Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics and Stochastic Analysis
VolumeΒ 2008Β (2008), Article IDΒ 254897, 22 pages
Research Article

HΓΆlder-Type Inequalities for Norms of Wick Products

1Dipartimento di Matematica, UniversitΓ  di Bari, Campus Universitario, Via E. Orabona 4, 70125 Bari, Italy
2Department of Mathematics, Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion, OH 43302, USA

Received 3 November 2007; Revised 21 January 2008; Accepted 26 February 2008

Academic Editor: EnzoΒ Orsingher

Copyright Β© 2008 Alberto Lanconelli and Aurel I. Stan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 π‘ž, such that (1/𝑝)+(1/π‘ž)=1, any normally distributed random variable 𝑋, and any 𝑓 and 𝑔 complex-valued Borel measurable functions, such that both random variables βˆšΞ“(𝑝𝐼)𝑓(𝑋) and βˆšΞ“(π‘žπΌ)(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|||βˆšβ‰€πΈΞ“(|||𝑝𝐼)𝑓(𝑋)2𝐸|||βˆšΞ“(|||π‘žπΌ)𝑔(𝑋)2ξ‚„.(1.1) Here Ξ“ 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/π‘ž)=1 strongly suggest that (1.1) is in fact a HΓΆlder-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 π‘ž. 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/π‘ž)=1 cannot be improved. In Section 2, we present a minimal background about the SzegΓΆ-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 (Ξ©, β„±, 𝑃) be a probability space and π‘‹βˆΆΞ©β†’β„ 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 βŸ¨π‘“(𝑋),𝑔(𝑋)⟩∢=𝐸[𝑓(𝑋)𝑔(𝑋)] for all 𝑓, π‘”βˆΆβ„β†’β„‚ measurable, such that 𝐸[|𝑓(𝑋)|2]<∞ and 𝐸[|𝑔(𝑋)|2]<∞. Also, 𝑓0,𝑓1,𝑓2,… are polynomial functions chosen, such that for all 𝑛β‰₯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 π‘˜ different values π‘Ž1,π‘Ž2,…,π‘Žπ‘˜ with positive probabilities, then 𝑓𝑛 is the null polynomial for all 𝑛β‰₯π‘˜. 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 𝑛β‰₯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}𝑛β‰₯0 and {πœ”π‘›}𝑛β‰₯1, such that for all 𝑛β‰₯0,𝑋𝑓𝑛(𝑋)=𝑓𝑛+1(𝑋)+𝛼𝑛𝑓𝑛(𝑋)+πœ”π‘›π‘“π‘›βˆ’1(𝑋).(2.1)When 𝑛=0, π‘“π‘›βˆ’1=π‘“βˆ’1∢=0 (the null polynomial) and we can choose πœ”0∢=0. Also, if 𝑋 is a discrete random variable taking on only π‘˜ different values with positive probabilities, then for 𝑛=π‘˜βˆ’1, the equality (2.1) must be understood in the almost-sure sense, and we can choose 𝛼𝑛=0 and πœ”π‘›=0 for all 𝑛β‰₯π‘˜. The sequences {𝛼𝑛}𝑛β‰₯0 and {πœ”π‘›}𝑛β‰₯1 are called the SzegΓΆ-Jacobi parameters of 𝑋. Moreover, πœ”1,πœ”2,… are called the principal SzegΓΆ-Jacobi parameters of 𝑋. It is well known that for all 𝑛β‰₯1, 𝐸[𝑓2𝑛(𝑋)]=πœ”1πœ”2β‹―πœ”π‘› (see, e.g., [2, 3]).

Let 𝑁=π‘˜ if 𝑋 is discrete and takes on only π‘˜ values with positive probabilities, and 𝑁=∞ otherwise. We define the Hilbert space βˆ‘β„‹βˆΆ={𝑛<𝑁𝑐𝑛𝑓𝑛(𝑋)∣{𝑐𝑛}𝑛<π‘βˆ‘βŠ‚β„‚,𝑛<π‘πœ”π‘›!|𝑐𝑛|2<∞}, where πœ”π‘›!∢=πœ”1πœ”2β€¦πœ”π‘› for all 1≀𝑛<𝑁 and πœ”0!∢=1. β„‹ is in fact the closure of the space 𝐹∢={𝑓(𝑋)βˆ£π‘“ispolynomial} in 𝐿2(Ξ©, β„±, 𝑃). For many classic probability measures β„‹=𝐿2(Ξ©, 𝜎(𝑋), 𝑃), where 𝜎(𝑋) denotes the smallest sub-sigma-algebra of β„± 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 πΊπ‘›βˆΆ=πΉπ‘›βŠ–πΉπ‘›βˆ’1, that is, 𝐺𝑛 is the orthogonal complement of πΉπ‘›βˆ’1 into 𝐹𝑛 for all 𝑛β‰₯0. For convenience, we define πΉβˆ’1 and πΊβˆ’1 to be the null space. For all 𝑛β‰₯0, 𝐺𝑛=ℝ𝑓𝑛 and 𝐺𝑛 is called the homogenous chaos space of order 𝑛 generated by 𝑋. We will also call β„‹ the chaos space generated by 𝑋.

