Exponential tilting of subweibull distributions
Authors: F. William Townes
Venue: Accepted by TMLR ()

Abstract: The class of subweibull distributions has recently been shown to generalize the important properties of subexponential and subgaussian random variables. We describe alternative characterizations of subweibull distributions, illustrate their application to concentration inequalities, and detail the conditions under which their tail behavior is preserved after exponential tilting.

Published in Transactions on Machine Learning Research (09/2025)
Exponential tilting of subweibull distributions
F. William Townes Department of Statistics and Data Science Carnegie Mellon University
foownes@andrees Gandrew. cmnu, eduu
Reviewed on OpenReview: /ttps://opperrvveii net/ forum? Bd=BQBk111
Abstract
The class of subweibull distributions has recently been shown to generalize the important properties of subexponential and subgaussian random variables. We describe alternative characterizations of subweibull distributions, illustrate their application to concentration inoqualitics, and detail the conditions under which their tail behavior is preserved after exponential ial lilting.
1 Introduction
Subexponential and subgaussian distributions are of fundamental importance in the application of high dimensional probability to machine learning (Vershyuin 2018 Wainwright 2019). Recently it has been shown that the subweibull class unifics the subexponential and subgaussian familics, while also incorporating distributions with heavier tails {Vladimirova al.) 2020 Kuchibhotla & Chakraborlty/ 2022). Informally, q-subweibull (q 0) random variable has survival function that decays at least fast exp(-Xz") for some À > 0. For example, the exponential distribution is 1-subweibull and the Gaussian distribution is 2- subweibull. Here, we provide alternative characterizations of the subweibull class and introduce distinction between strietly and broadly subweibull distributions. As an example, the Poisson distribution is shown to be strictly subexponential (q = 1) but not subweibull for any > We demonstrate how subweibull properties can be used to prove Bernstein concentration incqualitics in both heavy and light- settings. Finally, we detail the conditions under which the subweibull property is preserved after exponential tilting.
To motivate this last result, consider the setting of adapting model to target distribution that differs from its training distribution. Maily et. al. (2023) proposed an importance weight strategy based on the assumption that the target distribution is well approximated by an exponential tilt of the training distribution. Our results will clarify particular settings where this approximation breaks down, and when one may transfer tail bounds from the training distribution to the target distribution.
2 Preliminaries
2.1 Laplace-Stieltjes transforms
Definition 2.1. The bilateral Laplace-Stielljes transform (BLT) of a random variable X with distribution
function is
(x = B[exx(-(XX)]]] f exp(-t))dF((
We do not restrict X be nonnegative or to have a density function. In the special case that Lx (t) oc for all in an open interval around [ 0, then X has moment generating function (MGF) which is Mx (t) = E[ex)(tX)] Cx(-)). The BLT can characterize the distribution even if the MGF does not exist.
Lemma 2.0.1. the BLTs of random variables X and Y satisfy (x Cy (t) for all in any nonempty open interval (a, b) RR not necessarily containing zero, then X d Y.

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For proof refer to Mukherjea et al. (2006). A random variable X is considered subexponential iff the MGF exists (Vershyning 2018). If (4)
for all 1 >> 0 (respectively / < 0), X is said to have a heavy left (respectively, right) tail (Nair et al., 2022). If a tail is not heavy it is said to be light. It is well known that the one-sided Laplaco-Stieljfe transform characterizes nonnegative distributions (Feller 1971 Lemma 2.0.1
shows that the BLT characterizes any distribution with at least. one light tail.
2.2 Orlicz norms
An Orliez function db :h:RA R4+ + R2 is nondecreasing, convex function with (0)) 0. Unless explicitly stated below, we further restrict it to be strictly increasing.
Definition 2.2. The OOOriccz norm of a random variable X given by
| inf { t
[X]
0:
E
+
S
where [n]
Since ((xitt is a decreasing function of t, it clear that E [((((X/t) I 1 for all > || norms, Orlicz norms have the following properties.
Homogencity: ||aX]] lly = ||
Subadditivity: |X + Y ||||| + ||
Positive definiteness:
||X] ll 0 implies X = 0 almost surely.
Suppose b R is a constant. If X is a degenerate random variable with Pr(X = b) = 1, homogeneity implies [ibli (1), Finiteness of the v-Orllcz norm is preserved under location-scale transformations. If || < and a, be E R, then
||aX + bllp I |||||||| + |اblا oo
If |1 pg oc, then ||a as well. Here are some examples of Orlicz norms. Let (xx) = x" for p 2 Then ly [||x]] 1/p and X E L" has finite moments up to order iff ||
oc. Here, will focus primarily on the norm derived from the Orliez function (x)(x) E 1, which is convex for 1.
Definition 2.3. The Orlicc norm (g 1) of random variable X is given by
|X inf 1> 0: exp ( X
<
The condition that [Xlle < oo clearly implies that | is subexponential. The following lemma, modified from Pollard 2024 is useful in establishing when O-Oricz norm is finite.
Lemma 2.0.2 Let co, Ko be constants and va an Orlicz function. The following are equivalent.
(a)
X
1b
<
Ko
(b)
|| | co max max(Ko.))
2

