

$$F(t) = t \frac{G(b) - G(a)}{F(b) - F(a)}$$
So, conditioning on the event $X < a$, for any $a, b$ with $F(a) < F(b)$, $X$ is uniformly distributed over $[0, a]$. Since $X$ is a continuous random variable, then $F$ is continuous and for all $a$ with $F(a) < 1$, there is $b$ with $F(b) > F(a)$. It follows that $X \sim k \cdot \text{Uni}$ with $k = \sup\{a: F(a) < 1\}$, which implies also $0 < k < \infty$. $\square$

Finally, we extend the characterization from 2D to nD.

**Proposition 8.** Let Dist be a continuous distribution over $\mathbb{R}$ or $[0, \infty)$. For every $n \ge 2$, $[\text{Dist}]_{1..n}$ is p-isotropic if and only if $[\text{Dist}]_{1..2}$ is p-isotropic.

*Proof.* Due to symmetry, it suffices to prove for unsigned domain $[0, \infty)$.

(⇒) Suppose that $[\text{Dist}]_{1..n}$ is p-isotropic for some $n \ge 3$. We will show that $[\text{Dist}]_{1..n-1}$ is p-isotropic, hence by transitivity also $[\text{Dist}]_{1..n-2}$, $[\text{Dist}]_{1..n-3}$, ..., $[\text{Dist}]_{1..2}$.

Consider $n$ i.i.d. random variables $X_i \sim \text{Dist}$. We know that $\frac{X_{1..n}}{\|X_{1..n}\|_p} \sim \text{UniDir}(\Delta_p^n)$. We would like to show that $\frac{X_{1..n-1}}{\|X_{1..n-1}\|_p} \sim \text{UniDir}(\Delta_p^{n-1})$. Since $X_{1..n}$ is uniformly distributed on a $d$-dimensional volume (the non-negative part of a ball), then necessarily, $X_{1..n-1}$ is also uniformly distributed. This creates a recursive structure of unitary simplices, namely, if $S_{1..n} \sim \text{UniDir}(\Delta_p^n)$, then $\frac{S_{1..n-1}}{(1-(S_n)^p)^{1/p}} \sim \text{UniDir}(\Delta_p^{n-1})$. In particular for $S_i = \frac{X_i}{\|X_{1..n-1}\|_p}$, the following random variable is uniformly distributed over $\Delta_p^{n-1}$:

$$ \frac{S_{1..n-1}}{(1 - (S_n)^p)^{1/p}} = \frac{X_{1..n-1}}{\|X_{1..n-1}\|_p \cdot \left(1 - \left(\frac{X_{1..n-1}}{\|X_{1..n-1}\|_p}\right)^p\right)^{1/p}} = \frac{X_{1..n-1}}{\|X_{1..n-1}\|_p} $$

(⇔) For $p < \infty$, from Proposition 6, Dist has density $f(x) \propto \exp(-(x/b)^p)$, so $[\text{Dist}]_{1..n}$ has density $f(x_1) \cdots f(x_n) \propto \exp(-(\|x_{1..n}\|_p/b)^p)$, which is a function of $\|x_{1..n}\|_p$, so $[\text{Dist}]_{1..n}$ is p-isotropic. For $p = \infty$, from Propositions 7, Dist has density $f(x) \propto 1_{x \le b}$, so $[\text{Dist}]_{1..n}$ has density $f(x_1) \cdots f(x_n) \propto 1_{\max(x_1, \dots, x_n) \le b}$, which is a function of $\|x_{1..n}\|_p$, so $[\text{Dist}]_{1..n}$ is p-isotropic. $\square$

The following definition arises naturally from Proposition 8.

**Definition 3.** Let Dist be a continuous distribution over $\mathbb{R}$ or $[0, \infty)$. Dist is said to be p-iid-isotropic if $[\text{Dist}]_{1..d}$ is p-isotropic for any $d \ge 2$. In that case, from Proposition 8, p-isotropy holds for all $d \ge 2$.

Propositions 6, 6 and 8 can be summarized in the following claim.

**Proposition 9.** For $p \in [1, \infty]$, define the density functions $f_p$ over $x \ge 0$ as

$$ f_p(x) := \begin{cases} \infty e^{-x^p} & \text{if } p < \infty \\ 1_{x \le 1} & \text{if } p = \infty. \end{cases} $$
So, conditioning on the event $X < a$, for any $a, b$ with $F(a) < F(b)$, $X$ is uniformly distributed over $[0, a]$. Since $X$ is a continuous random variable, then $F$ is continuous and for all $a$ with $F(a) < 1$, there is $b$ with $F(b) > F(a)$. It follows that $X \sim k \cdot \text{Uni}$ with $k = \sup\{a: F(a) < 1\}$, which implies also $0 < k < \infty$. $\square$

Finally, we extend the characterization from 2D to nD.

**Proposition 8.** Let Dist be a continuous distribution over $\mathbb{R}$ or $[0, \infty)$. For every $n \ge 2$, $[\text{Dist}]_{1..n}$ is p-isotropic if and only if $[\text{Dist}]_{1..2}$ is p-isotropic.

*Proof.* Due to symmetry, it suffices to prove for unsigned domain $[0, \infty)$.

(⇒) Suppose that $[\text{Dist}]_{1..n}$ is p-isotropic for some $n \ge 3$. We will show that $[\text{Dist}]_{1..n-1}$ is p-isotropic, hence by transitivity also $[\text{Dist}]_{1..n-2}$, $[\text{Dist}]_{1..n-3}$, ..., $[\text{Dist}]_{1..2}$.

Consider $n$ i.i.d. random variables $X_i \sim \text{Dist}$. We know that $\frac{X_{1..n}}{\|X_{1..n}\|_p} \sim \text{UniDir}(\Delta_p^n)$. We would like to show that $\frac{X_{1..n-1}}{\|X_{1..n-1}\|_p} \sim \text{UniDir}(\Delta_p^{n-1})$. Since $X_{1..n}$ is uniformly distributed on a $d$-dimensional volume (the non-negative part of a ball), then necessarily, $X_{1..n-1}$ is also uniformly distributed. This creates a recursive structure of unitary simplices, namely, if $S_{1..n} \sim \text{UniDir}(\Delta_p^n)$, then $\frac{S_{1..n-1}}{(1-(S_n p)^{1/p})} \sim \text{UniDir}(\Delta_p^{n-1})$. In particular for $S_i = \frac{X_i}{\|X_{1..n-1}\|_p}$, the following random variable is uniformly distributed over $\Delta_p^{n-1}$:

$$ \frac{S_{1..n-1}}{(1 - (S_n p)^{1/p})} = \frac{X_{1..n-1}}{\|X_{1..n-1}\|_p \cdot \left(1 - \left(\frac{X_{1..n-1}}{\|X_{1..n-1}\|_p}\right)^p\right)^{1/p}} = \frac{X_{1..n-1}}{\|X_{1..n-1}\|_p} $$

(⇔) For $p < \infty$, from Proposition 6, Dist has density $f(x) \propto \exp(-(x/b)^p)$, so $[\text{Dist}]_{1..n}$ has density $f(x_1) \cdots f(x_n) \propto \exp(-(\|x_{1..n}\|_p/b)^p)$, which is a function of $\|x_{1..n}\|_p$, so $[\text{Dist}]_{1..n}$ is p-isotropic. For $p = \infty$, from Propositions 7, Dist has density $f(x) \propto 1_{x \le b}$, so $[\text{Dist}]_{1..n}$ has density $f(x_1) \cdots f(x_n) \propto 1_{\max(x_1, \dots, x_n) \le b}$, which is a function of $\|x_{1..n}\|_p$, so $[\text{Dist}]_{1..n}$ is p-isotropic. $\square$

The following definition arises naturally from Proposition 8.

**Definition 3.** Let Dist be a continuous distribution over $\mathbb{R}$ or $[0, \infty)$. Dist is said to be p-iid-isotropic if $[\text{Dist}]_{1..d}$ is p-isotropic for any $d \ge 2$. In that case, from Proposition 8, p-isotropy holds for all $d \ge 2$.

Propositions 6, 6 and 8 can be summarized in the following claim.

**Proposition 9.** For $p \in [1, \infty]$, define the density functions $f_p$ over $x \ge 0$ as

$$ f_p(x) := \begin{cases} \infty e^{-x^p} & \text{if } p < \infty \\ 1_{x \le 1} & \text{if } p = \infty. \end{cases} $$
A continuous random variable X is p-iid-isotropic if and only if $X/b \sim f_p$ for some $b > 0$.

*Proof.* From Proposition 8, it suffices to show that for any and any pair of i.i.d. continuous random variables X, Y in $[0, \infty)$, $[X, Y]$ is p-isotropic if and only if $X/b, Y/b \sim f_p$ for some $b > 0$.

(⇒) Follows from Propositions 6 and 7.

(⇔) For $p < \infty$, $e^{-x^p}e^{-y^p} = e^{-\|(x,y)\|_p^p}$ and for $p = \infty$, $1_{x \le 1}1_{y \le 1} = 1_{\max(x,y) \le 1}$. In both cases, Equation (3) holds (everywhere). □

A consequence of Proposition 9 is presented next.

**Proposition 10.** *Assumption 1 holds.*

*Proof.* If Equation (3) holds (almost everywhere), then by Proposition 9, the density for the distribution is $f_p$, which is differentiable at 0 and satisfies Equation (5) (everywhere). For $p = \infty$, any of the conventions $f_\infty(1) = g(1) = 0$ or $f_\infty(1) = g(1) = 1$ may be used. □

# 5 The independent-isotropic distribution

In this section, inspired by Proposition 9, we introduce the independent-isotropic distribution and extend it with two parameters to refer to its radius and its direction (projected on a line). We then show multiple connections that exist between two regimes that arise from this extended distribution: light-tailed and heavy-tailed regimes.

Proposition 9 states that the uniform distribution $b$ Uni is the only unsigned $\infty$-iid-isotropic distribution, and for $p < \infty$, the only unsigned $p$-iid-isotropic distribution has the density $\propto e^{-x^p}$, which is the density of a generalized gamma distribution, a power transformation of the gamma distribution. The generalized gamma distribution $b \cdot \text{GenGamma}(p, d)$ has density $f(x) = (1/D) x^{d-1}e^{-(x/b)^p}$ with $D = b^d \Gamma(d/p)/p$, which generalizes Equation (6). This distribution is related with the gamma distribution via $\text{Gamma}(d) \equiv \text{GenGamma}(p, d)$ and $\text{GenGamma}(p=1, d) \equiv b \text{Gamma}(d/p)^{1/p}$.

The generalized gamma has been discussed since 1924 [1] (p.p. 124) with the purpose of studying income curves and it has been studied in depth since then [20, 10, 13, 17] with varying names (like generalized Weibull), and different parametrizations and parameter names, including $a, \alpha, \beta, c, d, p$.