For any π‘š,𝑛β‰₯0, we define the Wick product π‘“π‘š(𝑋)⋄𝑓𝑛(𝑋) of π‘“π‘š(𝑋) and 𝑓𝑛(𝑋), as π‘“π‘š(𝑋)⋄𝑓𝑛(𝑋)∢=π‘“π‘š+𝑛(𝑋). Observe that if 𝑁=π‘˜ is finite, then π‘“π‘š(𝑋)⋄𝑓𝑛(𝑋)=0 for all π‘š and 𝑛, such that π‘š+𝑛β‰₯π‘˜. It is not hard to see that π‘“π‘š(𝑋)⋄𝑓𝑛(𝑋) is in fact the projection of the point-wise product π‘“π‘š(𝑋)𝑓𝑛(𝑋) on the space πΊπ‘š+𝑛. We extend now the Wick product by bilinearity, defining formally for all βˆ‘πœ‘=𝑛<π‘π‘π‘›π‘“π‘›βˆˆβ„‹ and βˆ‘πœ“=𝑛<π‘π‘‘π‘›π‘“π‘›βˆˆβ„‹,ξ“πœ‘β‹„πœ“βˆΆ=𝑛<𝑁(𝑝+π‘ž=π‘›π‘π‘π‘‘π‘ž)𝑓𝑛.(2.2) Since it is not guaranteed that βˆ‘π‘›<π‘πœ”π‘›βˆ‘!|𝑝+π‘ž=π‘›π‘π‘π‘‘π‘ž|2<∞, πœ‘β‹„πœ“ may not belong to β„‹.

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 β„‹, defined byΓ(𝑐𝐼)(0≀𝑛<𝑁𝑑𝑛𝑓𝑛(𝑋))∢=0≀𝑛<𝑁𝑐𝑛𝑑𝑛𝑓𝑛(𝑋),(2.3)where π‘‘π‘›βˆˆβ„‚ for all 0≀𝑛<𝑁.

A random variable βˆ‘πœ‘βˆΆ=0≀𝑛<𝑁𝑑𝑛𝑓𝑛(𝑋) belongs to the domain of Ξ“(𝑐𝐼) if and only if βˆ‘0≀𝑛<𝑁(1+|𝑐|2𝑛)|𝑑𝑛|2πœ”π‘›!<∞.

3. Wick-HΓΆlder Property

Definition 3.1. Let 𝑀 and 𝑑 be two fixed positive numbers. Let 𝑋 be a random variable, having finite moments of all orders, and let β„‹ denote the chaos space generated by 𝑋. 𝑋 is said to be (𝑀, 𝑑)-Wick-HΓΆlderian, if, for all positive numbers 𝑝 and π‘ž, such that (1/𝑝)+(1/π‘ž)=1/𝑑, and for all πœ‘(𝑋)βˆˆβ„‹ and πœ“(𝑋)βˆˆβ„‹, such that βˆšΞ“(𝑝𝐼)πœ‘(𝑋)βˆˆβ„‹ and βˆšΞ“(π‘žπΌ)πœ“(𝑋)βˆˆβ„‹, there exists πœ‘(𝑋)β‹„πœ“(𝑋)βˆˆβ„‹, and the following inequality holds:𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||βˆšβ‰€π‘€πΈΞ“(|||𝑝𝐼)πœ‘(𝑋)2𝐸|||βˆšΞ“(|||π‘žπΌ)πœ“(𝑋)2ξ‚„.(3.1)

Since for any fix πœ‘(𝑋)βˆˆβ„‹ the function π‘’βˆΆ[0, ∞)β†’[0, ∞], βˆšπ‘’(𝑑)=𝐸[|Ξ“(𝑑𝐼)πœ‘(𝑋)|2] is non-decreasing, if 𝑋 is an (𝑀,𝑑)-Wick-HΓΆlderian random variable, then 𝑋 is also (𝑀,π‘‘ξ…ž)-Wick-HΓΆlderian for all π‘‘ξ…ž>𝑑. By taking πœ‘=πœ“=1, we conclude from (3.1) that 𝑀β‰₯1.

Definition 3.2. Let 𝑑 be a fixed positive number. Let 𝑋 be a random variable, having finite moments of all orders. 𝑋 is said to be 𝑑-Wick-HΓΆlderian if 𝑋 is (1,𝑑)-Wick-HΓΆlderian.

Again, if 𝑋 is a 𝑑-Wick-HΓΆlderian random variable, then 𝑋 is also π‘‘ξ…ž-Wick-HΓΆlderian for all π‘‘ξ…ž>𝑑.

Definition 3.3. Let 𝑋 be a random variable, having finite moments of all orders. 𝑋 is said to be Wick-HΓΆlderian if there exists a positive number 𝑑, such that 𝑋 is 𝑑-Wick-HΓΆlderian.

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-HΓΆlderian;(2) there exist two positive numbers 𝑀 and 𝑑, such that 𝑋 is (𝑀,𝑑)-Wick-HΓΆlderian.

