# Hölder-Type Inequalities for Norms of  Wick Products

Authors: Alberto Lanconelli; Aurel I. Stan
Journal: Journal of Applied Mathematics and Stochastic Analysis
 (2008)

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## Abstract

Various upper bounds for theL2-norm of the Wick product of two
measurable functions of a random variable X, having finite moments
of any order, together with a universal minimal condition, are proven.
The inequalities involve the second quantization operator of a constant
times the identity operator. Some conditions ensuring that the constants
involved in the second quantization operators are optimal, and interesting
examples satisfying these conditions are also included.


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## Body

## 1. Introduction

It was proven in [1] that for any positive numbers p and q,
such that (1/p)+(1/q)=1,
any normally distributed random variable X,
and any f and g complex-valued Borel measurable functions,
such that both random variables Γ(pI)f(X) and Γ(qI)(X) are square integrable, the Wick product f(X)⋄g(X) is square integrable and the following
inequality holds:(1.1)E[|f(X)⋄g(X)|2]≤E[|Γ(pI)f(X)|2]E[|Γ(qI)g(X)|2].
Here Γ denotes the second quantization operator and I the identity operator of the one-dimensional
Hilbert space ℂX.
The authors' motivation was to find a Hausdorff-Young-type inequality for the
theory of Bosonian Fock spaces and they believed that (1.1) was indeed an inequality
of this type, based on their feeling that the Wick product is an analogue
concept of the convolution product from the theory of Fourier transform. After
discussing with other mathematicians and thinking more about it, they have
become convinced that the Wick product is in fact a simpler product, playing
for the theory of Bosonian Fock spaces a role similar to the classic product of
two series. Together this reconsideration and the condition (1/p)+(1/q)=1 strongly suggest that (1.1) is in fact a
Hölder-type inequality for the theory of Gaussian Hilbert spaces (Bosonian Fock
spaces).We will
generalize inequality (1.1) to other types of random variables X,
and in some cases find the optimal constants p and q.
Moreover, we will prove that no matter how we choose a nonconstant random
variable X,
having finite moments of any order, the condition (1/p)+(1/q)=1 cannot be improved. In Section 2, we present a
minimal background about the Szegö-Jacobi parameters of a random variable
having finite moments of any order. We define a set of basic properties and
prove some connections between these properties in Section 3.
Section 4 is
dedicated completely to proving a fundamental necessary condition that we call
the universal minimal (unimprovable) condition. The main inequalities of the
paper are proven in Section 5. Finally, in Section 6, we provide many examples
in support of the results proven in the previous section. Some of these
examples demonstrate that the estimates from Section 5 are optimal.


## 2. Background

Let(Ω, ℱ, P) be a probability space and X:Ω→ℝ a random variable having finite moments of all
orders. That means, for all p>0, E[|X|p]<∞,
where E denotes the expectation with respect to P.
Since X has finite moments of all orders, all the
terms of the sequence, 1,X,X2,…,
are square integrable, and thus we can apply the Gram-Schmidt
orthogonalization procedure to obtain a sequence of orthogonal polynomial
random variables f0(X)=1, f1(X), f2(X),….
The inner product that we are using is 〈f(X),g(X)〉:=E[f(X)g(X)¯] for all f, g:ℝ→ℂ measurable, such that E[|f(X)|2]<∞ and E[|g(X)|2]<∞.
Also, f0,f1,f2,…  are polynomial functions chosen, such that for
all n≥0,
if fn is not the null polynomial, then fn has the degree equal to n and a leading coefficient of 1.
In fact, if X is a discrete random variable taking on only k different values a1,a2,…,ak with positive probabilities, then fn is the null polynomial for all n≥k.
If X is not a discrete random variable, or X is a discrete random variable taking on a
countable set of values with positive probabilities, then for all n≥0, fn is a polynomial of degree n with a leading coefficient equal to 1.It is well
known that there exist two sequences of real numbers{αn}n≥0 and {ωn}n≥1,
such that for all n≥0,(2.1)Xfn(X)=fn+1(X)+αnfn(X)+ωnfn−1(X).When n=0, fn−1=f−1:=0 (the null polynomial) and we can choose ω0:=0.
Also, if X is a discrete random variable taking on only k different values with positive probabilities,
then for n=k−1,
the equality (2.1) must be understood in the almost-sure sense, and we can
choose αn=0 and ωn=0 for all n≥k.
The sequences {αn}n≥0 and {ωn}n≥1
are called the Szegö-Jacobi
parameters of X.
Moreover, ω1,ω2,… are called the principal Szegö-Jacobi parameters of X.
It is well known that for all n≥1, E[fn2(X)]=ω1ω2⋯ωn (see, e.g., [2, 3]).LetN=k if X is discrete and takes on only k values with positive probabilities, and N=∞ otherwise. We define the Hilbert space ℋ:={∑n<Ncnfn(X)∣{cn}n<N⊂ℂ,∑n<Nωn!|cn|2<∞},
where ωn!:=ω1ω2…ωn for all 1≤n<N and ω0!:=1. ℋ is in fact the closure of the space F:={f(X)∣fispolynomial} in L2(Ω, ℱ, P).
For many classic probability measures ℋ=L2(Ω, σ(X), P),
where σ(X) denotes the smallest sub-sigma-algebra of ℱ with respect to which X is measurable. We denote by Fn the space of all random variables of the form f(X),
where f is a polynomial of degree at most n,
and define Gn:=Fn⊖Fn−1,
that is, Gn is the orthogonal complement of Fn−1 into Fn for all n≥0.
For convenience, we define F−1 and G−1 to be the null space. For all n≥0, Gn=ℝfn and Gn is called the homogenous chaos space of ordern generated by X.
We will also call ℋ the chaos space generated by X.For anym,n≥0,
we define the Wick productfm(X)⋄fn(X) of fm(X) and fn(X),
as fm(X)⋄fn(X):=fm+n(X).
Observe that if N=k is finite, then fm(X)⋄fn(X)=0 for all m and n,
such that m+n≥k.
It is not hard to see that fm(X)⋄fn(X) is in fact the projection of the point-wise
product fm(X)fn(X) on the space Gm+n.
We extend now the Wick product by bilinearity, defining formally for all φ=∑n<Ncnfn∈ℋ and ψ=∑n<Ndnfn∈ℋ,(2.2)φ⋄ψ:=∑n<N(∑p+q=ncpdq)fn.
Since it is not
guaranteed that ∑n<Nωn!|∑p+q=ncpdq|2<∞, φ⋄ψ may not belong to ℋ.Definition 2.1.
For
any complex numberc,
define the second quantization operator ofcI,
where I denotes the identity operator of the
one-dimensional Hilbert space ℂX,
spanned by X,
as a densely defined operator on ℋ,
defined by(2.3)Γ(cI)(∑0≤n<Ndnfn(X)):=∑0≤n<Ncndnfn(X),where dn∈ℂ for all 0≤n<N.A random variableφ:=∑0≤n<Ndnfn(X) belongs to the domain of Γ(cI) if and only if ∑0≤n<N(1+|c|2n)|dn|2ωn!<∞.