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3 Subweibull random variables
Definition 3.1. A random variable X is g-subweibull if (xxxp(("XX]))) < for some À 0. X is strictly g-subweibull if the condition is satisfied for all X O. If X is q-subweibull but not strictly so, refer to it as broadly q-subweibull.
The first part of this dcfinition also proposed by Kuchibhotla & Chakrabortty 2022 and by Vladimirova et al. 2020p using parameterization equivalent to 1/g. Clearly X is (striclly) q-subweibull if and only if X is (strictly) subexponential. As an example, the Laplace distribution is broadly 1-subweibull (ie broadly subexponential).
Definition 3.2. The radius of convergence of q-subweibull random variable X is defined by
Rg = sup (A 0 [lexp(A*|x(9) < oo)
and if no such À > 0 exists we adopt the convention that Rg = 0.
In the case of strictly g-subweibull distributions, Rg 00. X has "heavy tails" (in the sense of Nair et al. 2022 iff it is not subexponential (R1 0).
Lemma 3.0.1 Random variable X with Pr(X <0) {0, 13 q-subweibull and only if the nonnegative [-X X < and B |X 2 are qssubweibull. Let Rans Rgas and Rib denote the radii of convergence for X, A, and B, respectively. Then Ry min Rub).
random variables
Proposition 3.1.
The following are equivalent characterizations of q-subweibull random variable where
q 0.
1. Tail bound:
3 K1a such that t 1220 0,
Pr[[X > S 2exp (- (1/Kia)9)
(b) K1s > 0 such that
lim sup Pr[[X] expp t1000
exp ((t/K1b9)) <0
Growth rate of absolute moments:
(a) 3 K2 > 0 such that V p 2 I,
(E|IP)) 1/p s Kap
(b)
(E[[X))1)) 1/p
lim limsup
< 00
///
p-000
3. MGF of || finite in open interval of zero: 3 K3 > 0 such that
(a) 04 < A 2//0
1 A0Kg/2
E [(exp(X9)X1)11
(6) <
E [cxp(X'[x]1)] < cxp(KXA9)
A similar result (excluding conditions 1b and 2b) was proven by Vladimirova al. 2020 Our proof provides explicit constants, which reveals the connection with the Orlicz norm.
3

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Proposition 3.2 A random variable X is q-subweibull (q 2 1) if and only if the O norm is finite.
Furthermore, when the constants in Proposition 3.1 are related to the norm by the global constant
21 2T(1/g
C = exp
exp
(1)
cq
(eq log 2)
such that
|X
< K2 S 1
C
for K3 E {Kia; K2, K3)
An important special case is
C1 log2
(2)
A quasinorm for q < 1 (heavy tails) is discussed in Kuchibhotla & Chakrabortty 2022). This relationship to the Orlicz norm allows us to restate Proposition 2.7.1 of Vershynin (0018) with explicit constants. Corollary 3.0.1 If X is a subexponential random variable with mean zero, then
E E((xp(AX)] S exp(22XX²)
for all |A| & 1/(2)) where
K = eC1 |
XX| 1l
log? 2
It was shown by Vladimirova et al. 2020 that a q-subweibull distribution is also r-subweibull for all < g. We now show that this also implies it is strictly r-subweibull.
Corollary 3.0.2 If X is q-subweibull then it is strictly r-subweibull for all (0, Corollary 3.0.3. Every bounded random variable is strictly q-subweibull for all q 0.
Proof. If X is bounded then there exists M 2 0 such that |X| < M. Then Elexp(A9 S exp(A9M9) < & for all À > 0 and q > 0.
Corollary 3.0.4. If X is not strictly q-subweibull with 2 1 then it is not r-subweibull for any >
3.1 Subweibull properties of the Poisson distribution
Corollaries 3.0.2 and 3.0.4 suggest a hierarchy of distributions based on the heaviness of the tails. Broadly subweibull distributions, which have finite but nonzero radius of convergence (Rg), serve as "critical points" " in the transition between the strictly r-subweibull regime (r < q), with Rg = co and the not r-subweibull regime (r > q) with Rg = 0. However, the transition from strictly subweibull to not subweibull can be immediate, without passing through the stage of broadly subweibull. Here we provide a simple example: the Poisson tail is lighter than any exponential tail, but heavier than any weibull tail with g > 1.
Proposition 3.3 The Poisson distribution is strictly q-subweibull for but not q-subweibull for any
>>1 1.
4 Concentration inequalities
Subweibull tail bounds can be straightforwardly used to improve the tightness of Bernstein's concentration inequality, which we restale here with explicit constants. Proposition 4.1. If XI,..., Xn are independent, subexponential random variables, then
"hy...,,,,
2 exp
0 1 K
Pr
Σ i=1
X
2 t
<
2K'
2 exp
t K +
2K
where 0 max; (eel /² En |x,,, and K 2cC1 26 log(2))
4

