Hölder-Type Inequalities for Norms of Wick Products

Abstract

Various upper bounds for the 𝐿²-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.

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:𝐸|||𝑓(𝑋)⋄𝑔(𝑋)|||²≤𝐸|||Γ(√𝑝𝐼)𝑓(𝑋)|||²𝐸|||Γ(√𝑞𝐼)𝑔(𝑋)|||².(1.1) 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 𝑓_𝑛 is not the null polynomial, then 𝑓_𝑛 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,…,𝑎_𝑘 with positive probabilities, then 𝑓_𝑛 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, 𝑓_𝑛 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 {𝛼_𝑛}_𝑛≥0 and {𝜔_𝑛}_𝑛≥1, such that for all 𝑛≥0,𝑋𝑓_𝑛(𝑋)=𝑓_𝑛+1(𝑋)+𝛼_𝑛𝑓_𝑛(𝑋)+𝜔_𝑛𝑓_𝑛−1(𝑋).(2.1)When 𝑛=0, 𝑓_𝑛−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 𝛼_𝑛=0 and 𝜔_𝑛=0 for all 𝑛≥𝑘. The sequences {𝛼_𝑛}_𝑛≥0 and {𝜔_𝑛}_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[𝑓^2𝑛(𝑋)]=𝜔₁𝜔₂⋯𝜔_𝑛 (see, e.g., [2, 3]).

Let 𝑁=𝑘 if 𝑋 is discrete and takes on only 𝑘 values with positive probabilities, and 𝑁=∞ otherwise. We define the Hilbert space ℋ∶={∑_𝑛<𝑁𝑐_𝑛𝑓_𝑛(𝑋)∣{𝑐_𝑛}_𝑛<𝑁⊂ℂ,∑_𝑛<𝑁𝜔_𝑛!|𝑐_𝑛|²<∞}, where 𝜔_𝑛!∶=𝜔₁𝜔₂…𝜔_𝑛 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 𝐹_𝑛 the space of all random variables of the form 𝑓(𝑋), where 𝑓 is a polynomial of degree at most 𝑛, and define 𝐺_𝑛∶=𝐹_𝑛⊖𝐹_𝑛−1, that is, 𝐺_𝑛 is the orthogonal complement of 𝐹_𝑛−1 into 𝐹_𝑛 for all 𝑛≥0. For convenience, we define 𝐹₋₁ and 𝐺₋₁ to be the null space. For all 𝑛≥0, 𝐺_𝑛=ℝ𝑓_𝑛 and 𝐺_𝑛 is called the homogenous chaos space of order 𝑛 generated by 𝑋. We will also call ℋ the chaos space generated by 𝑋.

For any 𝑚,𝑛≥0, we define the Wick product 𝑓_𝑚(𝑋)⋄𝑓_𝑛(𝑋) of 𝑓_𝑚(𝑋) and 𝑓_𝑛(𝑋), as 𝑓_𝑚(𝑋)⋄𝑓_𝑛(𝑋)∶=𝑓_𝑚+𝑛(𝑋). Observe that if 𝑁=𝑘 is finite, then 𝑓_𝑚(𝑋)⋄𝑓_𝑛(𝑋)=0 for all 𝑚 and 𝑛, such that 𝑚+𝑛≥𝑘. It is not hard to see that 𝑓_𝑚(𝑋)⋄𝑓_𝑛(𝑋) is in fact the projection of the point-wise product 𝑓_𝑚(𝑋)𝑓_𝑛(𝑋) on the space 𝐺_𝑚+𝑛. We extend now the Wick product by bilinearity, defining formally for all 𝜑=∑_𝑛<𝑁𝑐_𝑛𝑓_𝑛∈ℋ and 𝜓=∑_𝑛<𝑁𝑑_𝑛𝑓_𝑛∈ℋ,𝜑⋄𝜓∶=_𝑛<𝑁(_{𝑝+𝑞=𝑛}𝑐_𝑝𝑑_𝑞)𝑓_𝑛.(2.2) Since it is not guaranteed that ∑_𝑛<𝑁𝜔_𝑛!|∑_{𝑝+𝑞=𝑛}𝑐_𝑝𝑑_𝑞|²<∞, 𝜑⋄𝜓 may not belong to ℋ.

Definition 2.1. For any complex number 𝑐, define the second quantization operator of 𝑐𝐼, where 𝐼 denotes the identity operator of the one-dimensional Hilbert space ℂ𝑋, spanned by 𝑋, as a densely defined operator on ℋ, defined byΓ(𝑐𝐼)(_{0≤𝑛<𝑁}𝑑_𝑛𝑓_𝑛(𝑋))∶=_{0≤𝑛<𝑁}𝑐^𝑛𝑑_𝑛𝑓_𝑛(𝑋),(2.3)where 𝑑_𝑛∈ℂ for all 0≤𝑛<𝑁.

A random variable 𝜑∶=∑_{0≤𝑛<𝑁}𝑑_𝑛𝑓_𝑛(𝑋) belongs to the domain of Γ(𝑐𝐼) if and only if ∑_{0≤𝑛<𝑁}(1+|𝑐|^2𝑛)|𝑑_𝑛|²𝜔_𝑛!<∞.

3. Wick-Hölder Property

Definition 3.1. Let 𝑀 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:𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝑀𝐸|||Γ(√𝑝𝐼)𝜑(𝑋)|||²𝐸|||Γ(√𝑞𝐼)𝜓(𝑋)|||².(3.1)

Since for any fix 𝜑(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. Let 𝑡 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, if 𝑋 is a 𝑡-Wick-Hölderian random variable, then 𝑋 is also 𝑡^-Wick-Hölderian for all 𝑡^>𝑡.

Definition 3.3. Let 𝑋 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. If 𝑋 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 numbers 𝑀 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, let 𝑝>0 and 𝑞>0, such that (1/𝑝)+(1/𝑞)=1/𝑠. Let 𝜑(𝑋)=∑_𝑛<𝑁𝑐_𝑛𝑓_𝑛(𝑋)∈ℋ, and 𝜓(𝑋)=∑_𝑛<𝑁𝑑_𝑛𝑓_𝑛(𝑋)∈ℋ, such that Γ(√𝑝𝐼)𝜑(𝑋)∈ℋ and Γ(√𝑞𝐼)𝜓(𝑋)∈ℋ, where {𝑐_𝑛}_𝑛≥0⊂ℂ, {𝑑_𝑛}_𝑛≥0⊂ℂ, and {𝑓_𝑛}_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‖‖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.(3.2)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‖‖𝑔(𝑋)⋄ℎ(𝑋)‖‖≤𝑀‖‖‖Γ√𝑝/(3𝑀)𝐼𝑔(𝑋)‖‖‖‖‖‖Γ√𝑞/(3𝑀)𝐼ℎ(𝑋)‖‖‖.(3.3)Thus, since 3<9 and 𝑀≥1, we have‖‖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|||²=‖‖Γ(√1+‖‖Γ(√𝑞𝐼)ℎ‖‖²𝑝𝐼)𝜑‖‖²‖‖Γ(√𝑞𝐼)𝜓‖‖².(3.4)Hence, 𝑋 is 𝑠-Wick-Hölderian.

Definition 3.5. Let {𝑋_𝑖}_𝑖∈𝐼 be a family of random variables, having finite moments of all orders. The family {𝑋_𝑖}_𝑖∈𝐼 is said to be uniformly Wick-Hölderian if there exists a positive number 𝑡₀, such that for all 𝑖∈𝐼, 𝑋_𝑖 is 𝑡₀-Wick-Hölderian.