Proof. (1)β‡’(2) This implication is obvious.
(2)β‡’(1) Let us assume that 𝑋 is (𝑀, 𝑑)-Wick-HΓΆlderian for some 𝑀β‰₯1 and 𝑑>0. Let π‘ βˆΆ=max(3,3𝑀𝑑).
Claim 1. 𝑋 is 𝑠-Wick-HΓΆlderian.
Indeed, let 𝑝>0 and π‘ž>0, such that (1/𝑝)+(1/π‘ž)=1/𝑠. Let βˆ‘πœ‘(𝑋)=𝑛<𝑁𝑐𝑛𝑓𝑛(𝑋)βˆˆβ„‹, and βˆ‘πœ“(𝑋)=𝑛<𝑁𝑑𝑛𝑓𝑛(𝑋)βˆˆβ„‹, such that βˆšΞ“(𝑝𝐼)πœ‘(𝑋)βˆˆβ„‹ and βˆšΞ“(π‘žπΌ)πœ“(𝑋)βˆˆβ„‹, where {𝑐𝑛}𝑛β‰₯0βŠ‚β„‚, {𝑑𝑛}𝑛β‰₯0βŠ‚β„‚, and {𝑓𝑛}𝑛β‰₯0 represents the sequence of orthogonal polynomials, having a leading coefficient equal to 1, generated by 𝑋. Let β€–β‹…β€– denote the 𝐿2-norm. Let 𝑔(𝑋)∢=πœ‘(𝑋)βˆ’π‘01 and β„Ž(𝑋)∢=πœ“(𝑋)βˆ’π‘‘01. We have πœ‘(𝑋)=𝑐01+𝑔(𝑋) and πœ“(𝑋)=𝑑01+β„Ž(𝑋). Since 𝑔(𝑋)βŸ‚1, β„Ž(𝑋)βŸ‚1, and 𝑔(𝑋)β‹„β„Ž(𝑋)βŸ‚1, where βŸ‚ denotes the orthogonality relation, applying the Pythagorean theorem, we obtainβ€–β€–β€–β€–πœ‘(𝑋)β‹„πœ“(𝑋)2=‖‖𝑐0⋄𝑑1+𝑔(𝑋)0ξ‚„β€–β€–1+β„Ž(𝑋)2=‖‖𝑐0𝑑01+𝑐0β„Ž(𝑋)+𝑑0‖‖𝑔(𝑋)+𝑔(𝑋)β‹„β„Ž(𝑋)2=|||𝑐0|||2|||𝑑0|||2+‖‖𝑐0β„Ž(𝑋)+𝑑0‖‖𝑔(𝑋)+𝑔(𝑋)β‹„β„Ž(𝑋)2≀|||𝑐0|||2|||𝑑0|||2+|||𝑐0|||β€–β€–β€–β€–+|||π‘‘β„Ž(𝑋)0|||β€–β€–β€–β€–+‖‖‖‖𝑔(𝑋)𝑔(𝑋)β‹„β„Ž(𝑋)2≀|||𝑐0|||2|||𝑑0|||2+|||𝑐0|||β€–β€–β€–β€–+|||π‘‘β„Ž(𝑋)0|||β€–β€–β€–β€–+‖‖‖‖𝑔(𝑋)𝑔(𝑋)β‹„β„Ž(𝑋)2β€–β€–β€–β€–+3𝑔(𝑋)β‹„β„Ž(𝑋)2.(3.2)Because (1/𝑝)+(1/π‘ž)=1/𝑠≀1/3, we have 1/𝑝<1/3 and 1/π‘ž<1/3. Thus, 𝑝>3 and π‘ž>3. However, 𝑝>3 implies 3‖𝑔(𝑋)β€–2≀𝑝‖𝑔(𝑋)β€–2βˆšβ‰€β€–Ξ“(𝑝𝐼)𝑔(𝑋)β€–2. Similarly, we have 3β€–β„Ž(𝑋)β€–2βˆšβ‰€β€–Ξ“(π‘žπΌ)β„Ž(𝑋)β€–2. Since 1/[𝑝/(3𝑀)]+1/[π‘ž/(3𝑀)]=3𝑀/𝑠≀1/𝑑 and 𝑋 is an (𝑀,𝑑)-Wick-HΓΆlderian random variable, we haveβ€–β€–β€–β€–β€–β€–β€–Ξ“ξ‚€βˆšπ‘”(𝑋)β‹„β„Ž(𝑋)β‰€π‘€ξ‚β€–β€–β€–β€–β€–β€–Ξ“ξ‚€βˆšπ‘/(3𝑀)𝐼𝑔(𝑋)ξ‚β€–β€–β€–π‘ž/(3𝑀)πΌβ„Ž(𝑋).(3.3)Thus, since 3<9 and 𝑀β‰₯1, we haveβ€–β€–β€–β€–πœ‘(𝑋)β‹„πœ“(𝑋)2≀|||𝑐0|||2|||𝑑0|||2|||𝑐+30|||2β€–β€–β€–β€–β„Ž(𝑋)2|||𝑑+30|||2‖‖‖‖𝑔(𝑋)2β€–β€–β€–β€–+9𝑔(𝑋)β‹„β„Ž(𝑋)2≀|||𝑐0|||2|||𝑑0|||2+|||𝑐0|||2β€–β€–βˆšΞ“(β€–β€–π‘žπΌ)β„Ž(𝑋)2+|||𝑑0|||2β€–β€–βˆšΞ“(‖‖𝑝𝐼)𝑔(𝑋)2+9𝑀2β€–β€–β€–Ξ“ξ‚€βˆšξ‚β€–β€–β€–π‘/(3𝑀)𝐼𝑔(𝑋)2β€–β€–β€–Ξ“ξ‚€βˆšξ‚β€–β€–β€–π‘ž/(3𝑀)πΌβ„Ž(𝑋)2≀||𝑐0||2||𝑑0||2+||𝑐0||2β€–β€–βˆšΞ“(β€–β€–π‘žπΌ)β„Ž(𝑋)2+||𝑑0||2β€–β€–βˆšΞ“(‖‖𝑝𝐼)𝑔(𝑋)2+β€–β€–β€–Ξ“ξ‚€βˆšξ‚β€–β€–β€–3𝑀⋅𝑝/(3𝑀)𝐼𝑔(𝑋)2β€–β€–β€–Ξ“ξ‚€βˆšξ‚β€–β€–β€–3π‘€β‹…π‘ž/(3𝑀)πΌβ„Ž(𝑋)2=|||𝑐0|||2β€–β€–βˆš1+Ξ“(‖‖𝑝𝐼)𝑔2|||𝑑0|||2β€–β€–βˆš1+Ξ“(β€–β€–π‘žπΌ)β„Ž2ξ‚„=β€–β€–βˆšΞ“(‖‖𝑝𝐼)πœ‘2β€–β€–βˆšΞ“(β€–β€–π‘žπΌ)πœ“2.(3.4)Hence, 𝑋 is 𝑠-Wick-HΓΆlderian.