In this paper, we will use a custom parametrization (Definition 4) that differs from those in the literature in that it scales the distribution by $(p/d)^{1/p}$ so that the $p$-th moment of the distribution is equal to 1 (thus also $\sqrt[p]{E[X^p]} = 1$). This is convenient to extend the parameter domain to $p = \infty$ (Proposition 11), and is a scale that proves to be natural throughout the different theorems and
A continuous random variable X is p-iid-isotropic if and only if $X/b \sim f_p$ for some $b > 0$.

*Proof.* From Proposition 8, it suffices to show that for any and any pair of i.i.d. continuous random variables X, Y in $[0, \infty)$, $[X, Y]$ is p-isotropic if and only if $X/b, Y/b \sim f_p$ for some $b > 0$.

(⇒) Follows from Propositions 6 and 7.

(⇔) For $p < \infty$, $e^{-x^p}e^{-y^p} = e^{-\|(x,y)\|_p^p}$ and for $p = \infty$, $1_{x \le 1}1_{y \le 1} = 1_{\max(x,y) \le 1}$. In both cases, Equation (3) holds (everywhere). $\square$

A consequence of Proposition 9 is presented next.

**Proposition 10.** *Assumption 1 holds.*

*Proof.* If Equation (3) holds (almost everywhere), then by Proposition 9, the density for the distribution is $f_p$, which is differentiable at 0 and satisfies Equation (5) (everywhere). For $p = \infty$, any of the conventions $f_\infty(1) = g(1) = 0$ or $f_\infty(1) = g(1) = 1$ may be used. $\square$

# 5 The independent-isotropic distribution

In this section, inspired by Proposition 9, we introduce the independent-isotropic distribution and extend it with two parameters to refer to its radius and its direction (projected on a line). We then show multiple connections that exist between two regimes that arise from this extended distribution: light-tailed and heavy-tailed regimes.

Proposition 9 states that the uniform distribution $b$ Uni is the only unsigned $\infty$-iid-isotropic distribution, and for $p < \infty$, the only unsigned $p$-iid-isotropic distribution has the density $\propto e^{-x^p}$, which is the density of a generalized gamma distribution, a power transformation of the gamma distribution. The generalized gamma distribution $b \cdot \text{GenGamma}(p, d)$ has density $f(x) = (1/D) x^{d-1}e^{-(x/b)^p}$ with $D = b^d \Gamma(d/p)/p$, which generalizes Equation (6). This distribution is related with the gamma distribution via $\text{Gamma}(d) \equiv \text{GenGamma}(p, d)$ and $\text{GenGamma}(p=1, d) \equiv b \text{ Gamma}(d/p)^{1/p}$.

The generalized gamma has been discussed since 1924 [1] (p.p. 124) with the purpose of studying income curves and it has been studied in depth since then [20, 10, 13, 17] with varying names (like generalized Weibull), and different parametrizations and parameter names, including $a, \alpha, \beta, c, d, p$.

In this paper, we will use a custom parametrization (Definition 4) that differs from those in the literature in that it scales the distribution by $(p/d)^{1/p}$ so that the $p$-th moment of the distribution is equal to 1 (thus also $\sqrt[p]{E[X^p]} = 1$). This is convenient to extend the parameter domain to $p = \infty$ (Proposition 11), and is a scale that proves to be natural throughout the different theorems and
visualizations presented in this paper.

**Definition 4.** The independent-isotropic distribution $\text{IndIso}(p)$ with parameter $p \in (0, \infty]$ is the distribution over $[0, \infty)$ given by

$$ \text{IndIso}(p) \equiv \begin{cases} (p \text{ Gamma}(\frac{1}{p}))^{\frac{1}{p}} & \text{if } p < \infty \\ \text{Uni} & \text{if } p = \infty \end{cases}; \quad f(x) = \begin{cases} \frac{p^{1-1/p}}{\Gamma(1/p)} e^{-x^p/p} & \text{if } p < \infty \\ \mathbf{1}_{x<1} & \text{if } p = \infty. \end{cases} $$

The density function $f$ of $\text{IndIso}(p)$ can also be found in Table 1, case $d = 1, \tau = \infty$.

The name *independent-isotropic* comes from the fact that, from Proposition 9, a continuous distribution is *p*-iid-isotropic if and only if it is $\pm b \text{IndIso}(p)$ for some scale $b > 0$.

The $\text{IndIso}$ distribution can therefore be used for *drawing a random direction in $\mathbb{R}^d$ uniformly in the $p$-norm*. It is widely known, for instance that sampling from $\text{Sph}_2^d$ can be achieved with $\mathbf{X}/\|\mathbf{X}\|$ where $\mathbf{X} \sim [\text{Gauss}]_{1..d} = [\text{IndIso}(2)]_{1..d}$. More generally, we can sample from $\text{Sph}_p^d$ in the $p$-norm using $\mathbf{X}/\|\mathbf{X}\|$ where $\mathbf{X} \sim [\text{IndIso}(p)]_{1..d}$. This approach incurs the costs of sampling from the gamma distribution but avoids the possibly large rejection probabilities incurred in the method of sampling uniformly from a hyper-box, discarding points with norm larger than 1, and normalizing.

**Proposition 11.** $\text{IndIso}(p)$ converges in distribution to $\text{IndIso}(\infty)$ as $p \to \infty$.

*Proof.* Notice that $\Gamma(1/p) = \Gamma(1/p+1)/p$, and $\lim_{p \to \infty} p^{1-1/p} = 1$, and $\Gamma(1) = 0$, and $\lim_{p \to \infty} e^{-x^p/p} = 1_{x \le 1}$. It follows that

$$ \lim_{p \to \infty} \frac{p^{1-1/p}}{\Gamma(1/p)} e^{-x^p/p} = \lim_{p \to \infty} \frac{p^{-1/p}}{\Gamma(1+1/p)} e^{-x^p/p} = \mathbf{1}_{1 \le x}. $$

□

## 5.1 Light-tailed regime (2 parameters)

Let us generalize definition 4 to include a second parameter $d$ so as to measure the radius $\|X_{1..d}\|_p$ for $X_i \sim \text{IndIso}(p)$. Clearly, for $d = 1$, this is just $\text{IndIso}(p)$. To distinguish this distribution from the one in the next section, we will refer to it as light-tailed in the sense that it decays faster than any power-law even for the smallest values of $p$.

**Definition 5.** The (light-tailed) independent-isotropic distribution $\text{IndIso}(p, d)$ with parameters $p \in (0, \infty]$ and $d \in (0, \infty)$ is the distribution over $[0, \infty)$ given by

$$ \text{IndIso}(p, d) \stackrel{\text{def}}{=} \|[\text{IndIso}(p)]_{1..d}\|_p, $$

Its density is detailed in Table 1, and, if unspecified, $d = 1$ is assumed in accordance to Definition 4.
visualizations presented in this paper.

**Definition 4.** The independent-isotropic distribution $\text{IndIso}(p)$ with parameter $p \in (0, \infty]$ is the distribution over $[0, \infty)$ given by

$$ \text{IndIso}(p) = \begin{cases} (p \text{ Gamma}(\frac{1}{p}))^{\frac{1}{p}} & \text{if } p < \infty \\ \text{Uni} & \text{if } p = \infty \end{cases} ; \quad f(x) = \begin{cases} \frac{p^{1-1/p}}{\Gamma(1/p)} e^{-x^p/p} & \text{if } p < \infty \\ \mathbf{1}_{x<1} & \text{if } p = \infty. \end{cases} $$

The density function $f$ of $\text{IndIso}(p)$ can also be found in Table 1, case $d = 1, \tau = \infty$.

The name *independent-isotropic* comes from the fact that, from Proposition 9, a continuous distribution is *p*-iid-isotropic if and only if it is $\pm b \text{IndIso}(p)$ for some scale $b > 0$.

The $\text{IndIso}$ distribution can therefore be used for *drawing a random direction in $\mathbb{R}^d$ uniformly in the $p$-norm*. It is widely known, for instance that sampling from $\text{Sph}_2^d$ can be achieved with $\mathbf{X}/\|\mathbf{X}\|$ where $\mathbf{X} \sim [\text{Gauss}]_{1..d} = [\text{IndIso}(2)]_{1..d}$. More generally, we can sample from $\text{Sph}_p^d$ in the $p$-norm using $\mathbf{X}/\|\mathbf{X}\|$ where $\mathbf{X} \sim [\text{IndIso}(p)]_{1..d}$. This approach incurs the costs of sampling from the gamma distribution but avoids the possibly large rejection probabilities incurred in the method of sampling uniformly from a hyper-box, discarding points with norm larger than 1, and normalizing.

**Proposition 11.** $\text{IndIso}(p)$ converges in distribution to $\text{IndIso}(\infty)$ as $p \to \infty$.

*Proof.* Notice that $\Gamma(1/p) = \Gamma(1/p+1)/p$, and $\lim_{p \to \infty} p^{1-1/p} = 1$, and $\Gamma(1) = 0$, and $\lim_{p \to \infty} e^{-x^p/p} = 1_{x \le 1}$. It follows that

$$ \lim_{p \to \infty} \frac{p^{1-1/p}}{\Gamma(1/p)} e^{-x^p/p} = \lim_{p \to \infty} \frac{p^{-1/p}}{\Gamma(1+1/p)} e^{-x^p/p} = \mathbf{1}_{1 \le x}. $$

□

## 5.1 Light-tailed regime (2 parameters)

Let us generalize definition 4 to include a second parameter $d$ so as to measure the radius $\|X_{1..d}\|_p$ for $X_i \sim \text{IndIso}(p)$. Clearly, for $d = 1$, this is just $\text{IndIso}(p)$. To distinguish this distribution from the one in the next section, we will refer to it as light-tailed in the sense that it decays faster than any power-law even for the smallest values of $p$.

**Definition 5.** The (light-tailed) independent-isotropic distribution $\text{IndIso}(p, d)$ with parameters $p \in (0, \infty]$ and $d \in (0, \infty)$ is the distribution over $[0, \infty)$ given by

$$ \text{IndIso}(p, d) \stackrel{\text{def}}{=} \|[\text{IndIso}(p)]_{1..d}\|_p, $$

Its density is detailed in Table 1, and, if unspecified, $d = 1$ is assumed in accordance to Definition 4.
The parameter $d$ can be thought of as the “inherent dimensionality” of the distribution in the sense that $\text{IndIso}(p, d)$ can be decomposed as an aggregation (p-norm) of $d$ independent distributions.

Figure 5 shows the densities of several instances of the $\text{IndIso}$ distribution for different combinations of $p$ and $d$. In general terms, $d$ controls the accumulation of mass near 0 (small $d$) or 1 (large $d$), and $p$ controls the exponential decay rate of the tail: for small $p$ the decay is slow and the mass is more spread compared to larger $p$; for $p = \infty$, the tail is cut sharply and the distribution is bounded with domain $[0, 1]$.