It follows from the proof of the previous proposition that a family {𝑋_𝑖}_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, 𝑋_𝑖 is (𝑀₀,𝑡₀)-Wick-Hölderian.

From now on, to make the notation easier, we say that a random variable 𝑋 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 {𝑋_𝑖}_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. Let 𝑋 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: 𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||Γ(√𝑝𝐼)𝜑(𝑋)|||²𝐸|||Γ(√𝑞𝐼)𝜓(𝑋)|||²,(4.1)then one must have: 1𝑝+1𝑞≤1.(4.2)

Proof. The fact that the support of 𝑋 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. Since𝜑(𝑋)⋄𝜓(𝑋)==11++𝑐(𝑐1𝑓₁+(𝑋𝜆))𝑓₁⋄1+𝑐𝜆𝑓₁(𝑋)(𝑋)+𝑐²𝜆𝑓₂(𝑋),(4.3)the inequality,𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||Γ(√𝑝𝐼)𝜑(𝑋)|||²𝐸|||Γ(√𝑞𝐼)𝜓(𝑋)|||²,(4.4)means that1+𝑐²(1+𝜆)²𝜔₁+𝑐⁴𝜆²𝜔₁𝜔₂≤1+𝑝𝑐²𝜔₁1+𝑞𝑐²𝜆²𝜔₁(4.5)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,(1+𝜆)²+𝑐²𝜆²𝜔₂≤𝑝+𝑞𝜆²+𝑝𝑞𝑐²𝜆²𝜔₁,(4.6)holds for all 𝑐≠0 and 𝜆∈ℝ. Passing to the limit, as 𝑐→0, in the last inequality, we obtain(1+𝜆)²≤𝑝+𝑞𝜆²(4.7)for all real numbers 𝜆. Moving all terms to the right, we conclude that the quadratic trinomial(𝑞−1)𝜆²−2𝜆+(𝑝−1),(4.8)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 that1𝑝+1𝑞≤1.(4.9)

Corollary 4.2. If 𝑋 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.

Let 𝑋 be a random variable, having finite moments of any order, and let {𝑓_𝑛}_𝑛≥0 and ℋ be the sequence of orthogonal polynomials and chaos space generated by 𝑋, respectively. If 𝛼∶={𝛼_𝑛}_𝑛≥0 is a sequence of complex numbers, then we denote by ℳ_𝛼 the densely defined linear operator on ℋ that maps 𝑓_𝑛(𝑋)→𝛼_𝑛𝑓_𝑛(𝑋) for all 𝑛≥0. We call the sequence 𝛼 the multiplier of the operator ℳ_𝛼. It is clear that for all 𝑛≥0, ℳ_𝛼𝐺_𝑛⊂𝐺_𝑛. 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=ℳ_𝛼. If 𝛼₀=1, then ℳ_𝛼1=1, and we say that the operator ℳ_𝛼 (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. Let 𝑋 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 𝑓_𝑛 be the orthogonal polynomial of degree 𝑛, generated by 𝑋. If 𝑐={𝑐_𝑛}_𝑛≥0 and 𝑑={𝑑_𝑛}_𝑛≥0 are two sequences of complex numbers, such that 𝑐₀=𝑑₀=1, and for all 𝜑(𝑋), 𝜓(𝑋)∈ℂ𝑓₀(𝑋)+ℂ𝑓_𝑛(𝑋), the following inequality holds: 𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||ℳ^𝑐𝜑(𝑋)|||²𝐸|||ℳ^𝑑𝜓(𝑋)|||²,(4.10)then one must have 1|||𝑐_𝑛|||²+1|||𝑑_𝑛|||²≤1.(4.11)

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 of 𝜑(𝑋) and 𝜓(𝑋) is always bounded above by the product of the norms of ℳ_𝑐𝜑(𝑋) and ℳ_𝑑𝜓(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 the 𝐿² case to the 𝐿^𝑟 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 𝐿^𝑟 case in the same manner, based on two very simple observations.

Observation 1. If 𝑓 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 of 𝑓 and 𝑔, or of 𝑓−𝑔, and mathematically by using the formula𝑓^𝑥⁰=lim_ℎ→0𝑓𝑥₀+ℎ+𝑓𝑥₀−ℎ−2𝑓𝑥₀ℎ²,(4.12)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. If 𝑟 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 Lemma 4.1 to powers 𝑟>2.

Lemma 4.4. Let 𝑋 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: 𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||^𝑟≤𝐸|||Γ(√𝑝𝐼)𝜑(𝑋)|||^𝑟𝐸|||Γ(√𝑞𝐼)𝜓(𝑋)|||^𝑟,(4.13)then one must have 1𝑝+1𝑞≤1.(4.14)

Proof. Let 𝜑_𝑐(𝑋)∶=1+𝑐𝑓₁(𝑋) and 𝜓_𝑐,𝜆(𝑋)∶=1+𝑐𝜆𝑓₁(𝑋), where 𝑐 and 𝜆 are arbitrary real numbers. Let us consider the functions 𝑔_𝜆∶ℝ→ℝ, 𝑔_𝜆(𝑐)∶=𝐸[|𝜑_𝑐(𝑋)⋄𝜓_𝑐,𝜆(𝑋)|^𝑟] and ℎ_𝜆∶ℝ→ℝ, ℎ_𝜆(𝑐)∶=𝐸[|Γ(√𝑝𝐼)𝜑_𝑐(𝑋)|^𝑟]𝐸[|Γ(√𝑞𝐼)𝜓_𝑐,𝜆(𝑋)|^𝑟]. Since 𝑔_𝜆(𝑐)≤ℎ_𝜆(𝑐) for all 𝑐∈ℝ, and 𝑔_𝜆(0)=ℎ_𝜆(0), according to Observation 1, we have𝑔^𝜆(0)≤ℎ^𝜆(0).(4.15)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,𝑔^𝜆(𝑐)=𝑟(𝑟−1)𝐸|||1+𝑐(1+𝜆)𝑓₁+𝑐²𝜆𝑓₂|||^𝑟−2+2𝑟𝜆𝐸|||1+𝑐(1+𝜆)𝑓₁(1+𝜆)𝑓¹+2𝑐𝜆𝑓₂²+𝑐²𝜆𝑓₂|||^𝑟−2ℎ_𝜆1+𝑐(1+𝜆)𝑓¹+𝑐²𝜆𝑓₂𝑓²,(𝑐)=𝑟(𝑟−1)𝑝𝐸|||1+𝑐√𝑝𝑓₁|||^𝑟−2𝑓²¹𝐸|||1+𝑐𝜆√𝑞𝑓₁|||^𝑟+2𝑟₂𝜆√𝑝𝑞𝐸|||1+𝑐√𝑝𝑓₁|||^𝑟−2×𝐸|||1+𝑐𝜆√(1+𝑐√𝑝𝑓₁)𝑓₁𝑞𝑓₁|||^𝑟−2+𝑟(𝑟−1)𝑞𝜆₂(1+𝑐𝜆√𝑞𝑓₁)𝑓₁𝐸|||1+𝑐𝜆√𝑞𝑓₁|||^𝑟−2𝑓²¹𝐸|||1+𝑐√𝑝𝑓₁|||^𝑟.(4.16) Setting 𝑐=0, we get nowℎ𝑔^𝜆_𝜆(0)=𝑟(𝑟−1)(1+𝜆)²𝐸𝑓²¹+2𝑟𝜆𝐸𝑓²,(0)=𝑟(𝑟−1)𝑝𝐸𝑓²¹+2𝑟²𝜆√𝑝𝑞𝐸𝑓₁𝐸𝑓¹+𝑟(𝑟−1)𝑞𝜆²𝐸𝑓²¹.(4.17)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 𝑔^𝜆(0)≤ℎ^𝜆(0) becomes, after dividing both sides by the positive number 𝑟(𝑟−1)𝐸[𝑓²¹](1+𝜆)²≤𝑝+𝑞𝜆²(4.18)for 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 that1𝑝+1𝑞≤1.(4.19)

Our technique for proving the universal minimal condition for 𝑟≥2 is similar to the proof of Theorem 3 from [4].