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Published in Transactions on Machine Learning Research (09/2025)
This is standard result (eg, Theorem 2.8.1 of Vershynin (0188) so the proof is omitted. We now show that if the summands have lighter than exponential tails, the bound can be tightened.
Proposition 4.2 Light-Laited Bernsleön inequality If X1, Xn are independent, zero-mean q-subweibull random variables with > 1, then
2 exp
0 OSISK' K
Pr
Σ
X
zt
t K oj
2 eexp
where og = Ei= [ Cg is the global constant from Equation
and 0² K 0 are from Proposition 4.1
For sums of heavy-tailed subweibull distributions (q < 1), the MGF does not exist, but is still possible to produce uniform tail bound.
Proposition 4.3.
Heavy-tailed Bernstein inequality
If X1, XI...., X, are independent, zero-mean q-subweibull random variables with 1, then
ta where Ci is the global constant from Equation and ||| 11 is the subexponential norm of [X,]9
Pr
X
2t t
exp
(CCi/ log E 1
Similar results are also found in Vladimirova et al. 2020); Kuchibhotla & Chakrabortly (0222)
5 Exponential tilting
Definition 5.1. Let X be random variable with distribution function F. If the BLT satisfies Lx(-0) < oa for some + O, then the exponentially tilted distribution is given by
Fo(x) = I exp(Ol) L-(-0) dF((t
We adopt the convention of using - instead of so that the interpretation the tilting parameter is consistent with other works that assume X has an MGF, which case one could equivalently require Mx (0) oc.
From the Radon-Nikodynn theorem, Fe is absolutely continuous with respect to F. Since the density function 80 ( is also strictly positive, exponential tilting does not change the support. Generally speaking it is possible to produce subexponential distribution by exponential tilting of any distribution with at least one light lail.
Proposition 5. If X ~ F is a random variable having at least one light tail then exponential tilting is possible for all some open interval (-S,T) with S,T 2 0 and S+T > 0. The resulting tilted distribution Fo subexponential with MGF Mz(t) Lx (-6
finite for all E (-S 0,7 0).
As an example, X F nonnegative, heavy tailed random variable (T -0. its left tail strictly subexponential (S oo) so exponential tilting is possible for all 0. By Proposition 5.1 the resulting vilted distribution is subexponential and hence has lighter tails than the original distribution. On the other hand, if X is broadly subexponential, exponential tilting producos another broadly subexponential distribution, with a shifted interval of convergence.
While exponential tilting can alter the tail behavior of heavy tailed and broadly subexponential distributions, it does not affect the tail behavior of q-subweibull distributions with lighter than exponential tails (i.e., q > 1). Lemma 5.0.1 Prescrvation of nonnegative subwcibull tails under exponential tilting. Let 0 be any real number. If X is nonnegative and q-subweibull (q 1), then the exponentially tilted variable Z Fg is also nonnegative and qssubweibull with the same radius of convergence.

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1. B[exp(X"X*)] oo for all À Rg) implies B/exp(X9Z*)] < co for all E Ryy.
2. E[exp(A9X9)] = 00 for all À > Rg implies Elexp(A929) = x for all X > Rg.
We now exlend Lemma 5.0.1 to general random variables.
Theorem 5.1. Preservation of subweibull tails under exponential (ilting. Let be any real number.
1. If X F is q-subweibull (g>> 1) with radius of convergence Rg then the exponentially tilted variable Z ~ Fg is also q-subweibull and has the same radius of convergence.
If X F strictly q-subweibull (g 2 1), the exponentially tilted variable z Fg is also strictly q-subweibull.
3. If X F is not q-subweibull (q > 1), then Z ~ Fg is also not q-subweibull.
5.1 Application to domain adaptation
Consider a classification problem where labeled training data are drawn from distribution and the goal is to make accurate predictions on a different target distribution Q. We are given unlabeled samples from Q. Maity et al. (0233 proposed exponential tilt model to facilitate importance weighting:
qx,, Y k) exp (0[T(z) + ax p(z, Y = k))
where ob, ag are the tilting parameters. It can be straightforwardly shown that
ag = - log Ex-P
= log (0s))
For simplicity we consider univariate T(X). If its distribution has heavy tails on both sides, the MGF is not finite and exponential tilting is not possible. In practice, the method of [aaty et al. (2023) relies on minimizing a discrepancy such as KL divergence between P and Q using samples. If the distribution of T(X) is heavy tailed, it may not be possible to consistently estimate the tilting parameters. If T(X) has heavy lail on only one side, then the importance weights may still be validly estimated, but the tilting parameters involved must be constrained. On the other hand, if T'x) follows a q-subweibull distribution with 9 > 1, then any tail bounds available for T(X) may be readily transferred to the target distribution Q using Theorem 5.1
6 Discussion
The theory of subexponential and subgaussian distributions is key prerequisite to many results in theoret- ical machine learning and nonasymptotic statistics. That the important subexponential properties can be generalized to the broader subweibull class has been established by Vladimirova et al. (2020) and Kuchib- hotla. & Chakrabortty (2022). Our work differs from these in several ways: we provide explicit constants without requiring quasinorms, distinguish between strictly and broadly subweibull distributions, and address exponential tilting, which has not been previously examined to our knowledge. Exponential tilting is used in a variety of statil areas such as causal inference (MeClean et al. 20244) and Monte Carlo sampling (Fuh natural exponential family with parameter 0. The expo-
& Wang. 2.24). If Fo is a tiltod distribution, nential families are building blocks for generalized linear models (McCullagh & Nelder 1989) For another application of exponential tilting machine learning beyond domain adaptation, see Li et al. (023)).
Here, we have provided a brief overview of subwcibull distributions and their Orlicz norms. We showed that the Poisson distribution is strictly 1-subweibull but not q-subweibull for any > 1. We illustrated the application of subweibull properties to prove concentration inequalities. Finally, we detailed the conditions under which the subweibull property is preserved under exponential tilting. Specifically, distribution is subweibull with a lighter than exponential tail, then the tail of the exponentially tilted distribution decays at the same rate.