<figure><img src="image_5.png" /><figcaption>Figure 5: Density of $\text{IndIso}$ for different combinations of $p$ and $d$.</figcaption></figure>

Some properties of the $\text{IndIso}$ distribution, quick to verify from the definition and the properties of the generalized gamma distribution [10, 13] are the following:

1. $[\pm\text{IndIso}(p)]_{1..d}$ can be written as the product between independent direction $S \sim \|\text{Uni}(\text{Ball}_p^d)\|_p$ and $p$-radius $R = b \ d^{1/p} \ \text{IndIso}(p, d)$, with the convention $d^{1/p} = 1$ for $p = \infty$.

2. For $p < \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p)] = b^{\frac{\Gamma(2/p)}{\Gamma(1/p)}} p^{1/p}$.

3. For $p = \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p)] = b/2$ (limit of the expression above).

4. For $p < \infty$, $\mathbb{E}[X^p \mid X \sim \text{IndIso}(p)]^{1/p} = b$.

5. For $p = \infty$, $\sup[X \mid X \sim \text{IndIso}(p)] = b$ (limit of the expression above).

6. For $p < \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p, d)] = b^{\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}} (p/d)^{1/p}$.

7. For $p = \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p, d)] = b^{\frac{d}{d+1}}$ (limit of the expression above).

8. The sequence $X_n \sim b \cdot \text{IndIso}(p, n)$ converges in distribution to 1 (a Dirac-delta) as $n \to \infty$.
The parameter $d$ can be thought of as the “inherent dimensionality” of the distribution in the sense that $\text{IndIso}(p, d)$ can be decomposed as an aggregation (p-norm) of $d$ independent distributions.

Figure 5 shows the densities of several instances of the $\text{IndIso}$ distribution for different combinations of $p$ and $d$. In general terms, $d$ controls the accumulation of mass near 0 (small $d$) or 1 (large $d$), and $p$ controls the exponential decay rate of the tail: for small $p$ the decay is slow and the mass is more spread compared to larger $p$; for $p = \infty$, the tail is cut sharply and the distribution is bounded with domain $[0, 1]$.

<figure><img src="image_6.png" /><figcaption>Figure 5: Density of $\text{IndIso}$ for different combinations of $p$ and $d$.</figcaption></figure>

Some properties of the $\text{IndIso}$ distribution, quick to verify from the definition and the properties of the generalized gamma distribution [10, 13] are the following:

1. $[\pm\text{IndIso}(p)]_{1..d}$ can be written as the product between independent direction $S \sim \|\text{Uni}(\text{Ball}_p^d)\|_p$ and $p$-radius $R = b \ d^{1/p} \ \text{IndIso}(p, d)$, with the convention $d^{1/p} = 1$ for $p = \infty$.

2. For $p < \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p)] = b^{\frac{\Gamma(2/p)}{\Gamma(1/p)}} p^{1/p}$.

3. For $p = \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p)] = b/2$ (limit of the expression above).

4. For $p < \infty$, $\mathbb{E}[X^p \mid X \sim \text{IndIso}(p)]^{1/p} = b$.

5. For $p = \infty$, $\sup[X \mid X \sim \text{IndIso}(p)] = b$ (limit of the expression above).

6. For $p < \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p, d)] = b^{\frac{\Gamma((d+1)/p)}{\Gamma(d/p)}} (p/d)^{1/p}$.

7. For $p = \infty$, $\mathbb{E}[X \mid X \sim \text{IndIso}(p, d)] = b^{\frac{d}{d+1}}$ (limit of the expression above).

8. The sequence $X_n \sim b \cdot \text{IndIso}(p, n)$ converges in distribution to 1 (a Dirac-delta) as $n \to \infty$.
## 5.2 Heavy-tailed regime (3 parameters)

In this section we extend the definition of IndIso to a Paretian regime by means of a parameter $\tau$ and we will also extend the domain of the parameter $d$ to include $d = \infty$. The densities and generators of IndIso for all the different combinations of $p, d, \tau$ are found in Table 1.

Heavy tails are important in applications where the objective is to analyze the magnitude and chance of large risks, represented as outliers in a tail. This is typical in financial [5] and actuarial [13] applications. In these cases, the Central Limit Theorem, which states that the sum of a large number of independent and identically distributed random variables has a distribution that is approximately normal, is not applicable for two main reasons. First, the CLT requires the existence of the first two moments, which is not the case for heavy-tailed distributions with tail index $\tau \le 2$. Second, even if $\tau > 2$, the convergence rate in the CLT is faster in the center of the distribution than in the tails, so the asymptotic tail behavior (hence also the magnitude and chance of large risks) is not sufficiently well approximated with a Gaussian distribution. In particular, the probability of observing a value larger than $9.06\sigma$ is approximately 1 in $7.6 \times 10^{18}$, which is approximately the number of grains of sand on Earth [4]. This is inconsistent with market data, which has observed events even more extreme than this. For this reason, several authors [5, 15] advocate for the use of heavy-tailed distributions to model financial risk.

Two of the most prominent heavy-tailed distributions are Lomax and Cauchy, and just like Exp and Gauss (i.e. IndIso(1) and $\pm IndIso(2)$), they are related with the process of generating random directions in the unit spheres $Sph_1^2$ and $Sph_2^2$. Indeed, let us project on the line $x = 1$ uniformly chosen random directions in $[0, \infty)^2$ in the $p$-norm (with $\text{UniDir}(\Delta_p^2)$), producing Figure 6. The blue distributions that arise for $p = 1$ and $p = 2$ are Lomax and $|\text{Cauchy}|$. For $p = \infty$ we obtain a mixture that we call UniParetoMix and consists of a uniform distribution over $[0, 1]$ and a pareto distribution over $[1, \infty)$, each weighted with $1/2$.

Three important remarks can be made from Figure 6. First, projecting to $x = b$ would simply change the scale of the densities. Second, the projections can be written as the ratio IndIso($p$)/IndIso($p$). To see why, first notice that the projections are the same regardless of the radius of the sphere, so it is equivalent to draw a radius at random, then sample the sphere of this radius and project. In other words, the projections are the same for any isotropic distribution. In particular, the coordinate distributions could be i.i.d, i.e. $[\text{IndIso}(p)]_{1..2}$. Lastly, due to the geometric construction, for $T \sim \frac{\pi_p}{2}$ Uni we get

$$ \frac{\text{IndIso}(p)}{\text{IndIso}(p)} \equiv \frac{\sin_p(T)}{\cos_p(T)} = \tan_p(T). $$

More generally, if we consider ratios of the form IndIso($p$)/IndIso($p, \tau$), we will observe that the parameter $\tau$ corresponds to the tail index of the distribution,
## 5.2 Heavy-tailed regime (3 parameters)

In this section we extend the definition of IndIso to a Paretian regime by means of a parameter $\tau$ and we will also extend the domain of the parameter $d$ to include $d = \infty$. The densities and generators of IndIso for all the different combinations of $p, d, \tau$ are found in Table 1.

Heavy tails are important in applications where the objective is to analyze the magnitude and chance of large risks, represented as outliers in a tail. This is typical in financial [5] and actuarial [13] applications. In these cases, the Central Limit Theorem, which states that the sum of a large number of independent and identically distributed random variables has a distribution that is approximately normal, is not applicable for two main reasons. First, the CLT requires the existence of the first two moments, which is not the case for heavy-tailed distributions with tail index $\tau \le 2$. Second, even if $\tau > 2$, the convergence rate in the CLT is faster in the center of the distribution than in the tails, so the asymptotic tail behavior (hence also the magnitude and chance of large risks) is not sufficiently well approximated with a Gaussian distribution. In particular, the probability of observing a value larger than $9.06\sigma$ is approximately 1 in $7.6 \times 10^{18}$, which is approximately the number of grains of sand on Earth [4]. This is inconsistent with market data, which has observed events even more extreme than this. For this reason, several authors [5, 15] advocate for the use of heavy-tailed distributions to model financial risk.

Two of the most prominent heavy-tailed distributions are Lomax and Cauchy, and just like Exp and Gauss (i.e. IndIso(1) and $\pm IndIso(2)$), they are related with the process of generating random directions in the unit spheres $Sph_1^2$ and $Sph_2^2$. Indeed, let us project on the line $x = 1$ uniformly chosen random directions in $[0, \infty)^2$ in the $p$-norm (with $\text{UniDir}(\Delta_p^2)$), producing Figure 6. The blue distributions that arise for $p = 1$ and $p = 2$ are Lomax and $|\text{Cauchy}|$. For $p = \infty$ we obtain a mixture that we call UniParetoMix and consists of a uniform distribution over $[0, 1]$ and a pareto distribution over $[1, \infty)$, each weighted with $1/2$.

Three important remarks can be made from Figure 6. First, projecting to $x = b$ would simply change the scale of the densities. Second, the projections can be written as the ratio IndIso($p$)/IndIso($p$). To see why, first notice that the projections are the same regardless of the radius of the sphere, so it is equivalent to draw a radius at random, then sample the sphere of this radius and project. In other words, the projections are the same for any isotropic distribution. In particular, the coordinate distributions could be i.i.d, i.e. $[\text{IndIso}(p)]_{1..2}$. Lastly, due to the geometric construction, for $T \sim \frac{\pi_p}{2}$ Uni we get

$$ \frac{\text{IndIso}(p)}{\text{IndIso}(p)} \equiv \frac{\sin_p(T)}{\cos_p(T)} = \tan_p(T). $$

More generally, if we consider ratios of the form IndIso($p$)/IndIso($p, \tau$), we will observe that the parameter $\tau$ corresponds to the tail index of the distribution,
<figure><img src="image_6.png" /><figcaption>Figure 6: Geometric construction of Lomax, |Cauchy| and UniParetoMix (blue densities) by sampling orange points from UniDir(Sph<sub>p</sub><sup>2</sup>) and projecting them on x = 1 via red lines. For p = 2, this is known as a gnomonic projection.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 7: Decay of the survival function P[X > t] as t grows where X ~ IndIso(1)/IndIso(1, τ). Dashed lines represent fixed slopes that are asymptotes.</figcaption></figure>

which governs the survival function (one minus the CDF). That is, if X ~ IndIso(p)/IndIso(p, τ), then P(X > t) = o(1/t<sup>τ</sup>), which is shown in Figure 7 for p = 1 and various τ.

The smaller τ, the heavier the tail, and the larger τ, the lighter the tail. In the limit as τ → ∞, in consequence of property 8 (convergence to 1), the ratio IndIso(p)/IndIso(p, τ) converges to the light-tailed IndIso(p). More generally, as we prove later in Proposition 12, IndIso(p, d)/IndIso(p, τ) converges in distribution to IndIso(p, d) as τ → ∞.

This double relationship between the heavy and light-tailed regimes is the inspiration for Definition 6, in which we consider ratios IndIso(p, d)/IndIso(p, τ).