Finally, we can also see that for multipliers, the inequality𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||^𝑟≤𝐸|||ℳ^𝑐𝜑(𝑋)|||^𝑟𝐸|||ℳ^𝑑𝜓(𝑋)|||^𝑟,(4.20)for a fix 𝑟≥2, also implies the condition that1|||𝑐_𝑛|||²+1|||𝑑_𝑛|||²≤1(4.21)for all 𝑛≥1.

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

We present first the following lemma.

Lemma 5.1. Let {𝜔_𝑛}_𝑛≥1 be a sequence of positive numbers, such that there exist a number 𝑡, greater than or equal to 1, and a sequence {𝛼_𝑛}_𝑛≥2, of nonnegative numbers, such that the series ∑^∞𝑛=2𝛼_𝑛 is convergent, and for all 𝑛 and 𝑘 natural numbers, with 𝑛>𝑘, it holds that 𝜔_𝑛≤𝑡1+𝛼_𝑛𝜔^𝑘+𝜔_𝑛−𝑘.(5.1)Then, for all nonnegative numbers 𝑘≥𝑟≥0, it holds that 𝜔𝜔^𝑘_𝑟≤𝑀𝑡^𝑘𝑘𝑟,(5.2)where 𝑀∶=max{1,(1/𝑡)∏^∞𝑛=2(1+𝛼_𝑛)}, (^𝑘𝑟)∶=(1⋅2⋯𝑘)/[(1⋅2⋯𝑟)(1⋅2⋯(𝑘−𝑟))], and (^𝜔𝑘_𝜔𝑟)∶=(𝜔₁𝜔₂⋯𝜔_𝑘)/[(𝜔₁𝜔₂⋯𝜔_𝑟)(𝜔₁𝜔₂⋯𝜔_𝑘−𝑟)] for 0<𝑟<𝑘. If 𝑟=0 or 𝑟=𝑘, then (^𝑘𝑟)∶=1 and (^𝜔𝑘_𝜔𝑟)∶=1.

Proof. Since 𝛼_𝑛≥0 for all 𝑛≥2 and ∑^∞𝑛=2𝛼_𝑛<∞, we conclude that the product ∏^∞𝑛=2(1+𝛼_𝑛) is convergent. For 𝑘=0, the inequality (5.2) is obvious since (^𝜔0_𝜔0)=(⁰⁰)=1. We prove now by induction on 𝑘 that for all 𝑘≥1, we have𝜔𝜔^𝑘_𝑟≤[^𝑘_𝑖=21+𝛼^𝑖]𝑡^𝑘−1𝑘𝑟,(5.3)for all 𝑟∈{0,1,…,𝑘}. For 𝑘=1, we have nothing to prove, since (^𝜔𝑘_𝜔𝑟)=(^𝑘𝑟)=1, and ∏^𝑘𝑖=2(1+𝛼_𝑖) 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𝜔𝜔^𝑛+1_𝑟=𝜔^𝑛+1𝜔_𝑛!≤𝜔𝑡_𝑟1𝜔+_𝑟−1𝛼!𝜔_{𝑛+1−𝑟𝑛+1}𝜔_𝑛−𝑟!𝜔^𝑟+𝜔_{𝑛+1−𝑟}𝜔^𝑛!=𝑡1+𝜔𝛼_𝑟𝜔_{𝑟−1𝑛+1}!𝜔_{𝑛+1−𝑟}𝜔_𝑛−𝑟![𝜔^𝑛!𝜔_𝑟−1!𝜔_{𝑛+1−𝑟}!+𝜔^𝑛!=𝑡1+𝛼_𝑛+1𝜔_𝑟!𝜔_𝑛−𝑟!][𝜔𝜔^𝑛_𝑟−1+𝜔𝜔^𝑛_𝑟].(5.4)Using now this inequality, the induction hypothesis, and the classic Pascal identity (^𝑛𝑟−1)+(^𝑛𝑟)=(^𝑟𝑛+1), we get𝜔𝜔^𝑛+1_𝑟≤[^𝑛+1_𝑖=21+𝛼^𝑖]𝑡^𝑛=[_𝑛+1[𝑟−𝑛1+𝑛𝑟]_𝑖=21+𝛼^𝑖]𝑡^𝑛𝑛+𝑟1.(5.5)Hence, for all 𝑘≥1, we have𝜔𝜔^𝑘_𝑟≤≤^𝑘_𝑖=2∞1+𝛼^𝑖𝑡^𝑘−1𝑘𝑟=1_𝑖=21+𝛼^𝑖𝑡^𝑘−1𝑘𝑟𝑡^∞_𝑖=21+𝛼^𝑖𝑡^𝑘𝑘𝑟.(5.6)In order to make this inequality also true, for 𝑘=0, we have to replace (1/𝑡)[∏^∞𝑖=2(1+𝛼_𝑖)] by max{1,(1/𝑡)[∏^∞𝑖=2(1+𝛼_𝑖)]}.

We introduce now three definitions.

Definition 5.2. Given a number 𝑡, greater than or equal to 1, and a sequence of nonnegative numbers 𝛼={𝛼_𝑛}_𝑛≥2, such that the series ∑^∞𝑛=2𝛼_𝑛 is convergent, then a sequence of nonnegative numbers {𝜔_𝑛}_𝑛≥1 is said to be (𝑡, 𝛼)-subadditive if the following inequality,𝜔_𝑛≤𝑡1+𝛼_𝑛𝜔^𝑘+𝜔_𝑛−𝑘,(5.7)holds for all natural numbers 𝑛 and 𝑘, such that 𝑛>𝑘.

Definition 5.3. Given a number 𝑡, greater than or equal to 1, then a sequence of nonnegative numbers {𝜔_𝑛}_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,𝜔_𝑚+𝑛≤𝑡𝜔_𝑚+𝜔_𝑛,(5.8)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{𝛼_𝑛∣𝑛≥2}).

Definition 5.4. If 𝑡 is a number, greater than or equal to 1, and {𝜔_𝑛}_𝑛≥1 is a sequence of nonnegative numbers, then the sequence {𝜔_𝑛}_𝑛≥1 is said to be exp-𝑡-subadditive if𝜔^{1/𝑡𝑚+𝑛}≤𝜔^𝑚1/𝑡+𝜔^𝑛1/𝑡(5.9)for all 𝑚 and 𝑛 positive integers.

Observation 3. An exp-𝑡-subadditive sequence {𝜔_𝑛}_𝑛≥1 of nonnegative numbers is also exp-𝑠-subadditive for all 𝑠>𝑡.