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Acknowledgments
Thanks to Arun Kumar Kuchibhotla, Sam Power, Patrick Staples, Valerie Ventura, Larry Wasserman, and Matt Werenski for helpful comments and suggestions.
References
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A Proofs for Section 2 (Preliminaries)
Lemma 2.0.2
Let Co, Ko > 0 be constants and 36 an Orlicz function. The following are equivalent.
(a)
[
)
tb
Σ Ko
(b)
| & co max max(Ko:) 13
Proof. (a)
(b): if Ko s 1 the result is immediate. If Ko 1, let E (0, 1).
+ - (1-o)(0) + [X]
E w/
= E
Co/a
X
s E
(1 - a))(())))))))))))))))))))))))a + oxp
cu
< aKo
Sctting a = 1/Ko produces the result.
(b) (a): Tet |1 S t. Then
X
E
<
We need to show for every K1 O, there exists a Cj >> O such that
X
E
<
Ki
For all Ki 2 1 simply choose e₁ I. For K <1,
X
[X|
E
2b (1-K) (1-K))(0) + K1
t/Ki
t
[
$ (1 K1) Ki
< K₁
So we set ej 2 tax(t,t/Ki).
B Proofs for Section 3 (Subweibull random variables)
Lemma 3.0.1 Random variable X with Pr(X <) & (0, 1} is q-subweibull if and only if the nonnegative random variables A = [-x X < 0j and B = [X X 2 0J are q-subweibull. Let Rgas Rga: and Rub denote the radii of convergence for X, A, and B, respectively. Then Ryz = min Rga. Rab..
Proof. Let p = Pr((X < 0) and define nonnegative random variables = |-X x < 0 and B = [X
x 2 0..
E Elexp X
[ (X*I
E (exx X < 0pp (X X 2 (1(-p)
E E'exp =E'esp>>AA))p+ (/ep
The left hand side is finite if and only if both terms on the right hand side are finite. If Ryz is the radius of convergence for X then E[exp(>*|x]]') < oo for all [0, Ryw). Clearly E(exp(>AA9)] < 00 and ([(xp((>* < oo for all then there exists some X > Rye such that Therefore, min Rya, Ryp RGz- the radius of convergence of X, contradiction.
[0, Rgn) also, implying min Rga: R,o 2 Ryz- However, if min Ryq: R,d > R < 00, which by Definition 3.2 means Rgg is not
8

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Proposition 3.1 The following equivalent characterizations of q-subweibull random variable X where
0.
Tail bound:
(a) 3 Kia such that V & 0,
Pr([X] t) s 2xx ( (t/Kia)))
(b) 3 K1 > 0 such that
lim sup Pr(pP > exp ((1/K1b))) < <0 61000
2. Growth rate of absolute moments:
(a) K2 > 0 such that v pp 2 1,
(E 1/p I K2p'1
(b)
(1I 1/p
lim limsup
< 0o
pl/
1-000
3. MGF of | finite open interval of zero: E K3 > 0 such that
(a) 0A
2/ Kg
1
E [(exp(A9XX19]
S 1 A0Kg/2
(b) 1
E [(xx((>9\x]9] S exp(K319)
Proof. (1a) = (1b):
lim sup Pr([X] > 1) exp ((1/11)99) < supP) exp ((1/11)99) 2 < x t-000
120
So we can simply set K1a Kia,
(1b) = (la): Assume
limsup Pr( > t) exp ((t/K1)))) = 10000
Then, for every C K, there exists some T such that for all t> T,
Pr([X] > exp ((t/K1)9) <C
For all t [0, Pr([X] > t) S and exp (1)) < exp ((T/Ki))) K Therefore
sup Pr([X] 1) exp (11/ K1)) max {C, exp ((T/K16))} t20
Let U = max {C, exp ((T)K))))) U < 2 this directly implies (1a) with Kia = Kin- In the case that
U > 2, set
log U 1/9
K1. K1s
Kι8
log 2
Let f(t)= = U exp (- (t(KKK))))) g(t) = 2 exp (- (t/Kia)9), and T = (lo Since f(T*) = (T*) and g(t) a strictly decreasing function, this implics that Pr([X] > t) < S g(t) for E [0, T For t T g(t) 2 f() since K1s K1,, and f(t) 2 Pr([X) > t) by assumption therefore 2 exp (- (t/Kla)") 2 Pr([X} > 1) for all 2 0.

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(la) (2a):
B| P1
Pr([X]P u/du
Pr([X] t)ptt-1dt
2
dt
exp(s)) /0 (Kias Kig g exp(-s)ds
(2p)
2(p/g)K{o ( (
For 2 c> O, the function ((x) bounded above by [())(e/e)(<//e² Therefore, with & 1//
r(p/2) s (/1g
Substituting this into the previous expressions.
E| - (2p/g) (2/0 exp(ppp/
B/ < (2T(//)(e)) D)
1/v
Kialeq) 1/1 110(e4)-1/0711
Let "/ Consider the function f(x) = (ax) which we would like to upper bound.
f(x) = exp
log(ax)
f(x) =
f((x) = fxx
log(ax)
x² (1 (1- (1-log(ar)
x²
x²
("(x) (x)
[[(x) x3
(1 (1- (loo((aa)) f(x)
x² (1 log(az))
x
The global maximum occurs at e/a which is confirmed by checking f'(e/a) 0 and f"(e/a) 0. Therefore a suitable upper bound is (ax) < exp(a/e). Substitute this into the above expression.
(T(L/))ea) (14)-1/9K1op4/
// < exp
eq
Therefore, there exists some K2 S C2Kia such that ||| < Kop as required where
(2(1/g)(eq)// (eg)
C2 exp
exp
eg
Note that C2 for all q.
(2a) = (26):
(E 1/v
[E (E)) 1/P
lim sup
S sup
S K₂ < 00
///a
p//9
10000
p21
(2b) - (2a): Assume
(E(XX))) 1/8
lim smp 8-000
K <0
p-/2
10