## 3. Wick-Hölder Property

Definition 3.1.
LetM and t be two fixed positive numbers. Let X be a random variable, having finite moments of
all orders, and let ℋ denote the chaos space generated by X. X
is said to be
(M, t)-Wick-Hölderian, if, for all positive
numbers p and q,
such that (1/p)+(1/q)=1/t,
and for all φ(X)∈ℋ and ψ(X)∈ℋ,
such that Γ(pI)φ(X)∈ℋ and Γ(qI)ψ(X)∈ℋ,
there exists
φ(X)⋄ψ(X)∈ℋ,
and the following inequality holds:(3.1)E[|φ(X)⋄ψ(X)|2]≤ME[|Γ(pI)φ(X)|2]E[|Γ(qI)ψ(X)|2].Since for any
fixφ(X)∈ℋ the function u:[0, ∞)→[0, ∞], u(t)=E[|Γ(tI)φ(X)|2] is non-decreasing, if X is an (M,t)-Wick-Hölderian random variable, then X is also (M,t′)-Wick-Hölderian for all t′>t.
By taking φ=ψ=1, we conclude from (3.1) that M≥1.Definition 3.2.
Lett be a fixed positive number. Let X be a random variable, having finite moments of
all orders. X is said to be
t-Wick-Hölderian if X is (1,t)-Wick-Hölderian.Again, ifX is a t-Wick-Hölderian random variable, then X is also t′-Wick-Hölderian for all t′>t.Definition 3.3.
LetX be a random variable, having finite moments of
all orders. X is said to be
Wick-Hölderian if there exists a positive
number t,
such that X is t-Wick-Hölderian.Proposition 3.4.
IfX is a random variable, having finite moments of
all orders, then the following two conditions are equivalent:
(1)
X is Wick-Hölderian;(2)
there exist two positive numbersM and t,
such that X is (M,t)-Wick-Hölderian.Proof.
(1)⇒(2) This implication is obvious.
(2)⇒(1) Let us assume that X is (M, t)-Wick-Hölderian for some M≥1 and t>0.
Let s:=max(3,3Mt).
Claim 1.
X is s-Wick-Hölderian.
Indeed, letp>0 and q>0,
such that (1/p)+(1/q)=1/s.
Let φ(X)=∑n<Ncnfn(X)∈ℋ,
and ψ(X)=∑n<Ndnfn(X)∈ℋ,
such that Γ(pI)φ(X)∈ℋ and Γ(qI)ψ(X)∈ℋ,
where {cn}n≥0⊂ℂ, {dn}n≥0⊂ℂ,
and {fn}n≥0 represents the sequence of orthogonal
polynomials, having a leading coefficient equal to 1,
generated by X.
Let ∥⋅∥ denote the L2-norm. Let g(X):=φ(X)−c01 and h(X):=ψ(X)−d01.
We have φ(X)=c01+g(X) and ψ(X)=d01+h(X).
Since g(X)⊥1, h(X)⊥1,
and g(X)⋄h(X)⊥1,
where ⊥ denotes the orthogonality relation, applying
the Pythagorean theorem, we obtain(3.2)‖φ(X)⋄ψ(X)‖2=∥[c01+g(X)]⋄[d01+h(X)]∥2=∥c0d01+c0h(X)+d0g(X)+g(X)⋄h(X)∥2=|c0|2|d0|2+∥c0h(X)+d0g(X)+g(X)⋄h(X)∥2≤|c0|2|d0|2+[|c0|∥h(X)∥+|d0|∥g(X)∥+∥g(X)⋄h(X)∥]2≤|c0|2|d0|2+[|c0|∥h(X)∥+|d0|∥g(X)∥+∥g(X)⋄h(X)∥]2+3∥g(X)⋄h(X)∥2..Because (1/p)+(1/q)=1/s≤1/3,
we have 1/p<1/3 and 1/q<1/3.
Thus, p>3 and q>3.
However, p>3 implies 3∥g(X)∥2≤p∥g(X)∥2≤∥Γ(pI)g(X)∥2.
Similarly, we have 3∥h(X)∥2≤∥Γ(qI)h(X)∥2.
Since 1/[p/(3M)]+1/[q/(3M)]=3M/s≤1/t and X is an (M,t)-Wick-Hölderian random variable, we
have(3.3)∥g(X)⋄h(X)∥≤M∥Γ(p/(3M)I)g(X)∥∥Γ(q/(3M)I)h(X)∥.Thus, since 3<9 and M≥1,
we have(3.4)∥φ(X)⋄ψ(X)∥2≤|c0|2|d0|2+3|c0|2∥h(X)∥2+3|d0|2∥g(X)∥2+9∥g(X)⋄h(X)∥2≤|c0|2|d0|2+|c0|2∥Γ(qI)h(X)∥2+|d0|2∥Γ(pI)g(X)∥2+9M2∥Γ(p/(3M)I)g(X)∥2∥Γ(q/(3M)I)h(X)∥2≤|c0|2|d0|2+|c0|2∥Γ(qI)h(X)∥2+|d0|2∥Γ(pI)g(X)∥2+∥Γ(3M⋅p/(3M)I)g(X)∥2∥Γ(3M⋅q/(3M)I)h(X)∥2=[|c0|21+∥Γ(pI)g∥2][|d0|21+∥Γ(qI)h∥2]=∥Γ(pI)φ∥2∥Γ(qI)ψ∥2.Hence, X is s-Wick-Hölderian.Definition 3.5.
Let{Xi}i∈I be a family of random variables, having finite
moments of all orders. The family {Xi}i∈I is said to be
uniformly Wick-Hölderian if there exists
a positive number t0,
such that for all i∈I, Xi is t0-Wick-Hölderian.It follows from the proof of the previous proposition that a
family{Xi}i∈I is uniformly Wick-Hölderian if and only if
there exists a pair (M0,t0) of positive numbers, such that for all i∈I, Xi is (M0,t0)-Wick-Hölderian.From now on, to
make the notation easier, we say that a random variableX is of class (M,t)-W-H, class t-W-H, or class W-H if X is (M,t)-Wick-Hölderian, t-Wick-Hölderian, or Wick-Hölderian,
respectively. We also say that a uniformly Wick-Hölderian family {Xi}i∈I,
of random variables, is of class unif.-W-H.