Proof. Applying the inequality (𝑎+𝑏)^𝑟≤𝑎^𝑟+𝑏^𝑟 for all 𝑎≥0, 𝑏≥0, and 0<𝑟≤1, we get𝜔^{1/𝑠𝑚+𝑛}=𝜔^{1/𝑡𝑚+𝑛}^𝑡/𝑠≤𝜔^𝑚1/𝑡+𝜔^𝑛1/𝑡^𝑡/𝑠≤𝜔^𝑚1/𝑡^𝑡/𝑠+𝜔^𝑛1/𝑡^𝑡/𝑠=𝜔^𝑚1/𝑠+𝜔^𝑛1/𝑠(5.10)for all 𝑚 and 𝑛 positive integers.

Observation 4. For all 𝑡≥1, any exp-𝑡-subadditive sequence {𝜔_𝑛}_𝑛≥1 of nonnegative numbers is also 2^𝑡−1-subadditive.

Proof. Since 𝑡≥1, the function 𝑓∶[0, ∞)→ℝ, 𝑓(𝑥)=𝑥^𝑡 is convex. Thus, for all 𝑚,𝑛≥1, we have𝜔_𝑚+𝑛=𝜔^{1/𝑡𝑚+𝑛}^𝑡≤𝜔^𝑚1/𝑡+𝜔^𝑛1/𝑡^𝑡=2^𝑡12𝜔^𝑚1/𝑡+12𝜔^𝑛1/𝑡^𝑡≤2^𝑡12𝜔^𝑚1/𝑡^𝑡+12𝜔^𝑛1/𝑡^𝑡=2^𝑡−1𝜔^𝑚+𝜔_𝑛.(5.11)

We present now the main result of this paper.

Theorem 5.5. Let 𝑋 be a random variable, having finite moments of any order, and let {𝜔_𝑛}_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 {𝜔_𝑛}_𝑛≥1 satisfies the condition 𝜔𝜔^𝑘_𝑟≤𝑀𝑡^𝑘𝑘𝑟(5.12)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. Let 𝑝>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 {𝑓_𝑛}_𝑛<𝑁 be the orthogonal sequence of polynomials, with a leading coefficient of 1, generated by 𝑋. We distinguish between two cases.
Case 1. If 𝑁=∞, then there exist two unique sequences of complex numbers {𝑐_𝑛}_𝑛≥0 and {𝑑_𝑛}_𝑛≥0, such that𝜑=^∞_𝑛=0𝜓=_∞𝑐_𝑛𝑓_𝑛(𝑋),_𝑛=0𝑑_𝑛𝑓_𝑛(𝑋).(5.13)We have𝜑⋄𝜓=^∞_𝑘=0(_{𝑢+𝑣=𝑘}𝑐_𝑢𝑑_𝑣)𝑓_𝑘(𝑋).(5.14)Using the triangle inequality, we obtain𝐸|||𝜑⋄𝜓|||²=^∞_𝑘=0≤_∞|_{𝑢+𝑣=𝑘}𝑐_𝑢𝑑_𝑣|²𝜔_𝑘!_𝑘=0[_{𝑢+𝑣=𝑘}|||𝑐_𝑢𝑑_𝑣|||]²=_∞𝜔_𝑘!_𝑘=0𝜔_𝑘![_{𝑢+𝑣=𝑘}1√𝜔_𝑢!𝜔_𝑣!𝑝^𝑢𝑞^𝑣√𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||√𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||]²(5.15)From the Cauchy-Bunyakovsky-Schwarz inequality, inequality (5.2), and Newton's binomial formula, we obtain𝐸|||𝜑⋄𝜓|||²≤^∞_𝑘=0𝜔_𝑘![_{𝑢+𝑣=𝑘}1𝜔_𝑢!𝜔_𝑣!𝑝^𝑢𝑞^𝑣]×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=_∞]_𝑘=0[_{𝑢+𝑣=𝑘}𝜔_𝑘!𝜔_𝑢!𝜔_𝑣!𝑝1^𝑢1𝑞^𝑣]×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=_∞]_𝑘=0[_{𝑢+𝑣=𝑘}𝜔𝜔^𝑘_𝑢𝑝1^𝑢1𝑞^𝑣]×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²≤_∞]_𝑘=0[_{𝑢+𝑣=𝑘}𝑀𝑡^𝑘𝑘𝑢𝑝1^𝑢1𝑞^𝑣]×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀_∞]_𝑘=0𝑡^𝑘[_{𝑢+𝑣=𝑘}𝑘𝑢𝑝1^𝑢1𝑞^𝑣]×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀_∞]_𝑘=0𝑡^𝑘1𝑝+1𝑞^𝑘×[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀_∞]_𝑘=0𝑡^𝑘1𝑡^𝑘[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀_∞]_𝑘=01[_{𝑢+𝑣=𝑘}𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||²𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀_∞]_𝑢=0𝑝^𝑢𝜔_𝑢!|||𝑐_𝑢|||^2∞_𝑣=0𝑞^𝑣𝜔_𝑣!|||𝑑_𝑣|||²=𝑀𝐸|||Γ(√𝑝𝐼)𝑓|||²𝐸|||Γ(√𝑞𝐼)𝑔|||².(5.16)
Case 2. If 𝑁<∞, 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/𝑝^𝑢)(1/𝑞^𝑣) 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/𝑝^𝑢)(1/𝑞^𝑣)=1/𝑡^𝑘, we will have ∑_{𝑢+𝑣=𝑘}(^𝑘𝑢)(1/𝑝^𝑢)(1/𝑞^𝑣)<1/𝑡^𝑘. 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. Let 𝑡 be a number greater than or equal to 1, and 𝛼={𝛼_𝑛}_𝑛≥2 a sequence of nonnegative numbers producing a convergent series ∑^∞𝑛=2𝛼_𝑛. Let 𝑋 be a random variable, having finite moments of any order, and let {𝜔_𝑛}_𝑛≥1 be the principal Szegö-Jacobi parameters of 𝑋. If the sequence {𝜔_𝑛}_𝑛≥1 is (𝑡,𝛼)-subadditive, then 𝑋 is of class (𝑀,𝑡)-W-H, where 𝑀∶=max{1,(1/𝑡)∏^∞𝑛=2(1+𝛼_𝑛)}.

Corollary 5.7. If 𝑋 is a random variable, having finite moments of all orders, such that its principal Szegö-Jacobi sequence {𝜔_𝑛}_𝑛≥1 is 𝑡-subadditive, then 𝑋 is of class 𝑡-W-H.

Corollary 5.8. If 𝑋 is a random variable, having finite moments of all orders, such that its principal Szegö-Jacobi sequence {𝜔_𝑛}_𝑛≥1 is exp-𝑡-subadditive, then 𝑋 is of class 2^𝑡−1-W-H. In particular, by taking 𝑝=𝑞=𝜆=2^𝑡, it is concluded that if ℋ denotes the chaos space generated by 𝑋, and if 𝜑(𝑋), 𝜓(𝑋)∈ℋ, such that Γ(2^𝑡/2𝐼)𝜑(𝑋)∈ℋ and Γ(2^𝑡/2𝐼)𝜓(𝑋)∈ℋ, then 𝜑(𝑋)⋄𝜓(𝑋)∈ℋ and the following inequality holds: 𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||Γ(2^𝑡/2𝐼)𝜑(𝑋)|||²𝐸|||Γ(2^𝑡/2𝐼)𝜓(𝑋)|||².(5.17)

Observation 5. If 𝑝=𝑞=2𝑡, then to prove that the inequality,
𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||Γ(√2𝑡𝐼)𝜑(𝑋)|||²𝐸|||Γ(√2𝑡𝐼)𝜓(𝑋)|||²,(5.18)holds whenever both expectations from the right-hand side are finite, we do not need the condition that each 𝜔-binomial coefficient (^𝜔𝑘_𝜔𝑟) is less than or equal to 𝑀𝑡^𝑘(^𝑘𝑟), but it is enough to assume that for each 𝑘≥0, the sum of all binomial coefficients, having 𝜔_𝑘 on the top, is less than or equal to 𝑀𝑡^𝑘 times the sum of all classic binomial coefficients having 𝑘 on the top, that means^𝑘_𝑟=0𝜔𝜔^𝑘_𝑟≤𝑀𝑡^𝑘⋅2^𝑘(5.19)for all 𝑘≥0.