**Definition 6.** The independent-isotropic distribution IndIso(p, d, τ) with parameters p ∈ [1, ∞], d ∈ (0, ∞] and τ ∈ (0, ∞) is the distribution over [0, ∞) given by

$$ \mathrm{IndIso}(p, d, \tau) := \frac{\mathrm{IndIso}(p, d)}{\mathrm{IndIso}(p, \tau)}. $$

Its density function is detailed in Table 1 and the default values τ = ∞ and
<figure><img src="image_7.png" /><figcaption>Figure 6: Geometric construction of Lomax, |Cauchy| and UniParetoMix (blue densities) by sampling orange points from UniDir(Sph<sub>p</sub><sup>2</sup>) and projecting them on x = 1 via red lines. For p = 2, this is known as a gnomonic projection.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Decay of the survival function P[X > t] as t grows where X ~ IndIso(1)/IndIso(1, τ). Dashed lines represent fixed slopes that are asymptotes.</figcaption></figure>

which governs the survival function (one minus the CDF). That is, if X ~ IndIso(p)/IndIso(p, τ), then P(X > t) = o(1/t<sup>τ</sup>), which is shown in Figure 7 for p = 1 and various τ.

The smaller τ, the heavier the tail, and the larger τ, the lighter the tail. In the limit as τ → ∞, in consequence of property 8 (convergence to 1), the ratio IndIso(p)/IndIso(p, τ) converges to the light-tailed IndIso(p). More generally, as we prove later in Proposition 12, IndIso(p, d)/IndIso(p, τ) converges in distribution to IndIso(p, d) as τ → ∞.

This double relationship between the heavy and light-tailed regimes is the inspiration for Definition 6, in which we consider ratios IndIso(p, d)/IndIso(p, τ).

**Definition 6.** The independent-isotropic distribution IndIso(p, d, τ) with parameters p ∈ [1, ∞], d ∈ (0, ∞] and τ ∈ (0, ∞) is the distribution over [0, ∞) given by

$$ \mathrm{IndIso}(p, d, \tau) := \frac{\mathrm{IndIso}(p, d)}{\mathrm{IndIso}(p, \tau)}. $$

Its density function is detailed in Table 1 and the default values τ = ∞ and
$d = 1$ are understood if unspecified, in accordance to Definitions 4 and 5.

Definition 6 is consistent with Definition 5 because $\text{IndIso}(p, d, \infty) \equiv \text{IndIso}(p, d)$. This is why we use the same generic distribution “IndIso” for both regimes. Furthermore, it is clear that for $d = \infty$, the numerator converges to 1, so the distribution converges to an *inverse light-tailed* distribution $1/\text{IndIso}(p, \tau)$, also detailed in Table 1.

<figure><img src="image_8.png" /><figcaption>Figure 8: Lomax hyperbolas.</figcaption></figure>

The level curves of $[\pm\text{IndIso}(p, 1, 1)]_{1..2}$ for $p = 1, 2, \infty$ are depicted in Figure 9. These curves are not $p$-spheres anymore, and they are not even successive scales of the same shape. But they are approximately $p$-spheres for small radiuses, and the deformation they suffer for large radiuses is related with those produced by hyperbolic coordinates in the quadrant model system, as shown in Figure 5.2.

<figure><img src="image_9.png" /><figcaption>Figure 9: Level curves containing 25%, 50% and 75% of mass (in pink) for the $p = 1, 2, \infty$ with $d = \tau = 1$. The level curve passing through (0, 1) is shown in orange (compare to Figure 1).</figcaption></figure>

To conclude this section, we present a graphical way of interpreting the parameters $p$, $d$, and $\tau$, which have been regarded so far as related to norm, inherent dimensionality and tail index respectively.

The log-log plots in Figure 10 show that if the graph of the density of $\text{IndIso}(p, d, \tau)$ had to be divided into three parts, they would be the head $\ll 1$, the middle $\approx 1$ and the tail $\gg 1$. These three parts are governed by $d$, $p$ and $\tau$ respectively, and the graphs in Figure 10 show the degree of independence of the head and the tail behavior.

The head is governed by $d$ because of the factor $x^{d-1}$ in the density function. For $d = 1$, this factor is constant, the log-log plot of the head is flat, and the density at 0 is finite and non-zero. For $d < 1$, the log-log slope is negative and
$d = 1$ are understood if unspecified, in accordance to Definitions 4 and 5.

Definition 6 is consistent with Definition 5 because $\text{Indlso}(p, d, \infty) \equiv \text{Indlso}(p, d)$. This is why we use the same generic distribution “Indlso” for both regimes. Furthermore, it is clear that for $d = \infty$, the numerator converges to 1, so the distribution converges to an *inverse light-tailed* distribution $1/\text{Indlso}(p, \tau)$, also detailed in Table 1.

<figure><img src="image_9.png" /><figcaption>Figure 8: Lomax hyperbolas.</figcaption></figure>

The level curves of $[\pm\text{Indlso}(p, 1, 1)]_{1..2}$ for $p = 1, 2, \infty$ are depicted in Figure 9. These curves are not $p$-spheres anymore, and they are not even successive scales of the same shape. But they are approximately $p$-spheres for small radiuses, and the deformation they suffer for large radiuses is related with those produced by hyperbolic coordinates in the quadrant model system, as shown in Figure 5.2.

<figure><img src="image_10.png" /><figcaption>Figure 9: Level curves containing 25%, 50% and 75% of mass (in pink) for the $p = 1, 2, \infty$ with $d = \tau = 1$. The level curve passing through (0, 1) is shown in orange (compare to Figure 1).</figcaption></figure>

To conclude this section, we present a graphical way of interpreting the parameters $p$, $d$, and $\tau$, which have been regarded so far as related to norm, inherent dimensionality and tail index respectively.

The log-log plots in Figure 10 show that if the graph of the density of $\text{Indlso}(p, d, \tau)$ had to be divided into three parts, they would be the head $\ll 1$, the middle $\approx 1$ and the tail $\gg 1$. These three parts are governed by $d$, $p$ and $\tau$ respectively, and the graphs in Figure 10 show the degree of independence of the head and the tail behavior.

The head is governed by $d$ because of the factor $x^{d-1}$ in the density function. For $d = 1$, this factor is constant, the log-log plot of the head is flat, and the density at 0 is finite and non-zero. For $d < 1$, the log-log slope is negative and
<figure><img src="image_10.png" /><figcaption>Figure 10: Density of $\text{IndIso}(p, d, \tau)$ for various combinations of parameters. The log-log scale makes it clear that $\tau$ governs the tail and $d$ governs the head.</figcaption></figure>

the density diverges to infinity at 0 (faster for $d \ll 1$), and for $d > 1$, the density converges to 0 at 0 and takes off very slowly for $d \gg 1$.

Similarly, the tail is governed by $\tau$ because of the factor $\exp(-x^p)$ in the density function, and the coupling between the head and the tail is governed by $p$. For large $p$, the head and the tail are more independent (decoupled for $p = \infty$), and for small $p$, the head and the tail are more coupled, making the change from one slope to the other slower.

## 5.3 Summary and particularizations of the distribution

The densities and generators of the $\text{IndIso}$ distribution (Definitions 5 and 6) are summarized in Table 1 and visualized in Figure 11.

The $\text{IndIso}$ distribution has many important distributions as special cases because its definition is based in powers and quotients of the gamma and uniform distributions. Most of the particularizations mentioned next are simply inherited from the relationship of $\text{IndIso}$ with the generalized gamma distribution, its
<figure><img src="image_11.png" /><figcaption>Figure 10: Density of $\text{Indlso}(p, d, \tau)$ for various combinations of parameters. The log-log scale makes it clear that $\tau$ governs the tail and $d$ governs the head.</figcaption></figure>

the density diverges to infinity at 0 (faster for $d \ll 1$), and for $d > 1$, the density converges to 0 at 0 and takes off very slowly for $d \gg 1$.

Similarly, the tail is governed by $\tau$ because of the factor $\exp(-x^p)$ in the density function, and the coupling between the head and the tail is governed by $p$. For large $p$, the head and the tail are more independent (decoupled for $p = \infty$), and for small $p$, the head and the tail are more coupled, making the change from one slope to the other slower.

## 5.3 Summary and particularizations of the distribution

The densities and generators of the $\text{Indlso}$ distribution (Definitions 5 and 6) are summarized in Table 1 and visualized in Figure 11.

The $\text{Indlso}$ distribution has many important distributions as special cases because its definition is based in powers and quotients of the gamma and uniform distributions. Most of the particularizations mentioned next are simply inherited from the relationship of $\text{Indlso}$ with the generalized gamma distribution, its
<table><caption>Table 1: Sample generation and density function of the lndlso distribution. Γ is the gamma function (Γ(1 + n) = n!) and B is the beta function (B(a, b) = Γ(a)Γ(b)/Γ(a+b)).</caption><thead><tr><th></th><th>Case</th><th>Sample<br>generator</th><th>Density</th><th>Integration<br>constant</th></tr></thead><tbody><tr><td rowspan="3">Narrow tail (τ = ∞)</td><td>p &lt; ∞,<br>d &lt; ∞,<br>τ = ∞</td><td>b(Gamma(d/p)/d/p)<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup> exp(-d/x<sup>p</sup>/b<sup>p</sup>)</td><td>D = b<sup>d</sup>(<sup>p</sup>&frasl;<sub>d</sub>)<sup>d/p</sup>Γ(<sup>d</sup>&frasl;<sub>p</sub>)/p</td></tr><tr><td>p = ∞,<br>d &lt; ∞,<br>τ = ∞</td><td>b Uni<sup>1/d</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {(<sup>x</sup>&frasl;<sub>b</sub>)<sup>d-1</sup> if x ≤ b<br>0 if x ≥ b</td><td>D = b B(d, 1) = <sup>b</sup>&frasl;<sub>d</sub></td></tr><tr><td>p ≤ ∞,<br>d = ∞,<br>τ = ∞</td><td>b</td><td>Delta-Dirac at x = b<br>(discrete)</td><td>—</td></tr><tr><td rowspan="3">Heavy tail (power law regime)</td><td>p &lt; ∞,<br>d &lt; ∞,<br>τ &lt; ∞</td><td>b(τ/d Gamma(d/p)/Gamma(τ/p))<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup>(1 + d/x<sup>p</sup>)<sup>-τ+d/p</sup>, or<br><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup> exp<sub>q</sub>(-x<sup>p</sup>/d<sup>p</sup>(τ+d))</td><td>D = b<sup>d</sup>(<sup>p</sup>&frasl;<sub>d</sub>)<sup>d/p</sup>B(τ/p, d/p)/p<br>q = 1 + <sup>p</sup>&frasl;<sub>τ+d</sub></td></tr><tr><td>p = ∞,<br>d &lt; ∞,<br>τ &lt; ∞</td><td>b Uni<sup>1/d</sup>/Uni<sup>1/τ</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {(<sup>x</sup>&frasl;<sub>b</sub>)<sup>d-1</sup> if x ≤ b<br>(<sup>x</sup>&frasl;<sub>b</sub>)<sup>-τ-1</sup> if x ≥ b</td><td>D = b(<sup>1</sup>&frasl;<sub>τ</sub> + <sup>1</sup>&frasl;<sub>d</sub>)</td></tr><tr><td>p &lt; ∞,<br>d = ∞,<br>τ &lt; ∞</td><td>b(τ/p)/Gamma(τ/p)<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> <sup>1</sup>&frasl;<sub>x<sup>1+τ</sup></sub> exp(-τ/b<sup>p</sup>/x<sup>p</sup>)</td><td>D = b<sup>-τ</sup>(<sup>p</sup>&frasl;<sub>τ</sub>)<sup>τ/p</sup>Γ(<sup>τ</sup>&frasl;<sub>p</sub>)/p</td></tr><tr><td></td><td>p = ∞,<br>d = ∞,<br>τ &lt; ∞</td><td>b Uni<sup>1/τ</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {0 if x &lt; b<br>(<sup>b</sup>&frasl;<sub>x</sub>)<sup>τ+1</sup> if x ≥ b</td><td>D = <sup>b</sup>&frasl;<sub>τ</sub></td></tr></tbody></table>

inverse and the generalized beta prime distribution (or generalized beta of the second kind).