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Then for every C>> K, there exists some p" such that for all p > p²,
(I( 1/2
<C
p²/9
The Lp norm is increasing in so for p E [1,p"), (B(IIP)) // s (B((XP"') 'n 1/"
and p²/o 2 1, which
establishes
(BIIP)) 1/2
- max (EIX"')' c}
sup p21
p4/
(2a) (3a): The power series representation of the exponential function produces
20 (A9
8
X X
E [exp(>*| =
p!
p!
p=l
From (2) we have [[] S (pq)² and p! 2 (p/e)p by Stirling approximation.
[exp(>>|XX 9) 1++
(A9K)))
Σ
(A²))eq)*
1
(p/e)p
1 29 A9Kzed
where the last series converges when \9 K2gee
21/9
A K2
K2(26g)1/9
Since (2eq) e² for all o 0, and 1/(1 1-x) x) is increasing for 1, set C3 = e² that A scries convergence and
2// Cg
implies
E (exp((*9X(1)] < X9K
5 Xr(C3K2)9/2
Xo (a
21/0 Therefore there exists some K3 < C3K2 such that if X <
Kg
1
E (eep((>1)X11) T- Kgg2/2
Clearly C3 1 for all q as well.
(3a) (3b): Straightforward application of the numerical inequality 1 20 for 0 S x 1 1/2.
(35) = (la): Set Ci = (log 2)- 1/9 and note that C1 > 1 for all q > 0. Therefore we may choose
(log 2/
1
C₁K3 S
K3
K3
so that ((exp(A(Xx1)) < exp()*Kg) Then,
Pr[[X] > t) Pr (exp(>>|x(3) > exp())t))
< E/eexp(A exp(-((()))
< 2cxp (-(ckks))
So there exists some Kia C2 K3 such that condition (1a) is satisfied desired.
11

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Proposition 3.2. A random variable X is q-subweibull (q 2 1) if and only if the O norm is finite.
Furthermore, when
the constants in Proposition 3.1 are related to the norm by the global constant
21 (2(1/
e²
Cg exp
exp
(1)
eq
(eg log 2)
such that
|
K2 S C1
C
for Kj E {Kia K2, K3}
Proof. Proposition 3.1 (3b) implics finite norm: Let ())()) = cxp(x*) as before. This is convex function for 1. Rearranging terms produces
[X 1/2
]
E
-E [exp((>9)119) - 1 exp(A*K3)
From Lemma 2.0.2 this implies
|x I (1/A) max {1, exp(K3X9) 1}
where 0 < À < 1/K3. Observing that the second term in the max is increasing in À and ranges from 0 to e 1-1
> > 1, while 1/A is decreasing function, the tightest bound in terms of X is achieved when
(xp(K)X -1 = 1 which produces X = (log2) 1/9 KK3. This establishes || I C.K3 < co with C4 = 2) 1/4, -1/4.
Finite norm implies Proposition 3.1 (1a): Let K4 = || (1 + é) for some arbitrarily small e <> 0.
X
Pr([X] 2 1) Pr
exp
exp ((1/Ka)))
K4
((14
(IXI)
$ exp
exp (-(1/Ka)))
K4
S exp ( (- (1)
This shows there exists some K1. S K₁ = | (1 + e) such that the desired condition satisfied.
To show the relationship to the global constant, recall from the proof of Proposition that
C C4 (log 2) 2)
2T(1/q (eq) (1q)-1/9
C2 exp
eg
C3 2²
Set
(T(119)
C = CiC2Ca exp
eq
(eg log 19
Since C1, Cz, C3 1, C2 2 1 also.
We can choose E 5 C C1-11 -1-13 that
K1a S < S CgIl
and
| SGK SCCK S C.C.C.K1. = =Cy
By similar argument,
K2 C2Kia C2011/111 &C,lIXIIv
12

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Published in Transactions on Machine Learning Research (09/2025)
and
|x CiKk S C.CKK2 s C2
Finally,
K3 S C3C2Kla S C, = CgllXX1lv
|| S C11K3 S CgKs. and
Corollary 3.0.1 Tf X is subexponential random variable with mean zero, then
E E(exp(AX)] & exp(2K2X²)
for all | & 1/(2K) where
K= = = = e² log 2
|
Proof. By Propositions 3.1 and 3.2
there some K2 S CillX|le such that B|| I KPpp for all p 2 1
where Ci e/ 1og(2) (Equation
X
|1"K3p" (p/e)²
=1+ (AeK2)²
E[exp(Ax)] = 1 + (0) +
11+)
p!
1- 1lee22
2=
p=2
where the sum converges whenever |te22 < The function 1+ (x]) is bounded above by exp(2x²) whenever |z|| S 1/2, so
Elexp(Ax>] exp (2(AcK2)²) s exp (2)
whenever |JeKa] 112 which satisfied by
| S 2eCi ||
Setting K = |le yields the desired result.
Corollary 3.0.2 If X is q-subweibull then it is strictly r-subweibull for all (0,g).
Proof. If X is q-subweibull, by Proposition 3.1 may assume there exists K 0 such that V 1,
(E[ 1/p S Kp
Let E (0. The MGF of [X| is given by
E P
E [exp((A
1+
p!
pml
Ar KPT (pr)pr/9
11+
(p/e)p
(X' (N pplr/a-1) pptlooo)))))))
1
Apply the root test to the series to determine convergence.
R(p) X"K' X'
Since r < 9. then limp- R(p) 0 and the series converges regardless of the value of X, which shows X is strictly r-subweibull.
13