Proof. The proof of this observation follows line by line the proof of Theorem 5.5, 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 𝑝^𝑢𝑞^𝑣 is always the same (independent of the pair) and equal to (2𝑡)^𝑘.

The following proposition will be useful in showing later that 𝜆=2^𝑡 is optimal for some particular random variables 𝑋.

Proposition 5.9. Let 𝑋 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 {𝜔_𝑛}_𝑛≥1 the principal Szegö-Jacobi sequence of 𝑋. We define 𝑆_𝑛∶=∑^𝑛𝑘=0(^𝜔𝑛_𝜔𝑘) for all 𝑛≥0, and 𝜆₀∶=limsup_𝑛→∞𝑆^𝑛1/𝑛. Then,
(1) if there exists a positive number 𝜆 such that the inequality,𝐸|||𝜑(𝑋)⋄𝜓(𝑋)|||²≤𝐸|||Γ(√𝜆𝐼)𝜑(𝑋)|||²𝐸|||Γ(√𝜆𝐼)𝜓(𝑋)|||²,(5.20)holds for all 𝜑∈ℋ and 𝜓∈ℋ, such that Γ(√𝜆𝐼)𝜑∈ℋ and Γ(√𝜆𝐼)𝜓∈ℋ, then 𝜆≥𝜆₀;(2) if 𝜆₀ satisfies (5.20), then 𝜆₀ is optimal (i.e., the smallest among all positive 𝜆's satisfying this inequality);(3) if 𝑆_𝑛≤𝜆^𝑛0 for all 𝑛≥0, then 𝜆₀ is optimal.

Proof. (1) Let 𝜆>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∶=𝑓_𝑟(𝑋) and 𝜓∶=𝑓_𝑘−𝑟(𝑋). The inequality 𝐸[|𝜑⋄𝜓|²]≤𝐸[|Γ(√𝜆𝐼)𝜑|²]𝐸[|Γ(√𝜆𝐼)𝜓|²] is equivalent to 𝜔_𝑘!≤𝜆^𝑟𝜔_𝑟!𝜆^𝑘−𝑟𝜔_𝑘−𝑟!. This implies that (^𝜔𝑘_𝜔𝑟)≤𝜆^𝑘. Summing up from 𝑟=0 to 𝑟=𝑘, we obtain 𝑆_𝑘≤(𝑘+1)𝜆^𝑘 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 limsup√_𝑘→∞^𝑘𝑆≤𝜆. Thus, 𝜆₀≤𝜆.
(2) It follows from 1.(3) It follows from letting 𝑀∶=1 and 𝑡∶=𝜆₀/2 in Observation 5.

6. Examples

Example 6.1. Let 𝑠≥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 𝜔_𝑛=𝑛^𝑠>0 for all 𝑛≥1. It is clear that {𝜔_𝑛}_𝑛≥1 is an exp-𝑠-subadditive (in fact exp-𝑠-additive) sequence. Therefore, {𝜔_𝑛}_𝑛≥1 is also 2^𝑠−1-subadditive. Thus, 𝑋 is of class 2^𝑠−1-W-H. Moreover, the following lemma holds.

Lemma 6.2. For all 𝑠≥1, it holds that [^𝑛_𝑘=0𝑘𝑛^𝑠lim_𝑛→∞]^1/𝑛≤2^𝑠,(6.1)[^𝑛_𝑘=0𝑘𝑛^𝑠]^1/𝑛=2^𝑠.(6.2)0<𝑠<1Formula (6.2) holds even for 𝑠≥1.

Proof. If [^𝑛_𝑘=0𝑘𝑛^𝑠]^1/𝑛≤{[^𝑛_𝑘=0𝑘𝑛]^𝑠}^1/𝑛=2^𝑛^𝑠/𝑛=2^𝑠.(6.3), then we have𝑠≥1 Since ℎ∶(0, the function ∞)→ℝ, ℎ(𝑥)=𝑥^𝑠, [^𝑛_𝑘=0𝑘𝑛^𝑠]^1/𝑛=𝑛^1/𝑛[^𝑛_𝑘=01𝑛𝑘𝑛^𝑠]^1/𝑛≥𝑛^1/𝑛{[^𝑛_𝑘=01𝑛𝑘𝑛]^𝑠}^1/𝑛=𝑛^1/𝑛2^𝑛𝑛^𝑠/𝑛=1√^𝑛𝑛^𝑠−12^𝑠.(6.4) is convex, and thus1√^𝑛𝑛^𝑠−12^𝑠≤[^𝑛_𝑘=0𝑘𝑛^𝑠]^1/𝑛≤2^𝑠.(6.5) Hence,√^𝑛𝑛→1Since 𝑛→∞, as lim_𝑛→∞[∑^𝑛𝑘=0(^𝑛𝑘)^𝑠]^1/𝑛, we conclude that 2^𝑠 exists and is equal to 0<𝑠<1.
If 1√^𝑛𝑛^𝑠−12^𝑠≥[^𝑛_𝑘=0𝑘𝑛^𝑠]^1/𝑛≥2^𝑠,(6.6), then in a similar way, we can show thatlim_𝑛→∞[∑^𝑛𝑘=0(^𝑛𝑘)^𝑠]^1/𝑛=2^𝑠and it follows again that 𝑋.

From formula (6.2), inequality (6.1), Corollary 5.8, and Proposition 5.9, we obtain the following proposition.

Proposition 6.3. Let 𝜔_𝑛=𝑛^𝑠 be a random variable, having finite moments of any order whose principal Szegö-Jacobi parameters are 𝑛≥1 for all 𝑠, where 𝑠≥1 is a fixed real number, such that ℋ. Let 𝑋 be the chaos space generated by 𝑝.
(1) If 𝑞 and (1/𝑝)+(1/𝑞)≤(1/2^𝑠−1) are positive numbers, such that 𝜑∈ℋ, then for all 𝜓∈ℋ and Γ(√𝑝𝐼)𝜑∈ℋ, such that Γ(√𝑞𝐼)𝜓∈ℋ and 𝑓(𝑋)⋄𝑔(𝑋)∈ℋ, there exists 𝐸|||𝜑⋄𝜓|||²≤𝐸|||Γ(√𝑝𝐼)𝜑|||²𝐸|||Γ(√𝑞𝐼)𝜓|||².(6.7) and the following inequality holds:𝜆₀∶=2^𝑠(2)𝜆 is the smallest among all positive numbers 𝐸|||𝜑⋄𝜓|||²≤𝐸|||Γ(√𝜆𝐼)𝜑|||²𝐸|||Γ(√𝜆𝐼)𝜓|||²,(6.8), for which the inequality,𝜑∈ℋholds for all 𝜓∈ℋ and Γ(√𝜆𝐼)𝜑∈ℋ, such that Γ(√𝜆𝐼)𝜓∈ℋ and 𝑠.