## Most general cases

For $p < \infty$ and $\tau = \infty$, lndlso($p, d$) is a **GeneralizedGamma** distribution:

* Parametrized as in [20], GenGamma($a, d, p$) ≡ $a (d/p)^{1/p}$ lndlso($d, p$).

* Parametrized as in scipy's library for scientific computation in Python, (scipy.special.gengamma), GenGamma($a, c$) ≡ $(a)^{1/c}$ lndlso($ac, c$).
<table><caption>Table 1: Sample generation and density function of the lndlso distribution. Γ is the gamma function (Γ(1 + n) = n!) and B is the beta function (B(a, b) = Γ(a)Γ(b)/Γ(a+b)).</caption><thead><tr><th></th><th>Case</th><th>Sample<br>generator</th><th>Density</th><th>Integration<br>constant</th></tr></thead><tbody><tr><td rowspan="3">Narrow tail (τ = ∞)</td><td>p &lt; ∞,<br>d &lt; ∞,<br>τ = ∞</td><td>b(Gamma(d/p)/d/p)<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup> exp(-d/x<sup>p</sup>/b<sup>p</sup>)</td><td>D = b<sup>d</sup>(<sup>p</sup>&frasl;<sub>d</sub>)<sup>d/p</sup>Γ(<sup>d</sup>&frasl;<sub>p</sub>)/p</td></tr><tr><td>p = ∞,<br>d &lt; ∞,<br>τ = ∞</td><td>b Uni<sup>1/d</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {(<sup>x</sup>&frasl;<sub>b</sub>)<sup>d-1</sup> if x ≤ b<br>0 if x ≥ b</td><td>D = b B(d, 1) = <sup>b</sup>&frasl;<sub>d</sub></td></tr><tr><td>p ≤ ∞,<br>d = ∞,<br>τ = ∞</td><td>b</td><td>Delta-Dirac at x = b<br>(discrete)</td><td>—</td></tr><tr><td rowspan="3">Heavy tail (power law regime)</td><td>p &lt; ∞,<br>d &lt; ∞,<br>τ &lt; ∞</td><td>b(τ/d Gamma(d/p)/Gamma(τ/p))<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup>(1 + d/x<sup>p</sup>)<sup>-τ+d/p</sup>, or<br><sup>1</sup>&frasl;<sub>D</sub>x<sup>d-1</sup> exp<sub>q</sub>(-x<sup>p</sup>/d<sup>p</sup>(τ+d))</td><td>D = b<sup>d</sup>(<sup>p</sup>&frasl;<sub>d</sub>)<sup>d/p</sup>B(τ/p, d/p)/p<br>q = 1 + <sup>p</sup>&frasl;<sub>τ+d</sub></td></tr><tr><td>p = ∞,<br>d &lt; ∞,<br>τ &lt; ∞</td><td>b Uni<sup>1/d</sup>/Uni<sup>1/τ</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {(<sup>x</sup>&frasl;<sub>b</sub>)<sup>d-1</sup> if x ≤ b<br>(<sup>x</sup>&frasl;<sub>b</sub>)<sup>-τ-1</sup> if x ≥ b</td><td>D = b(<sup>1</sup>&frasl;<sub>τ</sub> + <sup>1</sup>&frasl;<sub>d</sub>)</td></tr><tr><td>p &lt; ∞,<br>d = ∞,<br>τ &lt; ∞</td><td>b(τ/p)/Gamma(τ/p)<sup>1/p</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> <sup>1</sup>&frasl;<sub>x<sup>1+τ</sup></sub> exp(-τ/b<sup>p</sup>/x<sup>p</sup>)</td><td>D = b<sup>-τ</sup>(<sup>p</sup>&frasl;<sub>τ</sub>)<sup>τ/p</sup>Γ(<sup>τ</sup>&frasl;<sub>p</sub>)/p</td></tr><tr><td></td><td>p = ∞,<br>d = ∞,<br>τ &lt; ∞</td><td>b Uni<sup>1/τ</sup></td><td><sup>1</sup>&frasl;<sub>D</sub> {0 if x &lt; b<br>(<sup>b</sup>&frasl;<sub>x</sub>)<sup>τ+1</sup> if x ≥ b</td><td>D = <sup>b</sup>&frasl;<sub>τ</sub></td></tr></tbody></table>

inverse and the generalized beta prime distribution (or generalized beta of the second kind).

### Most general cases

For $p < \infty$ and $\tau = \infty$, lndlso(p, d) is a **GeneralizedGamma** distribution:

* Parametrized as in [20], GenGamma(a, d, p) ≡ a (d/p)<sup>1/p</sup> lndlso(d, p).

* Parametrized as in scipy's library for scientific computation in Python, (scipy.special.gengamma), GenGamma(a, c) ≡ (a)<sup>1/c</sup> lndlso(ac, c).
<figure><img src="image_11.png" /><figcaption>Figure 11: Density function of the *Indlso* distribution varying *d*, *p* and *τ*.</figcaption></figure>

* Parametrized as in [13] (p.p. 148), GenGamma(a, β, p) ≡ β p¹/a *Indlso*(ap, a); and as in [17] (p.p. 11), GenGamma(α, τ, λ) ≡ λ τ¹/α *Indlso*(ατ, α).

For $p < \infty$ and $\tau < \infty$, *Indlso*(p, d, τ) is a generalized BetaPrime distribution scaled by $(\frac{d}{\tau})^{1/p}$.
<figure><img src="image_12.png" /><figcaption>Figure 11: Density function of the *Indlso* distribution varying *d*, *p* and *τ*.</figcaption></figure>

* Parametrized as in [13] (p.p. 148), GenGamma(a, β, p) ≡ β p¹/a *Indlso*(ap, a); and as in [17] (p.p. 11), GenGamma(α, τ, λ) ≡ λ τ¹/α *Indlso*(ατ, α).

For $p < \infty$ and $\tau < \infty$, *Indlso*(p, d, τ) is a generalized BetaPrime distribution scaled by $(\frac{d}{\tau})^{1/p}$.
**Cases with $p=1$**

* $\text{Exp} \equiv \text{Indlso}(1)$ and $\text{Erlang}(k) \equiv \text{Gamma}(k) \equiv k\text{Indlso}(1, k)$.

* $\text{Lomax}(\alpha) \equiv \text{Indlso}(1, 1, \alpha)$.

* $\text{InvGamma}(\alpha) \equiv (1/\alpha)\text{Indlso}(1, \infty, \alpha)$.

* $\text{Indlso}(1, 1, \tau)$ is the **Lomax** distribution with shape parameter $\alpha = \tau$.

* $\text{Indlso}(1, 1, \tau)$ is the **TsallisQExponential** distribution with shape parameter $q = 1 + \frac{p}{d+\tau}$.

* $\text{Indlso}(1, 1, \tau=\infty)$ is the **Exponential** distribution.

* $\text{Indlso}(1, d, \tau=\infty)$ is the **Gamma** distribution.

* $\text{Indlso}(1, d, \tau)$ is a **BetaPrime** distribution scaled by $\frac{d}{\tau}$.

**Cases with $p=2$**

* $\text{Cauchy} \equiv \pm\text{Indlso}(2, 1, 1)$.

* $\text{Student}(\nu) \equiv \pm\text{Indlso}(2, 1, \nu)$.

* $|\text{Gauss}| \equiv \text{Chi} \equiv \text{Indlso}(2)$ and $\text{Chi}(k) \equiv \sqrt{k} \text{Indlso}(2, k)$. So, $\text{Chi}^2(k) \equiv k \text{Indlso}(2, k)^2$.

* $\text{Indlso}(2, 1, \tau)$ is the **TsallisQGaussian** distribution with shape parameter $q = 1 + \frac{p}{d+\tau}$.

* $\text{Indlso}(2, 1, \tau)$ and $\pm\text{Indlso}(2, 1, \tau=\nu)$ are, respectively, the **HalfStudent** and **Student** distributions with shape $\nu = \tau$ degrees of freedom.

* $\text{Indlso}(2, 1, \tau=\infty)$ and $\pm\text{Indlso}(2, 2, \tau=\infty)$ are, respectively, the **HalfNormal** and **Normal** distributions.

* $\text{Indlso}(2, d, \tau=\infty)$ is the **Chi** distribution.

**Cases with $p = \infty$**

* $\text{Indlso}(\infty, 1, \tau)$ is the uniform ratio distribution.

* $\text{Indlso}(\infty, 1, \tau=\infty)$ is the **Uniform** distribution.

* For $p = \infty$ and $\tau = \infty$, $\text{Indlso}(p, d)$ is a **standard power** distribution: $\text{Indlso}(\infty, d, \tau=\infty)$ is the **StandardPower** distribution, i.e., $\text{Beta}(\alpha=d, \beta=1)$.

**Miscellaneous cases**

* $\text{Uni} \equiv \text{Indlso}(\infty)$ and $|\text{Uni}^k| \equiv \text{Beta}(d, 1) \equiv \text{Indlso}(\infty, k)$, i.e. a standard power.

* $\text{Levy}(\alpha) \equiv \text{Indlso}(1, \infty, 1/2)$.
**Cases with $p=1$**

* $\text{Exp} \equiv \text{Indlso}(1)$ and $\text{Erlang}(k) \equiv \text{Gamma}(k) \equiv k\text{Indlso}(1, k)$.

* $\text{Lomax}(\alpha) \equiv \text{Indlso}(1, 1, \alpha)$.

* $\text{InvGamma}(\alpha) \equiv (1/\alpha)\text{Indlso}(1, \infty, \alpha)$.

* $\text{Indlso}(1, 1, \tau)$ is the **Lomax** distribution with shape parameter $\alpha = \tau$.