## 4. A Universal Minimal Condition

In this section we prove a very important condition
about any two corresponding multipliers involved in a Wick product
inequality.Lemma 4.1.
LetX be a random variable, having finite moments of
all orders, such that the support of X contains at least two distinct points (that
means X is not almost surely constant). Let f0=1 and f1 be the first two orthogonal polynomials, with
a leading coefficient equal to 1,
generated by X.
If p and q are two positive numbers, such that for all φ(X), ψ(X)∈ℂf0(X)+ℂf1(X) (i.e., φ and ψ are polynomial functions of degree at most 1), the following inequality
holds:(4.1)E[|φ(X)⋄ψ(X)|2]≤E[|Γ(pI)φ(X)|2]E[|Γ(qI)ψ(X)|2],then
one must have:(4.2)1p+1q≤1.Proof.
The fact that the support ofX contains at least two points guarantees that f1(X)≠0.
Let f2 be next orthogonal polynomial and let ω1 and ω2 be the first two principal Szegö-Jacobi
parameters of X.
We have ω1=∥f1(X)∥2>0 and ∥f2(X)∥2=ω2!=ω1ω2≥0 (it is possible that f2(X)=0,
in which case ω2=0). As before, ∥⋅∥ denotes the L2-norm. Let us apply inequality (4.1) to the
random variables φ(X)=1+cf1(X) and ψ=1+cλf1(X),
where c and λ are arbitrary real numbers, such that c≠0.
Since(4.3)φ(X)⋄ψ(X)=[1+cf1(X)]⋄[1+cλf1(X)]=1+c(1+λ)f1(X)+c2λf2(X),the inequality,(4.4)E[|φ(X)⋄ψ(X)|2]≤E[|Γ(pI)φ(X)|2]E[|Γ(qI)ψ(X)|2],means that(4.5)1+c2(1+λ)2ω1+c4λ2ω1ω2≤[1+pc2ω1][1+qc2λ2ω1]for all c≠0 and all λ∈ℝ.
Subtracting first 1 from both sides of this inequality, and then
dividing both sides of the resulting inequality, by the strictly positive
number c2ω1,
we conclude that the inequality,(4.6)(1+λ)2+c2λ2ω2≤p+qλ2+pqc2λ2ω1,holds for all c≠0 and λ∈ℝ.
Passing to the limit, as c→0,
in the last inequality, we obtain(4.7)(1+λ)2≤p+qλ2for all real numbers λ.
Moving all terms to the right, we conclude that the quadratic
trinomial(4.8)(q−1)λ2−2λ+(p−1),must be nonnegative for all real
values of λ.
Therefore, q must be greater than one, and the discriminant Δ=4−4(p−1)(q−1) must be less than or equal to zero. This is
equivalent to 1≤(p−1)(q−1),
which in turn means p+q≤pq.
Dividing both sides of this inequality by the positive number pq,
we conclude that(4.9)1p+1q≤1.Corollary 4.2.
IfX is a nonconstant random variable of class t-W-H, then t is at least 1.We will call the condition(1/p)+(1/q)≤1 the universal minimal (unimprovable) condition. We will also say that
a nonconstant random variable X of class 1-W-S satisfies the best Wick-Hölder inequality. We know
from [1] that every
Gaussian random variable satisfies the best Wick-Hölder inequality. However,
there are many other random variables of class 1-W-H, and in the next section we will give
some sufficient conditions that guarantee this property.LetX be a random variable, having finite moments of
any order, and let {fn}n≥0 and ℋ be the sequence of orthogonal polynomials and
chaos space generated by X,
respectively. If α:={αn}n≥0 is a sequence of complex numbers, then we
denote by ℳα the densely defined linear operator on ℋ that maps fn(X)→αnfn(X) for all n≥0.
We call the sequence α the multiplier of the operator ℳα.
It is clear that for all n≥0, ℳαGn⊂Gn.
The converse is also true, namely, if T is a linear operator defined on the space of all
polynomial functions of X that leaves all homogenous chaos spaces
generated by X invariant, then there exists a sequence of
complex numbers α,
such that T=ℳα.
If α0=1,
then ℳα1=1,
and we say that the operator ℳα (or the multiplier α) respects the vacuum space G0.
Doing the same proof as in Lemma 4.1, we can prove the following
result.Lemma 4.3.
LetX be a random variable, having finite moments of
all orders, such that the support of X contains at least n+1 distinct points, where n≥1 is fixed. Let f0=1,
and fn be the orthogonal polynomial of degree n,
generated by X.
If c={cn}n≥0 and d={dn}n≥0 are two sequences of complex numbers, such
that c0=d0=1,
and for all φ(X), ψ(X)∈ℂf0(X)+ℂfn(X),
the following inequality holds:(4.10)E[|φ(X)⋄ψ(X)|2]≤E[|ℳcφ(X)|2]E[|ℳdψ(X)|2],then
one must have(4.11)1|cn|2+1|dn|2≤1.We
call inequality (4.11) the generalized
universal minimal (unimprovable) condition. Even though we will not be
using this generalized condition in this paper, we would like to reformulate it
in words, so that some other mathematicians might use it in the future.If the norm of the Wick product ofφ(X) and ψ(X) is always bounded above by the product of the
norms of ℳcφ(X) and ℳdψ(X),
where c and d are two multipliers respecting the vacuum
space, then the sum of the reciprocals of the square of the modulus of any two
corresponding terms of the sequences, c and d,
must be at most 1.We extend now
the universal minimal condition from theL2 case to the Lr case for r≥2.
If we pay attention to the proof of the universal minimal condition, when
dividing by c2 and then passing to the limit as c→0,
then we can observe that, in fact, we were differentiating an inequality twice
with respect to c.
Therefore, we will attack the Lr case in the same manner, based on two very
simple observations.Observation 1.
Iff and g are two functions from ℝ to ℝ that are twice differentiable, such that f(x)≤g(x) for all x∈ℝ,
and there exists an x0∈ℝ,
such that f(x0)=g(x0),
then f′(x0)=g′(x0),
and f′′(x0)≤g′′(x0).Proof.
This can be
seen intuitively by drawing a picture for the graphs off and g,
or of f−g,
and mathematically by using the formula(4.12)f′′(x0)=lim⁡h→0f(x0+h)+f(x0−h)−2f(x0)h2,and a similar relation for g′′(x0).We can
formulate this observation in the following way:we can differentiate twice an inequality
between two functions at the points where the functions are equal (touching
each other), and the inequality is preserved.Observation 2.
Ifr is a real number, then the function h(x)=|x|r is differentiable if and
only if r>1,
in which case h′(x)=rx|x|r−2.We are generalizing now Lemma4.1 to powers r>2.Lemma 4.4.
LetX be a random variable, having finite moments of
all orders, such that the support of X contains at least two distinct points. Let f0=1 and f1 be the first two orthogonal polynomials, with
a leading coefficient equal to 1,
generated by X.
Let r≥2 be fixed. If p and q are two positive numbers, such that for all φ(X), ψ(X)∈ℂf0(X)+ℂf1(X),
the following inequality holds:(4.13)E[|φ(X)⋄ψ(X)|r]≤E[|Γ(pI)φ(X)|r]E[|Γ(qI)ψ(X)|r],then
one must have(4.14)1p+1q≤1.Proof.
Letφc(X):=1+cf1(X) and ψc,λ(X):=1+cλf1(X),
where c and λ are arbitrary real numbers. Let us consider
the functions gλ:ℝ→ℝ, gλ(c):=E[|φc(X)⋄ψc,λ(X)|r] and hλ:ℝ→ℝ, hλ(c):=E[|Γ(pI)φc(X)|r]E[|Γ(qI)ψc,λ(X)|r].
Since gλ(c)≤hλ(c) for all c∈ℝ,
and gλ(0)=hλ(0),
according to Observation 1, we have(4.15)gλ′′(0)≤hλ′′(0).Because X has finite moments of all orders, and r≥2,
a simple application of dominated convergence theorem shows that we can put the
derivatives inside the expectations. Thus, applying Observation 2 twice and
the product rule of differentiation, we get that for all c∈ℝ,(4.16)gλ′′(c)=r(r−1)E[|1+c(1+λ)f1+c2λf2|r−2((1+λ)f1+2cλf2)2]+2rλE[|1+c(1+λ)f1+c2λf2|r−2(1+c(1+λ)f1+c2λf2)f2],hλ′′(c)=r(r−1)pE[|1+cpf1|r−2f12]E[|1+cλqf1|r]+2r2λpqE[|1+cpf1|r−2(1+cpf1)f1]×E[|1+cλqf1|r−2(1+cλqf1)f1]+r(r−1)qλ2E[|1+cλqf1|r−2f12]E[|1+cpf1|r].
Setting c=0,
we get now(4.17)gλ′′(0)=r(r−1)(1+λ)2E[f12]+2rλE[f2],hλ′′(0)=r(r−1)pE[f12]+2r2λpqE[f1]E[f1]+r(r−1)qλ2E[f12].Since f2(X)⊥f0(X), f1(X)⊥f0(X),
and f0(X)=1,
we have E[f2(X)]=0 and E[f1(X)]=0.
Thus, the gλ′′(0)≤hλ′′(0)
becomes, after dividing both sides by the
positive number r(r−1)E[f12](4.18)(1+λ)2≤p+qλ2for all λ∈ℝ.
Now moving all terms to the right and writing the condition that the
discriminant of the quadratic function in λ be nonpositive, we obtain as before
that(4.19)1p+1q≤1.Our technique for proving the universal minimal
condition forr≥2 is similar to the proof of Theorem 3 from
[4].Finally, we can
also see that for multipliers, the inequality(4.20)E[|φ(X)⋄ψ(X)|r]≤E[|ℳcφ(X)|r]E[|ℳdψ(X)|r],for a fix r≥2,
also implies the condition that(4.21)1|cn|2+1|dn|2≤1for all n≥1.