Example 6.4. This is a modification of the previous example. Let 𝑠>2 be a fixed real number, such that 𝑋_𝑟. Let 𝜔_𝑛=𝑛^𝑠+𝑟𝑛 be a random variable whose principal Szegö-Jacobi parameters are 𝑛≥0 for all 𝑟>−1, where 𝑟>−1. The condition 𝜔₁>0 follows from the inequality 𝜔_𝑛>0, and ensures that 𝑛≥2 for all 𝑠>2. The reason why we have chosen 𝑟≥0 will become more transparent later.

Claim 1. If {𝜔_𝑛}_𝑛≥1, then the sequence 2^𝑠−1 is 𝑚-subadditive.

Indeed, for all 𝑛 and (𝑚𝑚++𝑛)𝑛^𝑠<≤22^𝑠−1_𝑠−1𝑚^𝑠+𝑛^𝑠,𝑚+𝑛.(6.9) natural numbers, we have𝑟Multiplying the second inequality by 𝜔_𝑚+𝑛≤2^𝑠−1𝜔^𝑚+𝜔_𝑛.(6.10) and adding the resulting inequality to the first one, we get−1<𝑟<0

Claim 2. If {𝜔_𝑛}_𝑛≥1, then the sequence 2^𝑠−1 is not 𝑟^∶=−𝑟>0-subadditive.

Indeed, if we define 𝑛≥1, then for all 𝜔_2𝑛=(2𝑛)^𝑠−2𝑛𝑟^=2^𝑠−1𝑛^𝑠−𝑟^𝑛+𝑛^𝑠−𝑟^𝑛+2𝑛2^𝑠−1−1𝑟^=2^𝑠−1𝜔^𝑛+𝜔_𝑛+2𝑛2^𝑠−1−1𝑟^>2^𝑠−1𝜔^𝑛+𝜔_𝑛.(6.11), we have−1<𝑟<0

Claim 3. If 𝛼={𝛼_𝑛}_𝑛≥2, then there exists a sequence ∑^∞𝑛=2𝛼_𝑛 of positive numbers, such that {𝜔_𝑛}_𝑛≥1 is convergent, and the sequence 2^𝑠−1 is (𝛼, 𝑟^=−𝑟∈(0)-subadditive.

As before, let 1), 𝑛≥2. Let 𝛼_𝑛 be a fixed natural number. Let us find a positive number 𝜔_𝑛≤1+𝛼_𝑛2^𝑠−1𝜔^𝑘+𝜔_𝑛−𝑘(6.12), such that1≤𝑘<𝑛for all 𝑛^𝑠−𝑟^𝑛≤1+𝛼_𝑛2^𝑠−1𝑘^𝑠+(𝑛−𝑘)^𝑠−1+𝛼^𝑛2^𝑠−1𝑟^𝑛(6.13). This inequality is equivalent to1≤𝑘<𝑛for all 𝑛^𝑠≤2^𝑠−1[𝑘^𝑠+(𝑛−𝑘)^𝑠]. Since 𝛼_𝑛, if we choose 𝑛^𝑠−𝑟^𝑛=1+𝛼_𝑛𝑛^𝑠−1+𝛼_𝑛2^𝑠−1𝑟^𝑛,(6.14) such that1≤𝑘<𝑛then the last inequality holds, for all 𝑛. In fact, for 𝑘=𝑛−𝑘 even, if we choose 𝑘=𝑛/2 (that means 𝛼_𝑛), then we can see that 𝛼_𝑛 cannot be chosen smaller than the value of the solution of (6.14). Solving (6.14) for 𝛼_𝑛=𝑟^2^𝑠−1−1𝑛^𝑠−1−2^𝑠−1𝑟^.(6.15), we get𝑛≥2Since 0<𝑟^<1 and 𝛼_𝑛>0, we can see that 𝑛≥2 for all 𝑠>2. Moreover, since we have chosen ∑^∞𝑛=2𝛼_𝑛, the series ∑^∞𝑛=21/𝑛^𝑠−1 has the same nature as the series 𝑟≥0, which is convergent.

Therefore, according to Theorem 5.8 for 𝑋_𝑟, 2^𝑡−1 is of class −1<𝑟<0-W-H, while for 𝑋_𝑟, (𝑀,2^𝑡−1) is of class 𝑀≥1-W-H for some 𝑀 (𝑟 depends on {𝑋_𝑟}_𝑟>−1). Moreover, the family 𝑀 is not uniformly Wick-Hölderian (that means we cannot find the same 𝑟>−1 for all lim_{𝑟→(−1)}⁺𝜔₁=0). This follows from the observation that lim_{𝑟→(−1)}⁺𝜔₂=2^𝑠−2>0, while 𝑝. If we assume the existence of two positive numbers 𝑞 and 𝐸[|𝑓₁(𝑋_𝑟)⋄𝑓₁(𝑋_𝑟)|²]≤𝐸[|Γ(√𝑝𝐼)𝑓₁(𝑋_𝑟)|²]𝐸[|Γ(√𝑞𝐼)𝑓₁(𝑋_𝑟)|²], such that the inequality: 𝑟>−1 holds for all 𝑓₁(𝑋_𝑟)⋄𝑓₁(𝑋_𝑟)=𝑓₂(𝑋_𝑟) since 𝑓₁, where 𝑓₂ and 𝑋_𝑟, are the orthogonal polynomials of degree 1 and 2, respectively, generated by 𝜔₂!≤𝑝𝜔₁!⋅𝑞𝜔₁!.(6.16), then we would conclude that𝜔₂≤𝑝𝑞𝜔₁This inequality reduces to 𝑟>−1 for all 𝑟, which is impossible since the left-hand side converges to a positive number, while the right-hand side tends to zero as −1 goes to 𝑠=2.

Example 6.5. Let us take now 𝑋_𝑟, and consider the family of random variables 𝜔_𝑛∶=𝑛²+𝑟𝑛 whose principal Szegö-Jacobi parameters are 𝑛≥1 for all 𝑟>−1, where 𝛼_𝑛∶=𝛼𝑛. If we take the other Szegö-Jacobi parameters to be 𝑛≥0 for all 𝛼, where {𝑋_𝑟}_𝑟>−1 is a fixed real number, then we can see that 𝑟≥0 are exactly the centered and rescaled Meixner random variables.

We can see exactly as in Example 6.5 that for {𝜔_𝑛}_𝑛≥1, 2²⁻¹ is −1<𝑟<0-subadditive, while for 2, it is not 𝑟^∶=−𝑟-suadditive. If 𝛼_𝑛=𝑟^(2²⁻¹−1)/(𝑛²⁻¹−2²⁻¹𝑟^)=𝑟^/(𝑛−2𝑟^), unfortunately, for ∑^∞𝑛=2𝛼_𝑛, the series 𝑛 is not convergent. Moreover, as we saw in the previous example, since (6.14) cannot be avoided for {𝜔_𝑛}_𝑛≥1 even, we can see that the sequence 2 is not (𝛽, 𝛽={𝛽_𝑛}_𝑛≥2)-subadditive, for any nonnegative sequence ∑^∞𝑛=2𝛽_𝑛<∞, such that −1/2≤𝑟<0.

Claim 1. If 𝑛≥𝑘≥0, then for all 𝜔𝜔^𝑛_𝑘≤2^𝑛𝑘𝑛.(6.17), we have𝜔𝜔^𝑛_𝑘=𝜔𝜔^𝑛_𝑛−𝑘Since (𝑘𝑛)=(^{𝑛𝑛−𝑘}) and 𝑘≤𝑛/2, we may assume that 𝑘≥1.