* $\text{Indlso}(1, 1, \tau)$ is the **TsallisQExponential** distribution with shape pa- rameter $q = 1 + \frac{p}{d+\tau}$.

* $\text{Indlso}(1, 1, \tau=\infty)$ is the **Exponential** distribution.

* $\text{Indlso}(1, d, \tau=\infty)$ is the **Gamma** distribution.

* $\text{Indlso}(1, d, \tau)$ is a **BetaPrime** distribution scaled by $\frac{d}{\tau}$.

**Cases with $p=2$**

* $\text{Cauchy} \equiv \pm\text{Indlso}(2, 1, 1)$.

* $\text{Student}(\nu) \equiv \pm\text{Indlso}(2, 1, \nu)$.

* $|\text{Gauss}| \equiv \text{Chi} \equiv \text{Indlso}(2)$ and $\text{Chi}(k) \equiv \sqrt{k} \text{Indlso}(2, k)$. So, $\text{Chi}^2(k) \equiv k \text{Indlso}(2, k)^2$.

* $\text{Indlso}(2, 1, \tau)$ is the **TsallisQGaussian** distribution with shape parameter $q = 1 + \frac{p}{d+\tau}$.

* $\text{Indlso}(2, 1, \tau)$ and $\pm\text{Indlso}(2, 1, \tau=\nu)$ are, respectively, the **HalfStudent** and **Student** distributions with shape $\nu = \tau$ degrees of freedom.

* $\text{Indlso}(2, 1, \tau=\infty)$ and $\pm\text{Indlso}(2, 2, \tau=\infty)$ are, respectively, the **HalfNormal** and **Normal** distributions.

* $\text{Indlso}(2, d, \tau=\infty)$ is the **Chi** distribution.

**Cases with $p = \infty$**

* $\text{Indlso}(\infty, 1, \tau)$ is the uniform ratio distribution.

* $\text{Indlso}(\infty, 1, \tau=\infty)$ is the **Uniform** distribution.

* For $p = \infty$ and $\tau = \infty$, $\text{Indlso}(p, d)$ is a **standard power** distribution: $\text{Indlso}(\infty, d, \tau=\infty)$ is the **StandardPower** distribution, i.e., $\text{Beta}(\alpha=d, \beta=1)$.

**Miscellaneaus cases**

* $\text{Uni} \equiv \text{Indlso}(\infty)$ and $|\text{Uni}^k| \equiv \text{Beta}(d, 1) \equiv \text{Indlso}(\infty, k)$, i.e. a standard power.

* $\text{Levy}(\alpha) \equiv \text{Indlso}(1, \infty, 1/2)$.
• Weibull($k$) ≡ Exp$^k$ ≡ IndIso($k, k$).

## 5.4 Further light-heavy relationships

The heavy and light regimes are related through two rather different perspectives.

First the density of the heavy independent-isotropic distribution can be expressed using the q-exponential function [21]:

$$ \exp_q(x) = \begin{cases} (1 + (1 - q)x)^{\frac{1}{1-q}} & \text{if } q \neq 1 \\ \exp(x) & \text{if } q = 1. \end{cases} $$

The q-exponential function is the solution to the differential equation $dy/dx = y^q$ with boundary condition $y(0) = 1$ [22] (p.p. 48). It is an inverse Box-Cox transformation [6] (with parameter $\lambda = 1 - q$) and it is the building block of Tsallis entropy, used in statistical mechanics and other areas of physics.

As shown in Table 1, for $p, \tau, d < \infty$, the density of the IndIso distribution can be expressed as proportional to

$$ x^{d-1} \exp_q \left( -x^p / \frac{b^p p \tau}{d(\tau+d)} \right), \qquad (7) $$

where $q = 1 + \frac{p}{\tau+d}$. From this expression, it is easier (immediate) to corroborate that as $\tau \to \infty$, this distribution converges to the light-tailed distribution.

**Proposition 12.** IndIso($p, d, \tau$) converges in distribution to IndIso($p, d$) as $\tau \to \infty$.

*Proof.* It suffices to show convergence of the density function almost everywhere. The densities are shown in Table 1.

(Case $p < \infty$.) In Equation 7, as $\tau \to \infty$, the density becomes $x^{d-1} \exp(-x^p/b^p)$, because $q \to 1$ and $\frac{\tau}{\tau+d} \to 1$.

(Case $p = \infty, d < \infty$.) The integration constant $D = b(\frac{1}{\tau} + \frac{1}{d})$ converges to $b/d$ as $\tau \to \infty$. For $x \le b$, the distribution shape is unaltered, and for $x \ge b$, the function $(x/b)^{-\tau-1}$, converges to 0.

(Case $p = \infty, d = \infty$.) For $x < b$, the density remains at zero. For $x > b$, $(x/b)^{\tau+1}b/\tau$ converges also to zero. The singularity at $x = b$ diverges to infinity, forming a delta Dirac at $x = b$. $\square$

Second, some ancillary statistics of the light-tailed distribution follow the corresponding heavy-tailed distribution (same $p$). It is well known, for instance, that given a sample $[X_1, ..., X_n] \sim [\mu + \sigma \cdot \text{Gauss}]_{1..n}$ with empirical mean $\bar{X}$ and empirical standard deviation $S = (\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2)^{1/2}$, the t-statistic of a
• Weibull($k$) ≡ Exp$^k$ ≡ IndIso($k, k$).

## 5.4 Further light-heavy relationships

The heavy and light regimes are related through two rather different perspectives.

First the density of the heavy independent-isotropic distribution can be expressed using the q-exponential function [21]:

$$ \exp_q(x) = \begin{cases} (1 + (1 - q)x)^{\frac{1}{1-q}} & \text{if } q \neq 1 \\ \exp(x) & \text{if } q = 1. \end{cases} $$

The q-exponential function is the solution to the differential equation $dy/dx = y^q$ with boundary condition $y(0) = 1$ [22] (p.p. 48). It is an inverse Box-Cox transformation [6] (with parameter $\lambda = 1 - q$) and it is the building block of Tsallis entropy, used in statistical mechanics and other areas of physics.

As shown in Table 1, for $p, \tau, d < \infty$, the density of the IndIso distribution can be expressed as proportional to

$$ x^{d-1} \exp_q \left( -x^p / \frac{b^p p \tau}{d(\tau+d)} \right), \qquad (7) $$

where $q = 1 + \frac{p}{\tau+d}$. From this expression, it is easier (immediate) to corroborate that as $\tau \to \infty$, this distribution converges to the light-tailed distribution.

**Proposition 12.** IndIso($p, d, \tau$) converges in distribution to IndIso($p, d$) as $\tau \to \infty$.

*Proof.* It suffices to show convergence of the density function almost everywhere. The densities are shown in Table 1.

(Case $p < \infty$.) In Equation 7, as $\tau \to \infty$, the density becomes $x^{d-1} \exp(-x^p / \frac{b^p p}{d})$, because $q \to 1$ and $\frac{\tau}{\tau+d} \to 1$.

(Case $p = \infty, d < \infty$.) The integration constant $D = b(\frac{1}{\tau} + \frac{1}{d})$ converges to $b/d$ as $\tau \to \infty$. For $x \le b$, the distribution shape is unaltered, and for $x \ge b$, the function $(x/b)^{-\tau-1}$, converges to 0.

(Case $p = \infty, d = \infty$.) For $x < b$, the density remains at zero. For $x > b$, $(x/b)^{\tau+1}b/\tau$ converges also to zero. The singularity at $x = b$ diverges to infinity, forming a delta Dirac at $x = b$. $\square$

Second, some ancillary statistics of the light-tailed distribution follow the corresponding heavy-tailed distribution (same $p$). It is well known, for instance, that given a sample $[X_1, ..., X_n] \sim [\mu + \sigma \cdot \text{Gauss}]_{1..n}$ with empirical mean $\bar{X}$ and empirical standard deviation $S = (\frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2)^{1/2}$, the t-statistic of a
future observation $X_{n+1} \sim \text{Gauss}$ is distributed as

$$T = \frac{X_{n+1} - \bar{X}}{S\sqrt{(n+1)/n}} \sim \text{Student}(n-1) \equiv \pm \text{Indlso}(2, 1, n-1),$$

which is ancillary with respect to both $\mu$ and $\sigma$, as $\text{Indlso}(2, 1, n-1)$ does not depend on them.

In general, if $X_{1..n} \sim [b \cdot \text{Indlso}(p)]_{1..n}$ for an unknown scale parameter $b > 0$, then for a future observation $X_{n+1} \sim \text{Indlso}(p)$ we have

$$\frac{X_{n+1}}{\|X_{1..n}\|_p} \sim \frac{1}{d^{1/p}} \text{Indlso}(p, 1, n),$$

which is ancillary with respect to $b$. The explanation is straightforward: the numerator follows $b\text{Indlso}(p)$ and the denominator follows $b d^{1/p} \text{Indlso}(p, n)$, so the ratio is $d^{-1/p}\text{Indlso}(p, 1, n)$.

# 6 Summary

This article explores the concept of isotropy (uniform directionality) in different $p$-norms. This corresponds to distributions in $n$ dimensions whose density function can be written as a function of the $p$-norm.

In Section 3, we remark from [23] and from our own computer simulations (Figures 3 and 2) that sampling a point uniformly from the surface of a $p$-ball in $d$ dimensions is not always equivalent to sampling a point uniformly from its volume and projecting to the surface (with the line that passes by the origin). These two candidate procedures for the notion of sampling a random direction uniformly, happen to be equal if and only if $p$ is 1, 2 or $\infty$. We argue in Section 3.3 that for arbitrary $p$-norms, the volume-based procedure is the correct one.

In Section 4, we introduce the notion of independent-isotropic distribution by observing (Proposition 8) that the joint distribution of $n$ i.i.d. coordinates with distribution Dist over $\mathbb{R}^d$ is isotropic if and only if the joint distribution of $m$ of them is, for any $n \ge m \ge 2$. Then, we characterized the independent-isotropic distributions (Proposition 9).

In Section 5, we defined the family of Indlso distributions (Definition 4), and generalized it with two parameters (Definitions 5 and 6): the tail index and the inherent number of dimensions. We then showed that two different subfamilies can be differentiated (light and heavy-tailed) and we expose multiple connections that exist between them. In particular, Section 5.2 shows the geometric connection between the two, and Section 5.4 shows some analytical and statistical connections between the two. More precisely, they are connected on the analytical side, by the limit as $\tau \to \infty$ and the q-exponential function, and on the statistical side, by an ancillary statistic.
future observation $X_{n+1} \sim \text{Gauss}$ is distributed as

$$T = \frac{X_{n+1} - \bar{X}}{S\sqrt{(n+1)/n}} \sim \text{Student}(n-1) \equiv \pm \text{Indlso}(2, 1, n-1),$$

which is ancillary with respect to both $\mu$ and $\sigma$, as $\text{Indlso}(2, 1, n-1)$ does not depend on them.