## 5. Random Variables with(M,α)-Subadditive Omega Parameters

We present first the following lemma.Lemma 5.1.
Let{ωn}n≥1 be a sequence of positive numbers, such that
there exist a number t,
greater than or equal to 1,
and a sequence {αn}n≥2,
of nonnegative numbers, such that the series ∑n=2∞αn is convergent, and for all n and k natural numbers, with n>k,
it holds that
(5.1)ωn≤t(1+αn)(ωk+ωn−k).Then, for all nonnegative
numbers k≥r≥0,
it holds that(5.2)(ωkωr)≤Mtk(kr),where M:=max{1,(1/t)∏n=2∞(1+αn)},
(kr):=(1⋅2⋯k)/[(1⋅2⋯r)(1⋅2⋯(k−r))],
and
(ωkωr):=(ω1ω2⋯ωk)/[(ω1ω2⋯ωr)(ω1ω2⋯ωk−r)] for 0<r<k.
If r=0 or r=k,
then (kr):=1 and (ωkωr):=1.Proof.
Sinceαn≥0 for all n≥2 and ∑n=2∞αn<∞,
we conclude that the product ∏n=2∞(1+αn) is convergent. For k=0,
the inequality (5.2) is obvious since (ω0ω0)=(00)=1.
We prove now by induction on k that for all k≥1,
we have(5.3)(ωkωr)≤[∏i=2k(1+αi)]tk−1(kr),for all r∈{0,1,…,k}.
For k=1,
we have nothing to prove, since (ωkωr)=(kr)=1,
and ∏i=2k(1+αi) is defined to be 1 in this case.
Let us suppose now that (5.3) holds for k=n and all r∈{0,1,…,n} and prove that it continues to hold for k=n+1 and all r∈{0,1,…,n+1}.
Indeed, we may assume that 1≤r≤n since for r=0 and r=n+1,
(5.3) is trivial because t≥1.
It follows from (5.2) that(5.4)(ωn+1ωr)=ωn+1ωn!ωrωr−1!ωn+1−rωn−r!≤t(1+αn+1)[ωr+ωn+1−r]ωn!ωrωr−1!ωn+1−rωn−r!=t(1+αn+1)[ωn!ωr−1!ωn+1−r!+ωn!ωr!ωn−r!]=t(1+αn+1)[(ωnωr−1)+(ωnωr)].Using now this inequality, the
induction hypothesis, and the classic Pascal identity (nr−1)+(nr)=(n+1r),
we get(5.5)(ωn+1ωr)≤[∏i=2n+1(1+αi)]tn[(nr−1)+(nr)]=[∏i=2n+1(1+αi)]tn(n+1r).Hence, for all k≥1,
we have(5.6)(ωkωr)≤[∏i=2k(1+αi)]tk−1(kr)≤[∏i=2∞(1+αi)]tk−1(kr)=1t[∏i=2∞(1+αi)]tk(kr).In order to make this inequality
also true, for k=0,
we have to replace (1/t)[∏i=2∞(1+αi)] by max{1,(1/t)[∏i=2∞(1+αi)]}.We introduce
now three definitions.Definition 5.2.
Given a numbert,
greater than or equal to 1,
and a sequence of nonnegative numbers α={αn}n≥2,
such that the series ∑n=2∞αn is convergent, then a sequence of nonnegative
numbers {ωn}n≥1 is said to be
(t, α)-subadditive if the following
inequality,(5.7)ωn≤t(1+αn)(ωk+ωn−k),holds for all natural numbers n and k,
such that n>k.Definition 5.3.
Given a numbert,
greater than or equal to 1,
then a sequence of nonnegative numbers {ωn}n≥1 is said to be
t-subadditive, if it is (t, 0)-subadditive, where 0={0,0,…} denotes the identical zero sequence, that is,
if the following inequality,(5.8)ωm+n≤t(ωm+ωn),holds for all natural numbers m and n.
In particular, if t=1,
then a 1-subadditive sequence is
called simply subadditive.Of course(t,α)-subadditivity implies t′-subadditivity for t′:=t(1+sup{αn∣n≥2}).Definition 5.4.
Ift is a number, greater than or equal to 1,
and {ωn}n≥1 is a sequence of nonnegative numbers, then the
sequence {ωn}n≥1
is said to be
exp-t-subadditive if(5.9)ωm+n1/t≤ωm1/t+ωn1/tfor all m and n positive integers.Observation 3.
An exp-t-subadditive sequence {ωn}n≥1 of nonnegative numbers is also exp-s-subadditive for all s>t.Proof.
Applying the
inequality(a+b)r≤ar+br for all a≥0, b≥0,
and 0<r≤1,
we get(5.10)ωm+n1/s=[ωm+n1/t]t/s≤[ωm1/t+ωn1/t]t/s≤[ωm1/t]t/s+[ωn1/t]t/s=ωm1/s+ωn1/sfor all m and n positive integers.Observation 4.
For allt≥1,
any exp-t-subadditive sequence {ωn}n≥1 of nonnegative numbers is also 2t−1-subadditive.Proof.
Sincet≥1,
the function f:[0, ∞)→ℝ, f(x)=xt is convex. Thus, for all  m,n≥1,
we have(5.11)ωm+n=[ωm+n1/t]t≤[ωm1/t+ωn1/t]t=2t[12ωm1/t+12ωn1/t]t≤2t[12(ωm1/t)t+12(ωn1/t)t]=2t−1(ωm+ωn).We present now the main result of this paper.Theorem 5.5.
LetX be a random variable, having finite moments of
any order, and let {ωn}n≥1 be the principal Szegö-Jacobi parameters of X.
Let M and t be two numbers that are greater than or equal
to 1. If the sequence {ωn}n≥1 satisfies the condition(5.12)(ωkωr)≤Mtk(kr)for all 0≤r≤k<N,
where N denotes the dimension of the chaos space generated
by X,
then X is of class (M,t)-W-H.Proof.
Letp>0 and q>0,
such that (1/p)+(1/q)=(1/t).
Let ℋ be the chaos space generated by X.
Let φ∈ℋ and ψ∈ℋ,
such that Γ(pI)φ∈ℋ and Γ(qI)ψ∈ℋ.
Let {fn}n<N be the orthogonal sequence of polynomials,
with a leading coefficient of 1,
generated by X.
We distinguish between two cases.
Case 1.
IfN=∞,
then there exist two unique sequences of complex numbers {cn}n≥0 and {dn}n≥0,
such that(5.13)φ=∑n=0∞cnfn(X),ψ=∑n=0∞dnfn(X).We have(5.14)φ⋄ψ=∑k=0∞(∑u+v=kcudv)fk(X).Using the triangle inequality,
we obtain(5.15)E[|φ⋄ψ|2]=∑k=0∞|∑u+v=kcudv|2ωk!≤∑k=0∞[∑u+v=k|cudv|]2ωk!=∑k=0∞ωk![∑u+v=k1ωu!ωv!puqvpuωu!|cu|qvωv!|dv|]2From the