Definition 3.5. Let {𝑋𝑖}π‘–βˆˆπΌ be a family of random variables, having finite moments of all orders. The family {𝑋𝑖}π‘–βˆˆπΌ is said to be uniformly Wick-HΓΆlderian if there exists a positive number 𝑑0, such that for all π‘–βˆˆπΌ, 𝑋𝑖 is 𝑑0-Wick-HΓΆlderian.

It follows from the proof of the previous proposition that a family {𝑋𝑖}π‘–βˆˆπΌ is uniformly Wick-HΓΆlderian if and only if there exists a pair (𝑀0,𝑑0) of positive numbers, such that for all π‘–βˆˆπΌ, 𝑋𝑖 is (𝑀0,𝑑0)-Wick-HΓΆlderian.

From now on, to make the notation easier, we say that a random variable 𝑋 is of class (𝑀,𝑑)-W-H, class 𝑑-W-H, or class W-H if 𝑋 is (𝑀,𝑑)-Wick-HΓΆlderian, 𝑑-Wick-HΓΆlderian, or Wick-HΓΆlderian, respectively. We also say that a uniformly Wick-HΓΆlderian family {𝑋𝑖}π‘–βˆˆπΌ, 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 π‘ž are two positive numbers, such that for all πœ‘(𝑋), πœ“(𝑋)βˆˆβ„‚π‘“0(𝑋)+ℂ𝑓1(𝑋) (i.e., πœ‘ and πœ“ are polynomial functions of degree at most 1), the following inequality holds: 𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||βˆšβ‰€πΈΞ“(|||𝑝𝐼)πœ‘(𝑋)2𝐸|||βˆšΞ“(|||π‘žπΌ)πœ“(𝑋)2ξ‚„,(4.1)then one must have: 1𝑝+1π‘žβ‰€1.(4.2)

Proof. The fact that the support of 𝑋 contains at least two points guarantees that 𝑓1(𝑋)β‰ 0. Let 𝑓2 be next orthogonal polynomial and let πœ”1 and πœ”2 be the first two principal SzegΓΆ-Jacobi parameters of 𝑋. We have πœ”1=‖𝑓1(𝑋)β€–2>0 and ‖𝑓2(𝑋)β€–2=πœ”2!=πœ”1πœ”2β‰₯0 (it is possible that 𝑓2(𝑋)=0, in which case πœ”2=0). As before, β€–β‹…β€– denotes the 𝐿2-norm. Let us apply inequality (4.1) to the random variables πœ‘(𝑋)=1+𝑐𝑓1(𝑋) and πœ“=1+π‘πœ†π‘“1(𝑋), where 𝑐 and πœ† are arbitrary real numbers, such that 𝑐≠0. Sinceξ‚ƒπœ‘(𝑋)β‹„πœ“(𝑋)=1+𝑐𝑓1⋄(𝑋)1+π‘πœ†π‘“1ξ‚„(𝑋)=1+𝑐(1+πœ†)𝑓1(𝑋)+𝑐2πœ†π‘“2(𝑋),(4.3)the inequality,𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||βˆšβ‰€πΈΞ“(|||𝑝𝐼)πœ‘(𝑋)2𝐸|||βˆšΞ“(|||π‘žπΌ)πœ“(𝑋)2ξ‚„,(4.4)means that1+𝑐2(1+πœ†)2πœ”1+𝑐4πœ†2πœ”1πœ”2≀1+𝑝𝑐2πœ”11+π‘žπ‘2πœ†2πœ”1ξ‚„(4.5)for all 𝑐≠0 and all πœ†βˆˆβ„. Subtracting first 1 from both sides of this inequality, and then dividing both sides of the resulting inequality, by the strictly positive number 𝑐2πœ”1, we conclude that the inequality,(1+πœ†)2+𝑐2πœ†2πœ”2≀𝑝+π‘žπœ†2+π‘π‘žπ‘2πœ†2πœ”1,(4.6)holds for all 𝑐≠0 and πœ†βˆˆβ„. Passing to the limit, as 𝑐→0, in the last inequality, we obtain(1+πœ†)2≀𝑝+π‘žπœ†2(4.7)for all real numbers πœ†. Moving all terms to the right, we conclude that the quadratic trinomial(π‘žβˆ’1)πœ†2βˆ’2πœ†+(π‘βˆ’1),(4.8)must be nonnegative for all real values of πœ†. Therefore, π‘ž must be greater than one, and the discriminant Ξ”=4βˆ’4(π‘βˆ’1)(π‘žβˆ’1) must be less than or equal to zero. This is equivalent to 1≀(π‘βˆ’1)(π‘žβˆ’1), which in turn means 𝑝+π‘žβ‰€π‘π‘ž. Dividing both sides of this inequality by the positive number π‘π‘ž, we conclude that1𝑝+1π‘žβ‰€1.(4.9)

Corollary 4.2. If 𝑋 is a nonconstant random variable of class 𝑑-W-H, then 𝑑 is at least 1.