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Published in Transactions on Machine Learning Research (09/2025)
Corollary 3.0.4 If X is not strictly q-subweibull with 2 then it is not r-subweibull for any q.
Proof. From Proposition 3.1
may assuine À such that
sup Pr > t) exp(A(9) 11000
which implies there is an infinite sequence in such that
lim Pr([X] bn) exp(A()) 00 n-+00
Now let p > 0 and > q. Whenever 2 t (A/p) we have exp(pt") 2 exp(At9). Let {mm} be the infinite subsequence of { excluding the elements less than 1* Clearly Im well. Then
lim Pr([X] > fm) exp(pt,)) 2 lim Pr([X] > fm) exp(Atg,)) 0o +20
m=020
which implies X cannot be r-subweibull.
Proposition 3.3 The Poisson distribution strictly q-subweibull for q S but not q-subweibull for any q > 1.
Proof- Since the Poisson distribution has finite MGF with infinite radius of convergence, it is strictly subexponential and by Corollary 3.0.2 strictly q-subweibull for all qs 1. Let X ~ Poi(u). Without loss of generality assume >> 1 and let
[4] + with L < 1111.
Pr(X > E = p Pr(X n) 1" exp(-M) p)
n² exp(-H) nl((n)
n!!
j=n
Since (t) is increasing for 2 1, we have nn(n) S (t + 1)T +).. Also, [(n) S n for 1. For the p² term, is increasing for μ 2 1 and decreasing for < 1, so n² 2 minga t+1 n²) = μ² min {u, 1}. Combining these we obtain
Pr(X > 2)> 1 ff, 1)e 1¹ w'minfp,1ee 1e min(k4 1 (1 (++1(())**
(1+ 1)T((11+1) (1+1)())(1))
To assess whether the tail follows a subweibull rate decay, choose any > and y > 1, then
lim sup Pr(X > ()exp(A(() 2 min(u, 1le lim (t+ 1) exp(A19)
t0000
=min 1le "M exp lim llggu- log(t (t- + 1) - log - 1 log & + 21
The expression inside brackets is of the form 0o - Do so we rearrange terms and apply L L'Hopital's rule. Define
lim log log(t + 1) log tlog + X19
8-000
log p
log(t + 1)
At
lim (t log t) (+00
lim 1-000
+
log
logt t
t
t blogt
1/(t + 1)
Agt9-1
lim (1 t) 1
lim 0
(0) +
11000
1 log
+ log
>q(g - 1)19-27
lim (f 1+00
lim
00 = 00
1/t
Therefore
lim sup Pr(X exp()
t
Since this holds for all 0, X cannot satisfy Proposition 3.1 and therefore is not q-subweibull for any
q 1.
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Published in Transactions on Machine Learning Research (09/2025)
C Proofs for Section 4 (Concentration inequalities)
Proposition 4.2 Light-tailed Bernstein inequality
If . Xn are independent, zero mean q-subweibull random variables with > 1, then
2 exx
0 0sts K
Pr
Σ i=1
X 26
2k
t 2 K o
2 exx
C7no-lo
where 0g- E. | Cg is the global constant from Equation and 0², K, 0 are from Proposition 4.1
Proof. By Corollary 3.0.2 each X, is subexponential, so the bound for small deviations follows directly
from Proposition 4.1 For large deviations, let S Σ X₁. Since q the Orliez norm exists and by subadditivity,
1/4
X;
Xilllv 3 In
Σ
|
11/900
By Propositions 3.1 and 3.2
there exists some K1a S Cyll, such that
Pr([S] 2 S
- 2exp
Cano-to?
For all t 2 0. In particular, it holds for t Ko² /0.
Proposition 4.3. If Xi...., Xn are independent, zero-mean q-subweibull random variables with q 1, then
Heavy-tailed Bernstein incquality
Pr
Σ X 2t
s exxp
{ (C)) log 2) Σ ||
-t
where C1 is the global constant from Equation and || is the subexponential norm of |X
Proof. Since f(x) = x is concave for x 2 0 and g E (0, 1),
9
X
Σ |xi|
X
With À > 0 this implies
Pr
Σ
X;
Pr
X 49
<
E (exp [X]X,19]] exp ( -
If X; is q-subweibull, then |X,1 is subexponential. By Proposition 3.1 (3b), for each X; there some
K3 I 1,1%%,, such that
Elexp()|Xx11) exp()aiA)
if X < 1/K3. Therefore set 0g max; X ( 3. and restrict A < 1/(C) Plugging this back into the previous expression,
Pr
X; 2 S eexp Σ K3d x X19 s sexp (A(C1a,-1)) -())
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Published in Transactions on Machine Learning Research (09/2025)
||| The bound is decreasing in À when t t t" where 04 Ei=
(Ciog)) in which case the
tightest bound occurs with X = 1/(C)0g)- This leads to
1 X 2 exp (110g (G), (Cio, -10) exp
o ta
0g
010g
Referring to Proposition 3.1 (1b), we can see that
(
19 cxp exp ((t/Ku))) C10g )
lim sup exp t-
0g
with K1 (4, Therefore the sum of q-subweibull random variables is also q-subweibull. To get a bound over all 2 0, define
f(t) Tg (t/K16)
84
g(() = log(2) (1/Kia)"
Since 0g/0g 2 >>> log(2), can choose Kia > Kis such that f(t) 2 s(t) 2 for t i t and g(t) 2 f()) for 2 t where f(t*) g(t") O.
1/9
t² =
0g
K1O = (logg) (log2)1'911
0
therefore
Kia
og
1/9
(C,0,)2/9
Ciogg log 2
1/9
Og log
D Proofs for Section 5 (Exponential tilting)
If ~ F is random variable having at least one light tail then exponential tilting is possible for all in some open interval (-5, with S, T 2 0 and S+T> 0. The resulting tilted distribution
Proposition 5.1
Fg subexponential with MGF Mz(1) = ()
finite for all E (-S - 0,T O-)
Proof. Without loss of generality assume the right tail is light so Lx(-0) < oo for some > 0. For all #' (),
Cx(-0)) Blexp(0' X)] s Elexp(0XX)] 00
Sct T = sup Cx(-0) < oo) > O. X has a heavy left tail then Lx(-0) = co for all < 0, so the
interval of convergence is (-S.T) with = 0. If X has a light left tail then we can set S = iinf
(-(-0)= < oo) > This establishes the interval (-S,7) with S, T and T 0. Let Z Fg follow the tilted distribution with (-S,T). Its BLT is
Lz(() = =E(exp( -tz)] = 1 exp(-tz)dFF(3) = / 00 exp( -tx) exp(Ox)
(-(- dF(())
E[exp(-(C 0)x))/[x(-00) - -
Lx))
This finite when f-t (-S,T) or cquivalently E + 0 +). Since f E (-S,T), the interval of convergence for (z) open interval containing zero, which proves is subexponential and has the MGF
+
Mz(t) = -x(-0-t)((-
which is finite on the interval (-S 8, -).
Lemma 5.0.1 Prescrvation of nonnegative subwcibull tails under exponential tilting. Let 0 be any real number. If X is nonnegative and q-subweibull (q 1), then the exponentially tilted variable Z Fg is also nonnegative and qssubweibull with the same radius of convergence.
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1. =[exp(X"X*)] oo for all À Rg) implies B[exp(X9Z*)] < for all À E Ry).
X]xxp(A"X*)] do for all Rg implics E(cxp(AV29) 30 for all X Rg.
Proof. If X is q-subweibull with q 1 then by Corollary 3.0.2 it is strictly subexponential and Lx Ex(-)) 00 for all 0 E R. Let Pg. The MGF of 2 is
E/cxp(A*Z9)]
Lx(-0)
(1) case of Rg- T & 0 then
f exp(\9x*+00)ddFG)) (x)dF(x) & exp(> + (x)F(x) = Ejexp(A4X9)] < oc
If 0 > 0, choose E (A, Rg) and define
x²
pa X9
Then for x > xx, 1944 + Ox S p44. Therefore,
/ exp(X²> +z)dd((x)
// exp(A9x SpPA2**00=ddF(22 +z))dF(z) [ exp(> (z))
exp (Oz" +X'(=*)) cxp(p'>9)dF(x))
$ eexp (Ox* PT(x / exp(p)>")dF(x))
< 00
(2) case of A Rg- Tf 0 2 0 then
f exp(\"x++0))dF()>>> Ox)dF(x) expp(Aee +))F(() = Ejexp(A = oc
Tf 0 < 0. Choose p E (Rg, A) and define
-9
x²
19 69
Then for x > xx, 1949 + Or Ar9. Therefore,
/ expl\>x' 0r)dF(()= / exp(A +z))dF(s)+ [ exp(X">" "xp(X"*++Oe)ddF (x)
exp(A9a + (z)dF((x) [ exp(p'x")dF(z)
The first term is finite. We will show the second term is infinite. By assumption,
cxp(d'=9)dF(x)) o
exp(o&>9)dF(x) + l- exp(p&(9)dF(1)
But.
1 (xp(p2>))dF(z)) & exp (p'(x)))) Pr(x S 5x¹) x
17