Cauchy-Bunyakovsky-Schwarz inequality, inequality (5.2), and Newton's
binomial formula, we obtain(5.16)E[|φ⋄ψ|2]≤∑k=0∞ωk![∑u+v=k1ωu!ωv!puqv]×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=∑k=0∞[∑u+v=kωk!ωu!ωv!1pu1qv]×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=∑k=0∞[∑u+v=k(ωkωu)1pu1qv]×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]≤∑k=0∞[∑u+v=kMtk(ku)1pu1qv]×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=M∑k=0∞tk[∑u+v=k(ku)1pu1qv]×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=M∑k=0∞tk[1p+1q]k×[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=M∑k=0∞tk[1t]k[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=M∑k=0∞1[∑u+v=kpuωu!|cu|2qvωv!|dv|2]=M∑u=0∞puωu!|cu|2∑v=0∞qvωv!|dv|2=ME[|Γ(pI)f|2]E[|Γ(qI)g|2].
Case 2.
IfN<∞,
then all the inequalities used in Case 1, remain the same or become strict
inequalities, due to the fact that after a while, all ω's become zero. Thus, for
example, some complete sums like ∑u+v=k(ku)(1/pu)(1/qv) from Case 1 will become incomplete (that means
some of the pais (u, v), with u+v=k,
will be missing) in Case 2,
and therefore instead of ∑u+v=k(ku)(1/pu)(1/qv)=1/tk,
we will have ∑u+v=k(ku)(1/pu)(1/qv)<1/tk.
Therefore, all the inequalities from Case 1 will also remain true in
Case 2.
Hence, X is of class (M, t)-W-H in this case too.Corollary 5.6.
Lett be a number greater than or equal to 1,
and α={αn}n≥2 a sequence of nonnegative numbers producing a
convergent series ∑n=2∞αn.
Let X be a random variable, having finite moments of
any order, and let {ωn}n≥1 be the principal Szegö-Jacobi parameters of X.
If the sequence {ωn}n≥1 is (t,α)-subadditive, then X is of class (M,t)-W-H, where M:=max{1,(1/t)∏n=2∞(1+αn)}.Corollary 5.7.
IfX is a random variable, having finite moments of
all orders, such that its principal Szegö-Jacobi sequence {ωn}n≥1 is t-subadditive, then X is of class t-W-H.Corollary 5.8.
IfX is a random variable, having finite moments of
all orders, such that its principal Szegö-Jacobi sequence {ωn}n≥1 is exp-t-subadditive, then X is of class 2t−1-W-H. In particular, by taking p=q=λ=2t,
it is  concluded that if ℋ denotes the chaos space generated by X,
and if φ(X), ψ(X)∈ℋ,
such that Γ(2t/2I)φ(X)∈ℋ and Γ(2t/2I)ψ(X)∈ℋ,
then φ(X)⋄ψ(X)∈ℋ and the following inequality
holds:(5.17)E[|φ(X)⋄ψ(X)|2]≤E[|Γ(2t/2I)φ(X)|2]E[|Γ(2t/2I)ψ(X)|2].Observation 5.
Ifp=q=2t,
then to prove that the inequality,
(5.18)E[|φ(X)⋄ψ(X)|2]≤E[|Γ(2tI)φ(X)|2]E[|Γ(2tI)ψ(X)|2],holds whenever both expectations
from the right-hand side are finite, we do not need the condition that each ω-binomial coefficient (ωkωr) is less than or equal to Mtk(kr),
but it is enough to assume that for each k≥0,
the sum of all binomial coefficients, having ωk on the top, is less than or equal to Mtk times the sum of all classic binomial
coefficients having k on the top, that means(5.19)∑r=0k(ωkωr)≤Mtk⋅2kfor all k≥0.Proof.
The proof of this observation follows line by line the proof of Theorem5.5, and it uses the fact that for all pairs (u, v) such that u+v is equal to a
fixed number k since p=q=2t,
the product puqv is always the same (independent of the pair)
and equal to (2t)k.The following
proposition will be useful in showing later thatλ=2t is optimal for some particular random variables X.Proposition 5.9.
LetX be a random variable having finite moments of
all orders. We assume that the probability distribution of X has an infinite support. Let ℋ be the chaos space generated by X and {ωn}n≥1 the principal Szegö-Jacobi sequence of X.
We define Sn:=∑k=0n(ωnωk) for all n≥0,
and λ0:=limsupn→∞Sn1/n.
Then,
(1)
if there exists a positive numberλ such that the inequality,(5.20)E[|φ(X)⋄ψ(X)|2]≤E[|Γ(λI)φ(X)|2]E[|Γ(λI)ψ(X)|2],holds for all φ∈ℋ and ψ∈ℋ,
such that Γ(λI)φ∈ℋ and Γ(λI)ψ∈ℋ,
then λ≥λ0;(2)
ifλ0 satisfies (5.20), then λ0 is optimal (i.e., the smallest among all
positive λ's satisfying this inequality);(3)
ifSn≤λ0n for all n≥0,
then λ0 is optimal.Proof.
(1)  Letλ>0 such that (5.20) holds whenever both
expectations from its right-hand side are finite. For every 0≤r≤k,
let us choose φ:=fr(X) and ψ:=fk−r(X).
The inequality E[|φ⋄ψ|2]≤E[|Γ(λI)φ|2]E[|Γ(λI)ψ|2] is equivalent to ωk!≤λrωr!λk−rωk−r!.
This implies that
(ωkωr)≤λk.
Summing up from r=0 to r=k,
we obtain Sk≤(k+1)λk for all k≥0.
Taking the kth root in both sides of this inequality and
then passing to the superior limit as k→∞,
we get limsupk→∞Sk≤λ.
Thus, λ0≤λ.
(2)
It follows
from 1.(3)
It follows
from lettingM:=1 and t:=λ0/2 in Observation 5.