We will call the condition (1/𝑝)+(1/π‘ž)≀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-HΓΆlder inequality. We know from [1] that every Gaussian random variable satisfies the best Wick-HΓΆlder 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 {𝑓𝑛}𝑛β‰₯0 and β„‹ be the sequence of orthogonal polynomials and chaos space generated by 𝑋, respectively. If π›ΌβˆΆ={𝛼𝑛}𝑛β‰₯0 is a sequence of complex numbers, then we denote by ℳ𝛼 the densely defined linear operator on β„‹ that maps 𝑓𝑛(𝑋)→𝛼𝑛𝑓𝑛(𝑋) for all 𝑛β‰₯0. We call the sequence 𝛼 the multiplier of the operator ℳ𝛼. It is clear that for all 𝑛β‰₯0, β„³π›ΌπΊπ‘›βŠ‚πΊπ‘›. 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. 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 𝑛β‰₯1 is fixed. Let 𝑓0=1, and 𝑓𝑛 be the orthogonal polynomial of degree 𝑛, generated by 𝑋. If 𝑐={𝑐𝑛}𝑛β‰₯0 and 𝑑={𝑑𝑛}𝑛β‰₯0 are two sequences of complex numbers, such that 𝑐0=𝑑0=1, and for all πœ‘(𝑋), πœ“(𝑋)βˆˆβ„‚π‘“0(𝑋)+ℂ𝑓𝑛(𝑋), the following inequality holds: 𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||ℳ≀𝐸𝑐|||πœ‘(𝑋)2𝐸|||ℳ𝑑|||πœ“(𝑋)2ξ‚„,(4.10)then one must have 1|||𝑐𝑛|||2+1|||𝑑𝑛|||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 πœ‘(𝑋) and πœ“(𝑋) is always bounded above by the product of the norms of β„³π‘πœ‘(𝑋) and β„³π‘‘πœ“(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 πΏπ‘Ÿ case for π‘Ÿβ‰₯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 πΏπ‘Ÿ 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(π‘₯)≀𝑔(π‘₯) for all π‘₯βˆˆβ„, and there exists an π‘₯0βˆˆβ„, such that 𝑓(π‘₯0)=𝑔(π‘₯0), then π‘“ξ…ž(π‘₯0)=π‘”ξ…ž(π‘₯0), and π‘“ξ…žξ…ž(π‘₯0)β‰€π‘”ξ…žξ…ž(π‘₯0).

Proof. This can be seen intuitively by drawing a picture for the graphs of 𝑓 and 𝑔, or of π‘“βˆ’π‘”, and mathematically by using the formulaπ‘“ξ…žξ…žξ‚€π‘₯0=limβ„Žβ†’0𝑓π‘₯0π‘₯+β„Ž+𝑓0π‘₯βˆ’β„Žβˆ’2𝑓0ξ‚β„Ž2,(4.12)and a similar relation for π‘”ξ…žξ…ž(π‘₯0).

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 π‘Ÿ is a real number, then the function β„Ž(π‘₯)=|π‘₯|π‘Ÿ is differentiable if and only if π‘Ÿ>1, in which case β„Žξ…ž(π‘₯)=π‘Ÿπ‘₯|π‘₯|π‘Ÿβˆ’2.

We are generalizing now Lemma 4.1 to powers π‘Ÿ>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 π‘Ÿβ‰₯2 be fixed. If 𝑝 and π‘ž are two positive numbers, such that for all πœ‘(𝑋), πœ“(𝑋)βˆˆβ„‚π‘“0(𝑋)+ℂ𝑓1(𝑋), the following inequality holds: 𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)π‘Ÿξ‚„ξ‚ƒ|||βˆšβ‰€πΈΞ“(|||𝑝𝐼)πœ‘(𝑋)π‘Ÿξ‚„πΈξ‚ƒ|||βˆšΞ“(|||π‘žπΌ)πœ“(𝑋)π‘Ÿξ‚„,(4.13)then one must have 1𝑝+1π‘žβ‰€1.(4.14)