We fix 𝑛, and prove by induction on 𝑛≥2𝑘 that for all 𝜔𝜔^𝑛_𝑘≤2^𝑛(^𝑛𝑘), 𝑛=2𝑘.

Let us prove first this inequality for 𝜔_𝑚=𝑚(𝑚−𝑟^). Since 𝑚≥1 for all 𝜔𝜔^2𝑘_𝑘=2𝑘𝑘2𝑘𝑘−−𝑟𝑟^⋅2𝑘−1−𝑟^𝑘−𝑟^2𝑘𝑘−−12−−𝑟𝑟^_⋅2𝑘−3−𝑟^×2−𝑟_𝑘−1−𝑟^⋯1−𝑟^⋅1−𝑟^=2𝑘𝑘21+−𝑟^𝑟_×2+𝑟_𝑘−𝑟^2−𝑘1−−𝑟𝑟^×2+𝑘−𝑟_1−𝑟^2−𝑘−1−1−𝑟^𝑟^⋯=2𝑘𝑘14−+𝑟2_2𝑟_2−11−−𝑟𝑟^−1𝑘−𝑟^−𝑟^(1−𝑟^)×4+22𝑟_(𝑘−𝑟^)²−1𝑘−1−𝑟^−𝑟^(1−𝑟^)×4+22𝑟_(𝑘−1−𝑟^)²⋯−11−𝑟^−𝑟^(1−𝑟^)(1−𝑟^)².(6.18), we have−1/2≤𝑟<0If 𝑟^=−𝑟, since 0<𝑟^≤1/2, we have 2𝑟^−1≤0, and thus 4+22𝑟^−1𝑗−𝑟^−𝑟^(1−𝑟^)(𝑗−𝑟^)²<4,(6.19). Hence,𝑗∈{1,2,…,𝑘}for all 𝜔𝜔^2𝑘_𝑘<2𝑘𝑘4^𝑘=2^2𝑘2𝑘𝑘.(6.20). Hence, we conclude that(𝜔^𝑛𝜔_𝑘)≤2^𝑛(𝑘𝑛)Thus, we have succeeded in proving the induction hypothesis.

Let us suppose now that the inequality 𝑛≥2𝑘 holds for some (𝜔^𝑛+1𝜔_𝑘)≤2^𝑛+1(𝑛+𝑘1), and prove that the inequality (𝜔^𝑛+1𝜔_𝑘)/(𝜔𝜔^𝑛_𝑘)≤2(𝑛+𝑘1)/(𝑘𝑛) also holds. To prove this, it is enough to check that 𝜔_𝑛+1/𝜔_{𝑛+1−𝑘}≤2(𝑛+1)/(𝑛+1−𝑘). This is equivalent to (𝑛+1)(𝑛+1−𝑟^)/[(𝑛+1−𝑘)(𝑛+1−𝑘−𝑟^)]≤2(𝑛+1)/(𝑛+1−𝑘). This new inequality means that (𝑛+1−𝑟^)≤2(𝑛+1−𝑘−𝑟^), which reduces to checking that 2𝑘≤𝑛+1−𝑟^. Finally, this inequality is equivalent to 𝑛≥2𝑘, which is true since 1−𝑟^>0 and 𝑟≥1/2.

It follows now from Theorem 5.5 that for all 𝑋_𝑟, 2 is of class 𝑟∈(−1-W-S.

Claim 2. If −1/2), 𝜖>0, then for all 𝑀, there exists an 𝑟, which depends on both 𝜖 and 𝑛≥2𝑘, such that for all 𝜔𝜔^𝑛_𝑘≤𝑀(2+𝜖)^𝑛𝑘𝑛.(6.21), we have𝑛=2𝑘Indeed, we can see as before that for 𝜔𝜔^2𝑘_𝑘=2𝑘𝑘4+22𝑟^−1𝑘−𝑟^−𝑟^(1−𝑟^)×4+22𝑟_(𝑘−𝑟^)²−1𝑘−1−𝑟^−𝑟^(1−𝑟^)×4+22𝑟_(𝑘−1−𝑟^)²⋯−11−𝑟^−𝑟^(1−𝑟^)(1−𝑟^)².(6.22), we have4+2(2𝑟^−1)/(𝑗−1−𝑟^)−𝑟^(1−𝑟^)/(𝑗−1−𝑟^)²→4Since 𝑗→∞, as 𝑁, there exists a natural number 𝑗≥𝑁, such that for all 4+2(2𝑟^−1)/(𝑗−1−𝑟^)−𝑟^(1−𝑟^)/(𝑗−1−𝑟^)²≤(2+𝜖)², 𝑀∶=[1/(2+𝜖)^2|𝐽|]∏_𝑗∈𝐽[4+2(2𝑟^−1)/(𝑗−1−𝑟^)−𝑟^(1−𝑟^)/(𝑗−1−𝑟^)²] . Let 𝐽∶={𝑗∈{1,2,…, where ,𝑁−1}|4+2(2𝑟^−1)/(𝑗−1−𝑟^)−𝑟^(1−𝑟^)/(𝑗−1−𝑟^)²>(2+𝜖)²}|𝐽|, and 𝐽 denotes the cardinality of 𝑘≥1. Then, for all 𝜔𝜔^2𝑘_𝑘≤𝑀(2+𝜖)^2𝑘2𝑘𝑘.(6.23), we have𝑘≥1Now, if we fix 𝑛, we can prove, as before by induction on 𝑛≥2𝑘 that for all 𝑟∈(−1, the inequality (6.21) holds.

Therefore, for any −1/2), 𝜖>0 and any 𝑀>0, there exists 𝑋_𝑟, such that 𝑀 is of class (2+𝜖, 𝑟∈(−1)-W-H.

Claim 3. For all −1/2), 𝑋_𝑟, 1/√𝑟(1−𝑟) is of class 𝑞(𝑥)=(2+𝑟^𝑥)[2−(1−𝑟^)𝑥]-W-H.

To see this, let us observe that since the roots of the trinomial −2/𝑟^ are 2/(1−𝑟^) and 𝑥_𝑣=1/2[2/(1−𝑟^)−2/𝑟^]=1/(1−𝑟^)−1/𝑟^, the maximum of this trinomial is attained at 𝑞(𝑥_𝑣)=1/[𝑟^(1−𝑟^)] and is equal to 𝑥. The closer the value of 𝑥_𝑣 to 𝑞(𝑥), the greater the value of 𝑘≥1. As we saw before for all 𝜔𝜔^2𝑘_𝑘=×2𝑘𝑘2+2+𝑟_𝑘−𝑟^𝑟^2−𝑘1−−𝑟𝑟^×2+𝑘−𝑟_1−𝑟^2−𝑘−1−1−𝑟^𝑟^⋯=2𝑘𝑘𝑞1−1𝑟^2−11−−𝑟𝑟^≤2𝑘𝑘𝑞(𝑘𝑥−_𝑣𝑟^𝑞𝑘−11−𝑟^⋯𝑞1−1𝑟^)^𝑘=2𝑘𝑘1√𝑟^(1−𝑟^)^2𝑘.(6.24), we have1/√𝑟^(1−𝑟^)≥2Since 𝑛, we can prove now by induction on 𝑛≥2𝑘 that for all 𝜔𝜔^𝑛_𝑘≤𝑡^𝑛𝑘𝑛,(6.25), we have𝑡∶=1/√𝑟^(1−𝑟^)where 𝑋_𝑟. It follows now from Corollary 5.6 that 𝑡 is of class 𝑡-W-H.