## 6. Examples

Example 6.1.
Lets≥1 be a fixed real number and let X be a random variable, having finite moments of
all orders, and the principal Szegö-Jacobi parameters ωn=ns>0 for all n≥1.
It is clear that {ωn}n≥1 is an exp-s-subadditive (in fact exp-s-additive) sequence. Therefore, {ωn}n≥1 is also 2s−1-subadditive. Thus, X is of class 2s−1-W-H. Moreover, the following lemma
holds.Lemma 6.2.
For alls≥1,
it holds that(6.1)[∑k=0n(nk)s]1/n≤2s,(6.2)lim⁡n→∞[∑k=0n(nk)s]1/n=2s.Formula (6.2) holds even for 0<s<1.Proof.
Ifs≥1,
then we have(6.3)[∑k=0n(nk)s]1/n≤{[∑k=0n(nk)]s}1/n=[2n]s/n=2s.
Since s≥1,
the function h:(0, ∞)→ℝ, h(x)=xs is convex, and thus(6.4)[∑k=0n(nk)s]1/n=n1/n[∑k=0n1n(nk)s]1/n≥n1/n{[∑k=0n1n(nk)]s}1/n=n1/n[2nn]s/n=1(nn)s−12s.
Hence,(6.5)1(nn)s−12s≤[∑k=0n(nk)s]1/n≤2s.Since nn→1,
as n→∞,
we conclude that limn→∞[∑k=0n(nk)s]1/n exists and is equal to 2s.
If0<s<1,
then in a similar way, we can show that(6.6)1(nn)s−12s≥[∑k=0n(nk)s]1/n≥2s,and it follows again that limn→∞[∑k=0n(nk)s]1/n=2s.From formula (6.2), inequality
(6.1), Corollary
5.8, and Proposition 5.9, we obtain the following
proposition.Proposition 6.3.
LetX be a random variable, having finite moments of
any order whose principal Szegö-Jacobi parameters are ωn=ns for all n≥1,
where s is a fixed real number, such that s≥1.
Let ℋ be the chaos space generated by X.
(1)
Ifp and q are positive numbers, such that (1/p)+(1/q)≤(1/2s−1),
then for all φ∈ℋ and ψ∈ℋ,
such that Γ(pI)φ∈ℋ and Γ(qI)ψ∈ℋ,
there exists
f(X)⋄g(X)∈ℋ and the following inequality
holds:(6.7)E[|φ⋄ψ|2]≤E[|Γ(pI)φ|2]E[|Γ(qI)ψ|2].(2)
λ0:=2s is the smallest among all positive numbers λ,
for which the inequality,(6.8)E[|φ⋄ψ|2]≤E[|Γ(λI)φ|2]E[|Γ(λI)ψ|2],holds for all φ∈ℋ and ψ∈ℋ,
such that Γ(λI)φ∈ℋ and Γ(λI)ψ∈ℋ.Example 6.4.
This is a modification of the previous example. Lets be a fixed real number, such that s>2.
Let Xr be a random variable whose principal
Szegö-Jacobi parameters are ωn=ns+rn for all n≥0,
where r>−1.
The condition r>−1 follows from the inequality ω1>0,
and ensures that ωn>0 for all n≥2.
The reason why we have chosen s>2 will become more transparent later.Claim 1.
Ifr≥0,
then the sequence {ωn}n≥1 is 2s−1-subadditive.Indeed, for allm and n natural numbers, we have(6.9)(m+n)s≤2s−1(ms+ns),m+n<2s−1(m+n).Multiplying the second
inequality by r and adding the resulting inequality to the
first one, we get(6.10)ωm+n≤2s−1(ωm+ωn).Claim 2.
If−1<r<0,
then the sequence {ωn}n≥1 is not 2s−1-subadditive.Indeed, if we
definer′:=−r>0,
then for all n≥1,
we have(6.11)ω2n=(2n)s−2nr′=2s−1(ns−r′n+ns−r′n)+2n(2s−1−1)r′=2s−1(ωn+ωn)+2n(2s−1−1)r′>2s−1(ωn+ωn).Claim 3.
If−1<r<0,
then there exists a sequence α={αn}n≥2 of positive numbers, such that ∑n=2∞αn is convergent, and the sequence {ωn}n≥1 is (2s−1, α)-subadditive.As before, letr′=−r∈(0, 1).
Let n≥2 be a fixed natural number. Let us find a
positive number αn,
such that(6.12)ωn≤(1+αn)2s−1(ωk+ωn−k)for all 1≤k<n.
This inequality is equivalent to(6.13)ns−r′n≤(1+αn)2s−1[ks+(n−k)s]−(1+αn)2s−1r′nfor all 1≤k<n.
Since ns≤2s−1[ks+(n−k)s],
if we choose αn such that(6.14)ns−r′n=(1+αn)ns−(1+αn)2s−1r′n,then the last inequality holds,
for all 1≤k<n.
In fact, for n even, if we choose k=n−k (that means k=n/2), then we can see that αn cannot be chosen smaller than the value of the
solution of  (6.14). Solving
(6.14)  for αn,
we get(6.15)αn=r′(2s−1−1)ns−1−2s−1r′.Since n≥2 and 0<r′<1,
we can see that αn>0 for all n≥2.
Moreover, since we have chosen s>2,
the series ∑n=2∞αn has the same nature as the series ∑n=2∞1/ns−1,
which is convergent.Therefore,
according to Theorem 5.8 forr≥0, Xr is of class 2t−1-W-H, while for −1<r<0, Xr is of class (M,2t−1)-W-H for some M≥1 (M depends on r). Moreover, the family {Xr}r>−1 is not uniformly Wick-Hölderian (that means
we cannot find the same M for all r>−1). This follows from the observation that limr→(−1)+ω1=0,
while limr→(−1)+ω2=2s−2>0.