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Published in Transactions on Machine Learning Research (09/2025)
Therefore
(x)()))))dF(s) 00
implying
/ exp(>>>0 + (x)dF(() = 00
well.
Theorem 5.1. Preservation of subweibull tails under exponential tilting. Let 0 be any real number.
X F is q-subweibull (q > 1) with radius of convergence Rg. then the exponentially tilted variable Z Fe is also q-subwebbull and has the same radius of convergence.
2. If X ~ F is strictly q-subweibull (g 2 1), the exponentially tilted variable Z ~ Fø is also strictly q-subweibull
3. If X ~ F is not q-subweibull (q > 1), then Z ~ Fe is also not q-subweibull.
Proof. (1) By Corollary 3.0.2 X is strictly subexponential so Lx(-0) oc for all A E R. Choose any arbitrary 8 and Mj Ex (-0). Define nonnegative random variables A [-X X < 0] and B [X] X 2 0] with distributions F- and F+, respectively. By Lemma 3.0.1 both and B are q-subweibull and strictly subexponential. Let Rga and Ryb be the radii of convergence of A and B, respectively. Let Pr(X < 0) and assume Φ {0.1) 1} (otherwise simply apply Lemma 5.0.1 to X or -X). Note that
Mi Lx(0)p fn(-0)(1 p)
(3)
We have
exp fz
11 [(exp(X91219) =
exp exp(((|2)))dF((2) = exp(A'|x1))
Mi dF())
E[e (2 X + [xx]
Mi
P A 0A) + (1 p)
Eleep(A> +1488++00 +0B)
M,
M1
PLA(0)
exp( Oz)
exp(A?
(A O.F (I)...
Mi
(1
L)L
1 cxp(X²29) exp(0xx)
dF+
Lx(-0)
CB(-8)
= (p) exp(A²= ddl (2) + (1 --)) p) exp(X"=")dFīt (z)
= pElcxp(A9U9) + (1 P) p|E
where (sce Equation 3. (O) so that E 1). The nonnegative random variable U is distributed as "[-o) which is the exponentially tilting of A ~ F by -0 and V ~ Fst is similarly defined as the exponential tilting of B F+. By Lemma 5.0.1 this implies U and V are q-subweibull with radii of convergence Ryo and Rob, respectively. Let R2s be the radius of convergence of Z. Note that
1 dFo(z)= / exp(Ox) dF(()) Elexp(0X) X 0J Pr(X 0)
Pr(Z 2 0) Elexp((B))[[ -p) Lb( 0) (1- p)
Lx(-0)
Mi
M₁
Mi
=1-8
18