Proof. Let πœ‘π‘(𝑋)∢=1+𝑐𝑓1(𝑋) and πœ“π‘,πœ†(𝑋)∢=1+π‘πœ†π‘“1(𝑋), where 𝑐 and πœ† are arbitrary real numbers. Let us consider the functions π‘”πœ†βˆΆβ„β†’β„, π‘”πœ†(𝑐)∢=𝐸[|πœ‘π‘(𝑋)β‹„πœ“π‘,πœ†(𝑋)|π‘Ÿ] and β„Žπœ†βˆΆβ„β†’β„, β„Žπœ†βˆš(𝑐)∢=𝐸[|Ξ“(𝑝𝐼)πœ‘π‘(𝑋)|π‘Ÿβˆš]𝐸[|Ξ“(π‘žπΌ)πœ“π‘,πœ†(𝑋)|π‘Ÿ]. Since π‘”πœ†(𝑐)β‰€β„Žπœ†(𝑐) for all π‘βˆˆβ„, and π‘”πœ†(0)=β„Žπœ†(0), according to Observation 1, we haveπ‘”πœ†ξ…žξ…ž(0)β‰€β„Žπœ†ξ…žξ…ž(0).(4.15)Because 𝑋 has finite moments of all orders, and π‘Ÿβ‰₯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 π‘βˆˆβ„,π‘”πœ†ξ…žξ…žξ‚ƒ|||(𝑐)=π‘Ÿ(π‘Ÿβˆ’1)𝐸1+𝑐(1+πœ†)𝑓1+𝑐2πœ†π‘“2|||π‘Ÿβˆ’2ξ‚€(1+πœ†)𝑓1+2π‘πœ†π‘“22|||+2π‘Ÿπœ†πΈ1+𝑐(1+πœ†)𝑓1+𝑐2πœ†π‘“2|||π‘Ÿβˆ’2ξ‚€1+𝑐(1+πœ†)𝑓1+𝑐2πœ†π‘“2𝑓2ξ‚„,β„Žπœ†ξ…žξ…žξ‚ƒ|||√(𝑐)=π‘Ÿ(π‘Ÿβˆ’1)𝑝𝐸1+𝑐𝑝𝑓1|||π‘Ÿβˆ’2𝑓21𝐸|||√1+π‘πœ†π‘žπ‘“1|||π‘Ÿξ‚„+2π‘Ÿ2πœ†βˆšξ‚ƒ|||βˆšπ‘π‘žπΈ1+𝑐𝑝𝑓1|||π‘Ÿβˆ’2√(1+𝑐𝑝𝑓1)𝑓1|||βˆšΓ—πΈ1+π‘πœ†π‘žπ‘“1|||π‘Ÿβˆ’2√(1+π‘πœ†π‘žπ‘“1)𝑓1ξ‚„+π‘Ÿ(π‘Ÿβˆ’1)π‘žπœ†2𝐸|||√1+π‘πœ†π‘žπ‘“1|||π‘Ÿβˆ’2𝑓21𝐸|||√1+𝑐𝑝𝑓1|||π‘Ÿξ‚„.(4.16) Setting 𝑐=0, we get nowπ‘”πœ†ξ…žξ…ž(0)=π‘Ÿ(π‘Ÿβˆ’1)(1+πœ†)2𝐸𝑓21𝑓+2π‘Ÿπœ†πΈ2ξ‚„,β„Žπœ†ξ…žξ…žξ‚ƒπ‘“(0)=π‘Ÿ(π‘Ÿβˆ’1)𝑝𝐸21ξ‚„+2π‘Ÿ2πœ†βˆšξ‚ƒπ‘“π‘π‘žπΈ1𝐸𝑓1ξ‚„+π‘Ÿ(π‘Ÿβˆ’1)π‘žπœ†2𝐸𝑓21ξ‚„.(4.17)Since 𝑓2(𝑋)βŸ‚π‘“0(𝑋), 𝑓1(𝑋)βŸ‚π‘“0(𝑋), and 𝑓0(𝑋)=1, we have 𝐸[𝑓2(𝑋)]=0 and 𝐸[𝑓1(𝑋)]=0. Thus, the π‘”πœ†ξ…žξ…ž(0)β‰€β„Žπœ†ξ…žξ…ž(0) becomes, after dividing both sides by the positive number π‘Ÿ(π‘Ÿβˆ’1)𝐸[𝑓21](1+πœ†)2≀𝑝+π‘žπœ†2(4.18)for all πœ†βˆˆβ„. Now moving all terms to the right and writing the condition that the discriminant of the quadratic function in πœ† be nonpositive, we obtain as before that1𝑝+1π‘žβ‰€1.(4.19)

Our technique for proving the universal minimal condition for π‘Ÿβ‰₯2 is similar to the proof of Theorem 3 from [4].

Finally, we can also see that for multipliers, the inequality𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)π‘Ÿξ‚„ξ‚ƒ|||ℳ≀𝐸𝑐|||πœ‘(𝑋)π‘Ÿξ‚„πΈξ‚ƒ|||ℳ𝑑|||πœ“(𝑋)π‘Ÿξ‚„,(4.20)for a fix π‘Ÿβ‰₯2, also implies the condition that1|||𝑐𝑛|||2+1|||𝑑𝑛|||2≀1(4.21)for all 𝑛β‰₯1.

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

We present first the following lemma.

Lemma 5.1. Let {πœ”π‘›}𝑛β‰₯1 be a sequence of positive numbers, such that there exist a number 𝑑, greater than or equal to 1, and a sequence {𝛼𝑛}𝑛β‰₯2, of nonnegative numbers, such that the series βˆ‘βˆžπ‘›=2𝛼𝑛 is convergent, and for all 𝑛 and π‘˜ natural numbers, with 𝑛>π‘˜, it holds that πœ”π‘›ξ‚€β‰€π‘‘1+π›Όπ‘›πœ”ξ‚ξ‚€π‘˜+πœ”π‘›βˆ’π‘˜ξ‚.(5.1)Then, for all nonnegative numbers π‘˜β‰₯π‘Ÿβ‰₯0, it holds that ξƒ©πœ”π‘˜πœ”π‘Ÿξƒͺβ‰€π‘€π‘‘π‘˜ξƒ©π‘˜π‘Ÿξƒͺ,(5.2)where βˆπ‘€βˆΆ=max{1,(1/𝑑)βˆžπ‘›=2(1+𝛼𝑛)}, (π‘˜π‘Ÿ)∢=(1β‹…2β‹―π‘˜)/[(1β‹…2β‹―π‘Ÿ)(1β‹…2β‹―(π‘˜βˆ’π‘Ÿ))], and (πœ”π‘˜πœ”π‘Ÿ)∢=(πœ”1πœ”2β‹―πœ”π‘˜)/[(πœ”1πœ”2β‹―πœ”π‘Ÿ)(πœ”1πœ”2β‹―πœ”π‘˜βˆ’π‘Ÿ)] for 0<π‘Ÿ<π‘˜. If π‘Ÿ=0 or π‘Ÿ=π‘˜, then (π‘˜π‘Ÿ)∢=1 and (πœ”π‘˜πœ”π‘Ÿ)∢=1.