In general, if $X_{1..n} \sim [b \cdot \text{Indlso}(p)]_{1..n}$ for an unknown scale parameter $b > 0$, then for a future observation $X_{n+1} \sim \text{Indlso}(p)$ we have

$$\frac{X_{n+1}}{\|X_{1..n}\|_p} \sim \frac{1}{d^{1/p}} \text{Indlso}(p, 1, n),$$

which is ancillary with respect to $b$. The explanation is straightforward: the numerator follows $b\text{Indlso}(p)$ and the denominator follows $b d^{1/p} \text{Indlso}(p, n)$, so the ratio is $d^{-1/p}\text{Indlso}(p, 1, n)$.

# 6 Summary

This article explores the concept of isotropy (uniform directionality) in different $p$-norms. This corresponds to distributions in $n$ dimensions whose density function can be written as a function of the $p$-norm.

In Section 3, we remark from [23] and from our own computer simulations (Figures 3 and 2) that sampling a point uniformly from the surface of a $p$-ball in $d$ dimensions is not always equivalent to sampling a point uniformly from its volume and projecting to the surface (with the line that passes by the origin). These two candidate procedures for the notion of sampling a random direction uniformly, happen to be equal if and only if $p$ is 1, 2 or $\infty$. We argue in Section 3.3 that for arbitrary $p$-norms, the volume-based procedure is the correct one.

In Section 4, we introduce the notion of independent-isotropic distribution by observing (Proposition 8) that the joint distribution of $n$ i.i.d. coordinates with distribution Dist over $\mathbb{R}^d$ is isotropic if and only if the joint distribution of $m$ of them is, for any $n \ge m \ge 2$. Then, we characterized the independent-isotropic distributions (Proposition 9).

In Section 5, we defined the family of Indlso distributions (Definition 4), and generalized it with two parameters (Definitions 5 and 6): the tail index and the inherent number of dimensions. We then showed that two different subfamilies can be differentiated (light and heavy-tailed) and we expose multiple connections that exist between them. In particular, Section 5.2 shows the geometric connection between the two, and Section 5.4 shows some analytical and statistical connections between the two. More precisely, they are connected on the analytical side, by the limit as $\tau \to \infty$ and the q-exponential function, and on the statistical side, by an ancillary statistic.
References

[1] Luigi Amoroso. Ricerche intorno alla curva dei redditi. *Annali di matematica pura ed applicata*, 2:123-159, 1925.

[2] Miguel E Andrés, Nicolás E Bordenabe, Konstantinos Chatzikokolakis, and Catuscia Palamidessi. Geo-indistinguishability: Differential privacy for location-based systems. In *Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security*, pages 901–914, 2013.

[3] John A Baker. On the functional equation $f(x)g(y) = p(x+y)q(x/y)$. *aequationes mathematicae*, 14:493–506, 1976.

[4] David Blatner. *Spectrums: our mind-boggling universe from infinitesimal to infinity*. Bloomsbury Publishing USA, 2012.

[5] Jean-Philippe Bouchaud and Marc Potters. *Theory of financial risk and derivative pricing: from statistical physics to risk management*. Cambridge university press, 2003.

[6] George EP Box and David R Cox. An analysis of transformations. *Journal of the Royal Statistical Society Series B: Statistical Methodology*, 26(2):211–243, 1964.

[7] Keith Conrad. The gaussian integral. *University of Connecticut: Storrs, CT, USA*, pages 1–2, 2016.

[8] Gerald A Edgar and Gerald A Edgar. *Measure, topology, and fractal geometry*, volume 2. Springer, 2008.

[9] Edward Gaughan. *Introduction to analysis*, volume 1. American Mathematical Soc., 2009.

[10] Norman L. Johnson, Samuel Kotz, and Narayanaswamy Balakrishnan. *Continuous Univariate Distributions*, volume 1. Wiley-Interscience, 1994.

[11] Wolfgang B Jurkat. On cauchy's functional equation. *Proceedings of the American Mathematical Society*, 16(4):683–686, 1965.

[12] Palaniappan Kannappan. *Functional equations and inequalities with applications*. Springer Science & Business Media, 2009.

[13] Christian Kleiber and Samuel Kotz. *Statistical size distributions in economics and actuarial sciences*. John Wiley & Sons, 2003.

[14] Eugene Lukacs. A characterization of the gamma distribution. *The Annals of Mathematical Statistics*, 26(2):319–324, 1955.

[15] Benoit B Mandelbrot. *Fractals and scaling in finance: Discontinuity, concentration, risk. Selecta volume E*. Springer Science & Business Media, 2013.
References

[1] Luigi Amoroso. Ricerche intorno alla curva dei redditi. *Annali di matematica pura ed applicata*, 2:123-159, 1925.

[2] Miguel E Andrés, Nicolás E Bordenabe, Konstantinos Chatzikokolakis, and Catuscia Palamidessi. Geo-indistinguishability: Differential privacy for location-based systems. In *Proceedings of the 2013 ACM SIGSAC conference on Computer & communications security*, pages 901–914, 2013.

[3] John A Baker. On the functional equation $f(x) g(y) = p(x+ y) q(x/y)$. *aequationes mathematicae*, 14:493–506, 1976.

[4] David Blatner. *Spectrums: our mind-boggling universe from infinitesimal to infinity*. Bloomsbury Publishing USA, 2012.

[5] Jean-Philippe Bouchaud and Marc Potters. *Theory of financial risk and derivative pricing: from statistical physics to risk management*. Cambridge university press, 2003.

[6] George EP Box and David R Cox. An analysis of transformations. *Journal of the Royal Statistical Society Series B: Statistical Methodology*, 26(2):211–243, 1964.

[7] Keith Conrad. The gaussian integral. *University of Connecticut: Storrs, CT, USA*, pages 1–2, 2016.

[8] Gerald A Edgar and Gerald A Edgar. *Measure, topology, and fractal geometry*, volume 2. Springer, 2008.

[9] Edward Gaughan. *Introduction to analysis*, volume 1. American Mathematical Soc., 2009.

[10] Norman L. Johnson, Samuel Kotz, and Narayanaswamy Balakrishnan. *Continuous Univariate Distributions*, volume 1. Wiley-Interscience, 1994.

[11] Wolfgang B Jurkat. On cauchy's functional equation. *Proceedings of the American Mathematical Society*, 16(4):683–686, 1965.

[12] Palaniappan Kannappan. *Functional equations and inequalities with applications*. Springer Science & Business Media, 2009.

[13] Christian Kleiber and Samuel Kotz. *Statistical size distributions in economics and actuarial sciences*. John Wiley & Sons, 2003.

[14] Eugene Lukacs. A characterization of the gamma distribution. *The Annals of Mathematical Statistics*, 26(2):319–324, 1955.

[15] Benoit B Mandelbrot. *Fractals and scaling in finance: Discontinuity, concentration, risk. Selecta volume E*. Springer Science & Business Media, 2013.
[16] George Marsaglia. The x+ y, x/y characterization of the gamma distribution. *Contributions to Probability and Statistics: Essays in Honor of Ingram Olkin*, pages 91–98, 1989.

[17] Khodabin Morteza and Alireza Ahmadabadi. Some properties of generalized gamma distribution. *Mathematical Sciences Quarterly Journal*, 4, 03 2010.

[18] Jeffrey S Rosenthal. *First Look At Rigorous Probability Theory*, A. World Scientific Publishing Company, 2006.

[19] Grant Sanderson. Why $\pi$ is in the normal distribution (beyond integral tricks). Youtube, 2023.

[20] Edney W Stacy. A generalization of the gamma distribution. *The Annals of mathematical statistics*, pages 1187–1192, 1962.

[21] Constantino Tsallis. What are the numbers that experiments provide. *Quimica Nova*, 17(6):468–471, 1994.

[22] Constantino Tsallis. *Introduction to nonextensive statistical mechanics: approaching a complex world*, volume 1. Springer, 2009.

[23] William E Wood and Robert D Poodiack. Squigonometry: Trigonometry in the p-norm. *A Project-Based Guide to Undergraduate Research in Mathematics: Starting and Sustaining Accessible Undergraduate Research*, pages 263–286, 2020.

# 7 Appendix 1

In this section, we derive Proposition 9 with simpler proofs than those in Section 4.3, based on Assumption 1.

**Proposition 13.** Let Dist be a distribution over $[0, \infty)$. $[\text{Dist}]_{1...2}$ is continuous $p$-isotropic if and only if $f(0) > 0$ and the function $h(x) = f(x)/f(0)$ defined on $x \in [0, \infty)$ is Lebesgue measurable and satisfies

$$h(x)h(y) = h(\|(x, y)\|_p) \quad \forall x, y \ge 0. \qquad (8)$$

By symmetry, this holds if and only if $\pm \text{Dist}$ is $p$-isotropic.

*Proof.* (⇒) From Equation (5), we have $f(0)f(x) = g(|x|)$ for all $x \in \mathbb{R}$, so $g$ and $f$ are the same function with a different scale. Forcibly, $f(0) > 0$, since otherwise, we would have $f(x) = g(|x|) = g(\|(x, 0)\|_p) = f(|x|)f(0) = 0$ for all $x \in \mathbb{R}$, and $f$ would not be a density function. So $h(x) := f(x)/f(0)$ with domain $x \in [0, \infty)$ is well defined. $h$ inherits Lebesgue measurability from $f$ and it holds that $h(x)h(y) = f(x)f(y)/f(0)^2 = g(\|(x, y)\|_p)/f(0)^2 = f(\|(x, y)\|_p)f(0)/f(0)^2 = h(\|(x, y)\|_p)$.
[16] George Marsaglia. The x+ y, x/y characterization of the gamma distribution. *Contributions to Probability and Statistics: Essays in Honor of Ingram Olkin*, pages 91–98, 1989.

[17] Khodabin Morteza and Alireza Ahmadabadi. Some properties of generalized gamma distribution. *Mathematical Sciences Quarterly Journal*, 4, 03 2010.

[18] Jeffrey S Rosenthal. *First Look At Rigorous Probability Theory*, A. World Scientific Publishing Company, 2006.

[19] Grant Sanderson. Why $\pi$ is in the normal distribution (beyond integral tricks). Youtube, 2023.

[20] Edney W Stacy. A generalization of the gamma distribution. *The Annals of mathematical statistics*, pages 1187–1192, 1962.

[21] Constantino Tsallis. What are the numbers that experiments provide. *Quimica Nova*, 17(6):468–471, 1994.

[22] Constantino Tsallis. *Introduction to nonextensive statistical mechanics: approaching a complex world*, volume 1. Springer, 2009.

[23] William E Wood and Robert D Poodiack. Squigonometry: Trigonometry in the p-norm. *A Project-Based Guide to Undergraduate Research in Mathematics: Starting and Sustaining Accessible Undergraduate Research*, pages 263–286, 2020.

# 7 Appendix 1

In this section, we derive Proposition 9 with simpler proofs than those in Section 4.3, based on Assumption 1.