One can improve a little bit this −1<𝑟<−1/2, by observing that for all 1/2<𝑟^<1 (or, equivalently, 11−𝑟^>𝑥_𝑣>12−𝑟^>13−𝑟^>⋯.(6.26)), we have1/2<𝑟^<(7−√17)/4A simple computation shows that for 1/(2−𝑟^), 𝑥_𝑣 is closer to 1/(1−𝑟^) than 𝑥_𝑣 is to 𝑡∶=√𝑞(1/(2−𝑟^)). Therefore, in this case, we can choose a smaller (7−√17)/4≤𝑟^<1. If 1/(1−𝑟^)−𝑥_𝑣≤𝑥_𝑣−1/(2−𝑟^), then we have that 𝑡∶=√𝑞(1/(1−𝑟^)), and thus we can take 𝑋_𝑟 to conclude that 𝑡 is of class lim_{𝑟→(−1)}⁺𝜔₁=0-W-H.

Since lim_{𝑟→(−1)}⁺𝜔₂=2>0 and {𝑋_𝑟}, we can conclude, as in Example 6.5, that the family 𝑟^0>−1 is not unif.-W-H. On the other hand, since for any {1/√𝑟^(1−𝑟^)∣1/2<𝑟^≤𝑟^0}, the set 𝐵⊂(−1 is bounded above, we can see that for any ∞), {𝑋_𝑟}_𝑟∈𝐵, the family inf𝐵>−1 is unif.-W-H if and only if 𝑟>−1.

Claim 4. For any 𝑀, if 𝑡 and 𝑋_𝑟 are positive numbers, such that 𝑀 is of class (𝑡, 𝑡≥2)-W-H, then lim_𝑛→∞^𝑛⎷𝜔𝜔^2𝑛_𝑛=lim^𝑛→∞(^{𝜔2(𝑛+1)}_𝜔𝑛+1)=lim_𝑛→∞(^𝜔2𝑛_𝜔𝑛)(2𝑛+2)(2𝑛+1)(2𝑛+2+𝑟)(2𝑛+1+𝑟)(𝑛+1)²(𝑛+1+𝑡)²=16.(6.27).

To see this, let us observe first that by D'Alembert ratio theorem, we have𝑝=𝑞=2𝑡On the other hand, since for 𝐸[|𝑓_𝑛(𝑋_𝑟)⋄𝑓_𝑛(𝑋_𝑟)|²]≤𝑀𝐸[|Γ(√2𝑡)𝑓_𝑛(𝑋_𝑟)|²]², 𝑓_𝑛(𝑋_𝑟)⋄𝑓_𝑛(𝑋_𝑟)=𝑓_2𝑛(𝑋_𝑟) and 𝑛≥1, we conclude that for all 𝜔𝜔^2𝑛_𝑛≤𝑀(2𝑡)^2𝑛.(6.28),𝑛Taking radical or order 𝑛→∞ from both sides of this inequality and passing to the limit as 16≤(2𝑡)², we obtain that 𝑡≥2. Thus, we get 𝑟≥−1/2.

This last claim proves that at least for 𝑡, we got the best possible 2, which is 𝑋_𝑟, for the Wick-Hölder class of 𝑋_{𝛼,𝛽,𝑡}.

Let us consider now the family of nondegenerate centered Meixner random variables: 𝛼_𝑛=𝛼𝑛 whose Szegö-Jacobi parameters are 𝑛≥0 for all 𝜔_𝑛=𝛽𝑛²+(𝑡−𝛽)𝑛, and 𝑛≥1 for all 𝛼, where the parameters 𝛽, 𝑡, and 𝛼∈ℝ satisfy the conditions: 𝛽≥0, 𝑡>0, and 𝛽=0. We distinguish between two cases.

Case 1. If 𝑋_𝛼,0,𝑡, (i.e., 𝜔_𝑛=𝑡𝑛 is gaussian or poissonian), then 𝑛≥1 for all {𝜔_𝑛}_𝑛≥1, and therefore, 𝑋_𝛼,0,𝑡 is additive. In this case, 1 is of class 𝛽>0-W-H.

Case 2. If 𝜔_𝑛=𝛽𝜔^𝑛, then, 𝜔^𝑛=𝑛²+[(𝑡/𝛽)−1]𝑛, where (^𝜔𝑛_𝜔𝑘)=(^𝜔′𝑛_𝜔′𝑘), we can see that 𝑛≥𝑘≥0 for all 𝑋_{𝛼,𝛽,𝑡}. Therefore, 𝑋_𝑟 satisfies the same Wick-Hölder inequality as 𝑟=𝑡/𝛽, where 𝑡/𝛽=∞.

Defining 𝛽=0 for {𝑋_{𝛼,𝛽,𝑡}}_{𝛼∈ℝ,𝛽≥0,𝑡>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 𝑋_𝛼,0,𝑡 be the family of Meixner distributions. Then,
(1) for 1 is of class 𝛼∈ℝ-W-H for all 𝑡>0, and 𝛽>0;(2) if 𝑡/𝛽≥1/2 and 𝑋_{𝛼,𝛽,𝑡}, then 2 is of class 2-W-H, and 𝛽>0 is optimal;(3) if 𝑡/𝛽<1/2 and 𝑋_{𝛼,𝛽,𝑡}, then 1/√𝑟^(1−𝑟^) is of class 𝑟^=1−(𝑡/𝛽)-W-H, where {𝑋_{𝛼,𝛽,𝑡}}_{(𝛼,𝛽,𝑡)∈𝐼};(4) a family inf{𝑡/𝛽∣∃𝛼∈ℝ,(𝛼,𝛽,𝑡)∈𝐼}>0 of Meixner distributions is uniformly Wick Hölderian if and only if 𝑋_𝑞.

Example 6.7. Let 𝑞 be the 𝑞-Gaussian random variable with parameter 𝑞∈[−1, where 1], 𝑞∈(−1. It means that for any 1), 𝑋_𝑞, 𝛼_𝑛=0 is a symmetric random variable (so, 𝑛≥0 for all 𝜔_𝑛=(1−𝑞^𝑛)/(1−𝑞)) and 𝑛≥1 for all 𝑞=1. For 𝑋₁, 𝜔_𝑛=lim_𝑞→1⁻(1−𝑞^𝑛)/(1−𝑞)=𝑛 is the standard Gaussian since 𝑛≥1 for all 𝑞=−1. For 𝜔₁=1, 𝜔_𝑛=0 and 𝑛≥2 for all 𝑋₋₁. Thus, (1/2)[𝛿_{−1}+𝛿_{1}] is the centered Bernoulli random variable lim_𝑞→−1⁺𝜔_2𝑛=0. Despite the fact that 𝑛≥1 for all {𝑋_𝑞}_{−1≤𝑞≤1}, the family 𝑞 is uniformly Wick-Hölderian. In fact, every {𝜔_𝑛}_𝑛≥1-Gaussian random variable satisfies the best Wick-Hölder inequality since −1≤𝑞≤1 is subadditive for all 𝑞=−1. This statement is trivial for 𝑞=1 and −1<𝑞<1. For 𝑚 and all 𝑛 and 𝜔_𝑚+𝑛−𝜔_𝑚−𝜔_𝑛=1−𝑞^𝑚+𝑛1−𝑞−11−−𝑞𝑞^𝑚−11−−𝑞𝑞^𝑛=1−𝑞^𝑚𝑞^𝑛−11−𝑞<0.(6.29) natural numbers, we have

Acknowledgment

The research is supported by the NSF Grant no. DMS 0400526.