Proof. Since 𝛼𝑛β‰₯0 for all 𝑛β‰₯2 and βˆ‘βˆžπ‘›=2𝛼𝑛<∞, we conclude that the product βˆβˆžπ‘›=2(1+𝛼𝑛) is convergent. For π‘˜=0, the inequality (5.2) is obvious since (πœ”0πœ”0)=(00)=1. We prove now by induction on π‘˜ that for all π‘˜β‰₯1, we haveξƒ©πœ”π‘˜πœ”π‘Ÿξƒͺ≀[π‘˜ξ‘π‘–=2ξ‚€1+𝛼𝑖]π‘‘π‘˜βˆ’1ξƒ©π‘˜π‘Ÿξƒͺ,(5.3)for all π‘Ÿβˆˆ{0,1,…,π‘˜}. For π‘˜=1, we have nothing to prove, since (πœ”π‘˜πœ”π‘Ÿ)=(π‘˜π‘Ÿ)=1, and βˆπ‘˜π‘–=2(1+𝛼𝑖) is defined to be 1 in this case.
Let us suppose now that (5.3) holds for π‘˜=𝑛 and all π‘Ÿβˆˆ{0,1,…,𝑛} and prove that it continues to hold for π‘˜=𝑛+1 and all π‘Ÿβˆˆ{0,1,…,𝑛+1}. Indeed, we may assume that 1β‰€π‘Ÿβ‰€π‘› since for π‘Ÿ=0 and π‘Ÿ=𝑛+1, (5.3) is trivial because 𝑑β‰₯1. It follows from (5.2) thatξƒ©πœ”π‘›+1πœ”π‘Ÿξƒͺ=πœ”π‘›+1πœ”π‘›!πœ”π‘Ÿπœ”π‘Ÿβˆ’1!πœ”π‘›+1βˆ’π‘Ÿπœ”π‘›βˆ’π‘Ÿ!≀𝑑1+𝛼𝑛+1πœ”ξ‚ξ‚ƒπ‘Ÿ+πœ”π‘›+1βˆ’π‘Ÿξ‚„πœ”π‘›!πœ”π‘Ÿπœ”π‘Ÿβˆ’1!πœ”π‘›+1βˆ’π‘Ÿπœ”π‘›βˆ’π‘Ÿ!ξ‚€=𝑑1+𝛼𝑛+1[πœ”π‘›!πœ”π‘Ÿβˆ’1!πœ”π‘›+1βˆ’π‘Ÿ!+πœ”π‘›!πœ”π‘Ÿ!πœ”π‘›βˆ’π‘Ÿ!]ξ‚€=𝑑1+𝛼𝑛+1[ξƒ©πœ”π‘›πœ”π‘Ÿβˆ’1ξƒͺ+ξƒ©πœ”π‘›πœ”π‘Ÿξƒͺ].(5.4)Using now this inequality, the induction hypothesis, and the classic Pascal identity (π‘›π‘Ÿβˆ’1)+(π‘›π‘Ÿ)=(π‘Ÿπ‘›+1), we getξƒ©πœ”π‘›+1πœ”π‘Ÿξƒͺ≀[𝑛+1𝑖=2ξ‚€1+𝛼𝑖]𝑑𝑛[𝑛ξƒͺ+ξƒ©π‘›π‘Ÿξƒͺ]π‘Ÿβˆ’1=[𝑛+1𝑖=2ξ‚€1+𝛼𝑖]π‘‘π‘›ξƒ©π‘Ÿξƒͺ.𝑛+1(5.5)Hence, for all π‘˜β‰₯1, we haveξƒ©πœ”π‘˜πœ”π‘Ÿξƒͺβ‰€ξ‚ƒπ‘˜ξ‘π‘–=2ξ‚€1+π›Όπ‘–π‘‘ξ‚ξ‚„π‘˜βˆ’1ξƒ©π‘˜π‘Ÿξƒͺβ‰€ξ‚ƒβˆžξ‘π‘–=2ξ‚€1+π›Όπ‘–π‘‘ξ‚ξ‚„π‘˜βˆ’1ξƒ©π‘˜π‘Ÿξƒͺ=1π‘‘ξ‚ƒβˆžξ‘π‘–=2ξ‚€1+π›Όπ‘–π‘‘ξ‚ξ‚„π‘˜ξƒ©π‘˜π‘Ÿξƒͺ.(5.6)In order to make this inequality also true, for π‘˜=0, we have to replace ∏(1/𝑑)[βˆžπ‘–=2(1+𝛼𝑖)] by ∏max{1,(1/𝑑)[βˆžπ‘–=2(1+𝛼𝑖)]}.

We introduce now three definitions.

Definition 5.2. Given a number 𝑑, greater than or equal to 1, and a sequence of nonnegative numbers 𝛼={𝛼𝑛}𝑛β‰₯2, such that the series βˆ‘βˆžπ‘›=2𝛼𝑛 is convergent, then a sequence of nonnegative numbers {πœ”π‘›}𝑛β‰₯1 is said to be (𝑑, 𝛼)-subadditive if the following inequality,πœ”π‘›ξ‚€β‰€π‘‘1+π›Όπ‘›πœ”ξ‚ξ‚€π‘˜+πœ”π‘›βˆ’π‘˜ξ‚,(5.7)holds for all natural numbers 𝑛 and π‘˜, such that 𝑛>π‘˜.