If we assume the existence of two positive numbers p and q,
such that the inequality: E[|f1(Xr)⋄f1(Xr)|2]≤E[|Γ(pI)f1(Xr)|2]E[|Γ(qI)f1(Xr)|2] holds for all r>−1 since f1(Xr)⋄f1(Xr)=f2(Xr),
where f1 and f2,
are the orthogonal polynomials of degree 1 and 2, respectively, generated by Xr,
then we would conclude that(6.16)ω2!≤pω1!⋅qω1!.This inequality reduces to ω2≤pqω1 for all r>−1,
which is impossible since the left-hand side converges to a positive number,
while the right-hand side tends to zero as r goes to −1.Example 6.5.
Let us take nows=2,
and consider the family of random variables Xr whose principal Szegö-Jacobi parameters are ωn:=n2+rn for all n≥1,
where r>−1.
If we take the other Szegö-Jacobi parameters to be αn:=αn for all n≥0,
where α is a fixed real number, then we can see that {Xr}r>−1 are exactly the centered and rescaled Meixner
random variables.We can see
exactly as in Example6.5 that for r≥0, {ωn}n≥1 is 22−1-subadditive, while for −1<r<0,
it is not 2-suadditive. If r′:=−r,
unfortunately, for αn=r′(22−1−1)/(n2−1−22−1r′)=r′/(n−2r′),
the series ∑n=2∞αn is not convergent. Moreover, as we saw in the
previous example, since (6.14) cannot be avoided for n even, we can see that the sequence {ωn}n≥1 is not (2, β)-subadditive, for any nonnegative sequence β={βn}n≥2,
such that ∑n=2∞βn<∞.Claim 1.
If−1/2≤r<0,
then for all n≥k≥0,
we have(6.17)(ωnωk)≤2n(nk).Since (ωnωk)=(ωnωn−k) and (nk)=(nn−k),
we may assume that k≤n/2.We fixk≥1,
and prove by induction on n that for all n≥2k, (ωnωk)≤2n(nk).Let us prove
first this inequality forn=2k.
Since ωm=m(m−r′) for all m≥1,
we have(6.18)(ω2kωk)=(2kk)[2k−r′k−r′⋅2k−1−r′k−r′][2k−2−r′k−1−r′⋅2k−3−r′k−1−r′]⋯×[2−r′1−r′⋅1−r′1−r′]=(2kk)[(2+r′k−r′)(2−1−r′k−r′)]×[(2+r′k−1−r′)(2−1−r′k−1−r′)]⋯×[(2+r′1−r′)(2−1−r′1−r′)]=(2kk)[4+22r′−1k−r′−r′(1−r′)(k−r′)2]×[4+22r′−1k−1−r′−r′(1−r′)(k−1−r′)2]⋯×[4+22r′−11−r′−r′(1−r′)(1−r′)2].If −1/2≤r<0,
since r′=−r,
we have 0<r′≤1/2,
and thus 2r′−1≤0.
Hence,(6.19)4+22r′−1j−r′−r′(1−r′)(j−r′)2<4,for all j∈{1,2,…,k}.
Hence, we conclude that(6.20)(ω2kωk)<(2kk)4k=22k(2kk).Thus, we have succeeded in
proving the induction hypothesis.Let us suppose
now that the inequality(ωnωk)≤2n(nk) holds for some n≥2k,
and prove that the inequality (ωn+1ωk)≤2n+1(n+1k) also holds. To prove this, it is enough to
check that (ωn+1ωk)/(ωnωk)≤2(n+1k)/(nk).
This is equivalent to ωn+1/ωn+1−k≤2(n+1)/(n+1−k).
This new inequality means that (n+1)(n+1−r′)/[(n+1−k)(n+1−k−r′)]≤2(n+1)/(n+1−k),
which reduces to checking that (n+1−r′)≤2(n+1−k−r′).
Finally, this inequality is equivalent to 2k≤n+1−r′,
which is true since n≥2k and 1−r′>0.It follows now
from Theorem5.5 that for all r≥1/2, Xr is of class 2-W-S.Claim 2.
Ifr∈(−1, −1/2),
then for all ϵ>0,
there exists an M,
which depends on both r and ϵ,
such that for all n≥2k,
we have(6.21)(ωnωk)≤M(2+ϵ)n(nk).Indeed, we can see
as before that for n=2k,
we have(6.22)(ω2kωk)=(2kk)[4+22r′−1k−r′−r′(1−r′)(k−r′)2]×[4+22r′−1k−1−r′−r′(1−r′)(k−1−r′)2]⋯×[4+22r′−11−r′−r′(1−r′)(1−r′)2].Since 4+2(2r′−1)/(j−1−r′)−r′(1−r′)/(j−1−r′)2→4,
as j→∞,
there exists a natural number N,
such that for all j≥N, 4+2(2r′−1)/(j−1−r′)−r′(1−r′)/(j−1−r′)2≤(2+ϵ)2	.
Let M:=[1/(2+ϵ)2|J|]∏j∈J[4+2(2r′−1)/(j−1−r′)−r′(1−r′)/(j−1−r′)2],
where J:={j∈{1,2,…,N−1}|4+2(2r′−1)/(j−1−r′)−r′(1−r′)/(j−1−r′)2>(2+ϵ)2},
and |J| denotes the cardinality of J.
Then, for all k≥1,
we have(6.23)(ω2kωk)≤M(2+ϵ)2k(2kk).Now, if we fix k≥1,
we can prove, as before by induction on n that for all n≥2k,
the inequality (6.21) holds.Therefore, for
anyr∈(−1, −1/2) and any ϵ>0,
there exists M>0,
such that Xr is of class (M, 2+ϵ)-W-H.Claim 3.
For allr∈(−1, −1/2), Xr is of class 1/r(1−r)-W-H.To see this,
let us observe that since the roots of the trinomialq(x)=(2+r′x)[2−(1−r′)x] are −2/r′ and 2/(1−r′),
the maximum of this trinomial is attained at xv=1/2[2/(1−r′)−2/r′]=1/(1−r′)−1/r′ and is equal to q(xv)=1/[r′(1−r′)].
The closer the value of x to xv,