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Published in Transactions on Machine Learning Research (09/2025)
So Pr(Z < 0) p. By Lemma 3.0.1 this implics Z is q-subweibull with radius of convergence min Rya: Ryb): which is also the radius of convergence of X.
(2) For apply Proposition 5.1 with S = oo and 7' = co. For > 1, apply (1) with Rg = oo.
(3) This direct corollary of (1) obtained in the case of Rg = 0.
19


Reviews:
[{"review_id": "1JBT2U68f6", "reviewer": "Reviewer_ghv5", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "1. The paper provides a concise, well‐organized summary of known results on $q$-subweibull distribution: tail‐decay bounds, moment‐growth rates, and Laplace‐transform characterizations (originally due to Vladimirova et al. 2020; Kuchibhotla & Chakrabortty 2022)—and frames them in a unified “strict vs. broad” radius‐of‐convergence perspective.\n\n2. By showing the Poisson law is strictly 1-subweibull but not $q$-subweibull for any $q>1$, the authors give a clean, concrete example of how the subweibull classes transition from “light” to “too heavy.”\n\n3. The main contribution is the result that exponential tilting preserves the $q$-subweibull property (strict or broad) and **exactly** the same radius $R_q$ for all $q>1$.  While tilt-invariance is well known for subgaussian/subexponential cases, this result is extended to a more general class.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Definitions, lemmas, and proofs are concise and well-written, and organized.", "audience_yn": "Yes", "audience_explanation": "Yes, the results would be of interest to people who use tilted measures and concentration inequalities for distributions in the q-subweibull class.", "requested_changes": "* The manuscript in its current form reads more like an advanced coursework exercise than a full research paper. Not sure if it meets TMLR's expectations, but the authors can consider augmenting Section 5 with at least one concrete application of the tilt-invariance result. \n\n* The equivalence with the Orlicz norm is briefly mentioned, but not elaborated upon. Since Orlicz norms are often used in concentration inequalities and learning theory it might help to more clearly explain this connection, especially for readers less familiar with the $q$-subweibull literature.", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "sMD5xHfusz", "reviewer": "Reviewer_q7uH", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper surveys the class of q-sub-Weibull distributions as natural generalizations of sub-exponential and sub-Gaussian distributions. Among others, the authors show some monotonicity properties with respect to q, the parameter determining the tails, and preservation under tilting.", "claims_and_evidence_yn": "Yes", "claims_explanation": "All claims come equipped with proofs, which I have verified to be true.", "audience_yn": "Yes", "audience_explanation": "The so-called q-sub-Weibull distributions have attracted some attention recently in various contexts, especially since they can model some heavier-tailed distributions. I believe a survey on these distributions can be of service to the community.", "requested_changes": "As I indicated above, a survey on q-sub-Weibull distributions could be a useful resource. However, in its current form, I am not sure how to classify the main purpose of the paper. On the one hand, it contains a short list of results on q-sub-Weibull distributions, which seems too lean to be called a survey. On the other hand, I don't completely see the motivation for the specific results that are proven and stated in the paper. Without understanding the purpose of the paper, I will not be able to recommend its acceptance.\n\nI have two main suggestions for the paper, which should be understood as a *broad restructuring of the paper*:\n\n1. If it is indeed intended to be a survey, please survey the wealth of possible results on q-sub-Weibull.  Just as an example (only one among many), one could prove concentration inequalities a la Bernstein, and there is already a big body of relevant literature on Orlicz spaces. Another point would be to discuss applications and appearances in Statistics or other relevant fields.\n\n2. Please expand on the results that already appear in the paper. If the authors had a specific application for proving, say, Theorem 4.1, then that application should appear, or at least be described in the paper. The same thing is also true for the results appearing in Section 3.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Ei5HQZc7R7", "reviewer": "Reviewer_sxKF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors studied the properties of sub-Weibull distributions, a class of distributions that interpolates between sub-exponential and sub-Gaussian. In particular, exponentially tilting preserves three separate senses of sub-Weibull distributions. The contributions are mainly technical, without direct discussions of applications, but I believe this is a great fit for TMLR. This type of technical articles will become greatly useful when one searches for the exact type of concentration results needed for the application, and therefore deserve to be published.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors provided clear and concise proof of the main results in the appendix.", "audience_yn": "Yes", "audience_explanation": "Concentration inequalities in general are ubiquitous in ML theory, and this particularly nice class of interpolation between sub-exponential and sub-Gaussian is very interesting on its own. Certainly any additional properties of exponential tilting will prove to be interesting as well.", "requested_changes": "N/A", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)