**Proposition 13.** Let Dist be a distribution over $[0, \infty)$. $[\text{Dist}]_{1..2}$ is continuous $p$-isotropic if and only if $f(0) > 0$ and the function $h(x) = f(x)/f(0)$ defined on $x \in [0, \infty)$ is Lebesgue measurable and satisfies

$$h(x)h(y) = h(\|(x, y)\|_p) \quad \forall x, y \ge 0. \qquad (8)$$

By symmetry, this holds if and only if $\pm \text{Dist}$ is $p$-isotropic.

*Proof.* (⇒) From Equation (5), we have $f(0)f(x) = g(|x|)$ for all $x \in \mathbb{R}$, so $g$ and $f$ are the same function with a different scale. Forcibly, $f(0) > 0$, since otherwise, we would have $f(x) = g(|x|) = g(\|(x, 0)\|_p) = f(|x|)f(0) = 0$ for all $x \in \mathbb{R}$, and $f$ would not be a density function. So $h(x) := f(x)/f(0)$ with domain $x \in [0, \infty)$ is well defined. $h$ inherits Lebesgue measurability from $f$ and it holds that $h(x)h(y) = f(x)f(y)/f(0)^2 = g(\|(x, y)\|_p)/f(0)^2 = f(\|(x, y)\|_p)f(0)/f(0)^2 = h(\|(x, y)\|_p)$.
(⇔) If $h$ satisfies Equation (8) and is measurable, then so is $f(x) = f(0)h(|x|)$, and we have $f(x)f(y) = h(\|(x, y)\|_p)/f(0)^2$, which is a function of $\|(x, y)\|_p$. $\square$

We will use the functional equation 8 first to introduce iid-isotropy (Definition 3) by showing that p-isotropy with independent marginals does not depend on the number of dimensions (Proposition 14). Then, we will use the solutions of the equation to characterize the iid-isotropic distributions.

**Proposition 14.** Let Dist be a distribution over $[0, \infty)$ or $\mathbb{R}$. The following statements are equivalent:

* $[\text{Dist}]_{1..2}$ is p-isotropic.

* $[\text{Dist}]_{1..d}$ is p-isotropic for some integer $d \ge 2$.

* $[\text{Dist}]_{1..d}$ is p-isotropic for all integers $d \ge 2$.

*Proof.* Fix $p, n \ge 3$ and let the domain be $[0, \infty)$ (the result follows by symmetry for $\mathbb{R}$). We will show that (A) 2-dimensional isotropy implies $n$-dimensional isotropy and (B) $n$-dimensional isotropy implies $(n-1)$-dimensional isotropy, which concludes the proof.

(A) Let $h$ be the function of Proposition 13. Notice that $\|(x, \|(y, z)\|_p)\|_p = \|(x, y, z)\|_p$, so $f(x)f(\|(y, z)\|_p) = f(0)^2h(x)h(\|(y, z)\|_p) = h(\|(x, y, z)\|_p)$. Continuing in this fashion,

$$f(x_1) \cdots f(x_n) = h(x_1) \cdots h(x_n) f(0)^n = h(\|\mathbf{x}\|_p) f(0)^n,$$

which is a function of $\|\mathbf{x}\|_p$.

(B) Now, suppose that $[\text{Dist}]_{1..n}$ is p-isotropic. We will prove that $[\text{Dist}]_{1..n-1}$ is also p-isotropic, which concludes the proof. Assumption 1 at $(x_1, \dots, x_{n-1}, 0)$ implies that $f(x_1) \cdots f(x_{n-1})f(0) = g(\|(x_1, \dots, x_{n-1}, 0)\|_p)f(0)$, which is a function of the norm $\|(x_1, \dots, x_{n-1})\|_p$. $\square$

We will now use the solutions of the functional equation 8 to characterize the iid-isotropic distributions.

**Proposition 15.** If a distribution $[\text{Dist}]_{1..2}$ is $\infty$-isotropic, its marginal density function must satisfy $f(x) = \frac{1}{2b}$ whenever $|x| < b$ and $f(x) = 0$ whenever $|x| > b$, for some $b > 0$.

*Proof.* Equation (8) becomes $h(x)h(y) = h(\max(x, y))$ for $p = \infty$. If $h(y) > 0$ for some $y > 0$, then for all $x < y$ we have $h(x)h(y) = h(y)$, thus $h(x) = 1$. Letting $b \stackrel{\text{def}}{=} \sup\{b \in [0, \infty] : f(b) > 0\}$, the solutions to Equation (8) become either $h(x) = 0$ (if $b = \infty$), $h(x) = 1$ (if $b = 0$) and, if $b \in (0, \infty)$,

$$h(x) = \begin{cases} 0 & \text{if } x > b \\ 1 & \text{if } x < b \\ c & \text{if } x = b \end{cases}$$
(⇔) If $h$ satisfies Equation (8) and is measurable, then so is $f(x) = f(0)h(|x|)$, and we have $f(x)f(y) = h(\|(x, y)\|_p)/f(0)^2$, which is a function of $\|(x, y)\|_p$. $\square$

We will use the functional equation 8 first to introduce iid-isotropy (Definition 3) by showing that p-isotropy with independent marginals does not depend on the number of dimensions (Proposition 14). Then, we will use the solutions of the equation to characterize the iid-isotropic distributions.

**Proposition 14.** Let Dist be a distribution over $[0, \infty)$ or $\mathbb{R}$. The following statements are equivalent:

* $[\text{Dist}]_{1..2}$ is p-isotropic.

* $[\text{Dist}]_{1..d}$ is p-isotropic for some integer $d \ge 2$.

* $[\text{Dist}]_{1..d}$ is p-isotropic for all integers $d \ge 2$.

*Proof.* Fix $p, n \ge 3$ and let the domain be $[0, \infty)$ (the result follows by symmetry for $\mathbb{R}$). We will show that (A) 2-dimensional isotropy implies $n$-dimensional isotropy and (B) $n$-dimensional isotropy implies $(n-1)$-dimensional isotropy, which concludes the proof.

(A) Let $h$ be the function of Proposition 13. Notice that $\|(x, \|(y, z)\|_p)\|_p = \|(x, y, z)\|_p$, so $f(x)f(\|(y, z)\|_p) = f(0)^2h(x)h(\|(y, z)\|_p) = h(\|(x, y, z)\|_p)$. Continuing in this fashion,

$$f(x_1) \cdots f(x_n) = h(x_1) \cdots h(x_n) f(0)^n = h(\|\mathbf{x}\|_p) f(0)^n,$$

which is a function of $\|\mathbf{x}\|_p$.

(B) Now, suppose that $[\text{Dist}]_{1..n}$ is p-isotropic. We will prove that $[\text{Dist}]_{1..n-1}$ is also p-isotropic, which concludes the proof. Assumption 1 at $(x_1, \dots, x_{n-1}, 0)$ implies that $f(x_1) \cdots f(x_{n-1})f(0) = g(\|(x_1, \dots, x_{n-1}, 0)\|_p)f(0)$, which is a function of the norm $\|(x_1, \dots, x_{n-1})\|_p$. $\square$

We will now use the solutions of the functional equation 8 to characterize the iid-isotropic distributions.

**Proposition 15.** If a distribution $[\text{Dist}]_{1..2}$ is $\infty$-isotropic, its marginal density function must satisfy $f(x) = \frac{1}{2b}$ whenever $|x| < b$ and $f(x) = 0$ whenever $|x| > b$, for some $b > 0$.

*Proof.* Equation (8) becomes $h(x)h(y) = h(\max(x, y))$ for $p = \infty$. If $h(y) > 0$ for some $y > 0$, then for all $x < y$ we have $h(x)h(y) = h(y)$, thus $h(x) = 1$. Letting $b \stackrel{\text{def}}{=} \sup\{b \in [0, \infty] : f(b) > 0\}$, the solutions to Equation (8) become either $h(x) = 0$ (if $b = \infty$), $h(x) = 1$ (if $b = 0$) and, if $b \in (0, \infty)$,

$$h(x) = \begin{cases} 0 & \text{if } x > b \\ 1 & \text{if } x < b \\ c & \text{if } x = b \end{cases}$$
for any $c \in [0, \infty)$. For $f$ to be a density function, the form of $h$ must be the last one and $f(0) = \frac{1}{\int_{\mathbb{R}} h(x)dx} = \frac{1}{2b}$ must hold. $\square$

**Proposition 16.** For finite $p$, if a distribution $[\text{Dist}]_{1..2}$ is $p$-isotropic, its marginal density function must be $f(x) = c e^{a|x|^p}$ for some $a < 0$.

*Proof.* Denote by $h_p(x)$ the function $h_p(x) := h(x^{1/p})$, so that $h_p(x^p) = h(x)$. Equation (8) holds if and only if $h_p(x^p)h_p(y^p) = h_p(x^p + y^p)$ for all $x, y \ge 0$, in other words, if and only if $h_p(x)h_p(y) = h_p(x + y)$ for all $x, y \ge 0$. Provided that $h_p$ is measurable (because $h$ and $(x \mapsto x^{1/p})$ are), it was shown in Corollary 1.36a of [12] that the solution must be of the form $h_p(x) = 0$, $h_p(x) = e^{ax}$ with $a \in \mathbb{R}$, or $h_p(x) = \mathbf{1}\{x=0\}$. So, $h(x)$ is either $h(x) = 0$, or $h(x) = e^{ax}$, or $h(x) = \mathbf{1}\{x=0\}$. Of these, only the exponential solution with $a < 0$ produces a density $f$. $\square$
for any $c \in [0, \infty)$. For $f$ to be a density function, the form of $h$ must be the last one and $f(0) = \frac{1}{\int_{\mathbb{R}} h(x)dx} = \frac{1}{2b}$ must hold. $\square$

**Proposition 16.** For finite $p$, if a distribution $[\text{Dist}]_{1..2}$ is $p$-isotropic, its marginal density function must be $f(x) = c e^{a|x|^p}$ for some $a < 0$.

*Proof.* Denote by $h_p(x)$ the function $h_p(x) := h(x^{1/p})$, so that $h_p(x^p) = h(x)$. Equation (8) holds if and only if $h_p(x^p)h_p(y^p) = h_p(x^p + y^p)$ for all $x, y \ge 0$, in other words, if and only if $h_p(x)h_p(y) = h_p(x + y)$ for all $x, y \ge 0$. Provided that $h_p$ is measurable (because $h$ and $(x \mapsto x^{1/p})$ are), it was shown in Corollary 1.36a of [12] that the solution must be of the form $h_p(x) = 0$, $h_p(x) = e^{ax}$ with $a \in \mathbb{R}$, or $h_p(x) = \mathbf{1}\{x=0\}$. So, $h(x)$ is either $h(x) = 0$, or $h(x) = e^{ax}$, or $h(x) = \mathbf{1}\{x=0\}$. Of these, only the exponential solution with $a < 0$ produces a density $f$. $\square$