the greater the value of q(x).
As we saw before for all k≥1,
we have(6.24)(ω2kωk)=(2kk)[(2+r′k−r′)(2−1−r′k−r′)]×[(2+r′k−1−r′)(2−1−r′k−1−r′)]⋯×[(2+r′1−r′)(2−1−r′1−r′)]=(2kk)q(1k−r′)q(1k−1−r′)⋯q(11−r′)≤(2kk)[q(xv)]k=(2kk)[1r′(1−r′)]2k.Since 1/r′(1−r′)≥2,
we can prove now by induction on n that for all n≥2k,
we have(6.25)(ωnωk)≤tn(nk),where t:=1/r′(1−r′).
It follows now from Corollary 5.6 that Xr is of class t-W-H.One can improve
a little bit thist,
by observing that for all −1<r<−1/2 (or, equivalently, 1/2<r′<1), we have(6.26)11−r′>xv>12−r′>13−r′>⋯.A simple computation shows that
for 1/2<r′<(7−17)/4, 1/(2−r′) is closer to xv than 1/(1−r′) is to xv.
Therefore, in this case, we can choose a smaller t:=q(1/(2−r′)).
If (7−17)/4≤r′<1,
then we have that 1/(1−r′)−xv≤xv−1/(2−r′),
and thus we can take t:=q(1/(1−r′)) to conclude that Xr is of class t-W-H.Sincelimr→(−1)+ω1=0 and limr→(−1)+ω2=2>0,
we can conclude, as in Example 6.5, that the family {Xr} is not unif.-W-H. On the other hand, since
for any r0′>−1,
the set {1/r′(1−r′)∣1/2<r′≤r0′} is bounded above, we can see that for any B⊂(−1, ∞),
the family {Xr}r∈B is unif.-W-H if and only if infB>−1.Claim 4.
For anyr>−1,
if M and t are positive numbers, such that Xr is of class (M, t)-W-H, then t≥2.To see this,
let us observe first that by D'Alembert ratio theorem, we have(6.27)lim⁡n→∞(ω2nωn)n=lim⁡n→∞(ω2(n+1)ωn+1)(ω2nωn)=lim⁡n→∞(2n+2)(2n+1)(2n+2+r)(2n+1+r)(n+1)2(n+1+t)2=16.On the other hand, since for p=q=2t, E[|fn(Xr)⋄fn(Xr)|2]≤ME[|Γ(2t)fn(Xr)|2]2 and fn(Xr)⋄fn(Xr)=f2n(Xr),
we conclude that for all n≥1,(6.28)(ω2nωn)≤M(2t)2n.Taking radical or order n from both sides of this inequality and passing
to the limit as n→∞,
we obtain that 16≤(2t)2.
Thus, we get t≥2.This last claim
proves that at least forr≥−1/2,
we got the best possible t,
which is 2,
for the Wick-Hölder class of Xr.Let us consider
now the family of nondegenerate centered Meixner random variables:Xα,β,t whose Szegö-Jacobi parameters are αn=αn for all n≥0,
and ωn=βn2+(t−β)n for all n≥1,
where the parameters α, β,
and t satisfy the conditions: α∈ℝ, β≥0,
and t>0.
We distinguish between two cases.Case 1.
Ifβ=0,
(i.e., Xα,0,t is gaussian or poissonian), then ωn=tn for all n≥1,
and therefore, {ωn}n≥1 is additive. In this case, Xα,0,t is of class 1-W-H.Case 2.
Ifβ>0,
then, ωn=βωn′,
where ωn′=n2+[(t/β)−1]n,
we can see that (ωnωk)=(ωn′ωk′) for all n≥k≥0.
Therefore, Xα,β,t satisfies the same Wick-Hölder inequality as Xr,
where r=t/β.Definingt/β=∞ for β=0,
we can apply all the claims from this example to the family of Meixner
distributions, and formulate the following theorem.Theorem 6.6.
Let{Xα,β,t}α∈ℝ,β≥0,t>0 be the family of Meixner distributions. Then,
(1)
forXα,0,t
is of class
1-W-H for all α∈ℝ,
and t>0;(2)
ifβ>0 and t/β≥1/2,
then Xα,β,t is of class 2-W-H, and 2 is optimal;(3)
ifβ>0 and t/β<1/2,
then Xα,β,t is of class 1/r′(1−r′)-W-H, where r′=1−(t/β);(4)
a family{Xα,β,t}(α,β,t)∈I of Meixner distributions is uniformly Wick
Hölderian if and only if inf{t/β∣∃α∈ℝ,(α,β,t)∈I}>0.Example 6.7.
LetXq be the q-Gaussian random variable with parameter q,
where q∈[−1, 1].
It means that for any q∈(−1, 1), Xq is a symmetric random variable (so, αn=0 for all n≥0) and ωn=(1−qn)/(1−q) for all n≥1.
For q=1, X1 is the standard Gaussian since ωn=limq→1−(1−qn)/(1−q)=n for all n≥1.
For q=−1, ω1=1 and ωn=0 for all n≥2.
Thus, X−1 is the centered Bernoulli random variable (1/2)[δ{−1}+δ{1}].
Despite the fact that limq→−1+ω2n=0 for all n≥1,
the family {Xq}−1≤q≤1 is uniformly Wick-Hölderian. In fact, every q-Gaussian random variable satisfies the best
Wick-Hölder inequality since {ωn}n≥1 is subadditive for all −1≤q≤1.
This statement is trivial for q=−1 and q=1.
For −1<q<1 and all m and n natural numbers, we have(6.29)ωm+n−ωm−ωn=1−qm+n1−q−1−qm1−q−1−qn1−q=(1−qm)(qn−1)1−q<0.



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*Source: 254897-2008-05-08.xml*