Definition 5.3. Given a number 𝑑, greater than or equal to 1, then a sequence of nonnegative numbers {πœ”π‘›}𝑛β‰₯1 is said to be 𝑑-subadditive, if it is (𝑑, 0)-subadditive, where 0={0,0,…} denotes the identical zero sequence, that is, if the following inequality,πœ”π‘š+π‘›ξ‚€πœ”β‰€π‘‘π‘š+πœ”π‘›ξ‚,(5.8)holds for all natural numbers π‘š and 𝑛. In particular, if 𝑑=1, then a 1-subadditive sequence is called simply subadditive.

Of course (𝑑,𝛼)-subadditivity implies π‘‘ξ…ž-subadditivity for π‘‘ξ…žβˆΆ=𝑑(1+sup{π›Όπ‘›βˆ£π‘›β‰₯2}).

Definition 5.4. If 𝑑 is a number, greater than or equal to 1, and {πœ”π‘›}𝑛β‰₯1 is a sequence of nonnegative numbers, then the sequence {πœ”π‘›}𝑛β‰₯1 is said to be exp-𝑑-subadditive ifπœ”1/π‘‘π‘š+π‘›β‰€πœ”π‘š1/𝑑+πœ”π‘›1/𝑑(5.9)for all π‘š and 𝑛 positive integers.

Observation 3. An exp-𝑑-subadditive sequence {πœ”π‘›}𝑛β‰₯1 of nonnegative numbers is also exp-𝑠-subadditive for all 𝑠>𝑑.

Proof. Applying the inequality (π‘Ž+𝑏)π‘Ÿβ‰€π‘Žπ‘Ÿ+π‘π‘Ÿ for all π‘Žβ‰₯0, 𝑏β‰₯0, and 0<π‘Ÿβ‰€1, we getπœ”1/π‘ π‘š+𝑛=ξ‚ƒπœ”1/π‘‘π‘š+𝑛𝑑/π‘ β‰€ξ‚ƒπœ”π‘š1/𝑑+πœ”π‘›1/𝑑𝑑/π‘ β‰€ξ‚ƒπœ”π‘š1/𝑑𝑑/𝑠+ξ‚ƒπœ”π‘›1/𝑑𝑑/𝑠=πœ”π‘š1/𝑠+πœ”π‘›1/𝑠(5.10)for all π‘š and 𝑛 positive integers.

Observation 4. For all 𝑑β‰₯1, any exp-𝑑-subadditive sequence {πœ”π‘›}𝑛β‰₯1 of nonnegative numbers is also 2π‘‘βˆ’1-subadditive.

Proof. Since 𝑑β‰₯1, the function π‘“βˆΆ[0, ∞)→ℝ, 𝑓(π‘₯)=π‘₯𝑑 is convex. Thus, for all π‘š,𝑛β‰₯1, we haveπœ”π‘š+𝑛=ξ‚ƒπœ”1/π‘‘π‘š+π‘›ξ‚„π‘‘β‰€ξ€Ίπœ”π‘š1/𝑑+πœ”π‘›1/𝑑𝑑=2𝑑12πœ”π‘š1/𝑑+12πœ”π‘›1/𝑑𝑑≀2𝑑12ξ‚€πœ”π‘š1/𝑑𝑑+12ξ‚€πœ”π‘›1/𝑑𝑑=2π‘‘βˆ’1ξ‚€πœ”π‘š+πœ”π‘›ξ‚.(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 {πœ”π‘›}𝑛β‰₯1 be the principal SzegΓΆ-Jacobi parameters of 𝑋. Let 𝑀 and 𝑑 be two numbers that are greater than or equal to 1. If the sequence {πœ”π‘›}𝑛β‰₯1 satisfies the condition ξƒ©πœ”π‘˜πœ”π‘Ÿξƒͺβ‰€π‘€π‘‘π‘˜ξƒ©π‘˜π‘Ÿξƒͺ(5.12)for all 0β‰€π‘Ÿβ‰€π‘˜<𝑁, where 𝑁 denotes the dimension of the chaos space generated by 𝑋, then 𝑋 is of class (𝑀,𝑑)-W-H.

Proof. Let 𝑝>0 and π‘ž>0, such that (1/𝑝)+(1/π‘ž)=(1/𝑑). Let β„‹ be the chaos space generated by 𝑋. Let πœ‘βˆˆβ„‹ and πœ“βˆˆβ„‹, such that βˆšΞ“(𝑝𝐼)πœ‘βˆˆβ„‹ and βˆšΞ“(π‘žπΌ)πœ“βˆˆβ„‹. 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. If 𝑁=∞, then there exist two unique sequences of complex numbers {𝑐𝑛}𝑛β‰₯0 and {𝑑𝑛}𝑛β‰₯0, such thatπœ‘=βˆžξ“π‘›=0𝑐𝑛𝑓𝑛(𝑋),πœ“=βˆžξ“π‘›=0𝑑𝑛𝑓𝑛(𝑋).(5.13)We haveπœ‘β‹„πœ“=βˆžξ“π‘˜=0(𝑒+𝑣=π‘˜π‘π‘’π‘‘π‘£)π‘“π‘˜(𝑋).(5.14)Using the triangle inequality, we obtain𝐸||||||πœ‘β‹„πœ“2ξ‚„=βˆžξ“π‘˜=0|𝑒+𝑣=π‘˜π‘π‘’π‘‘π‘£|2πœ”π‘˜!β‰€βˆžξ“π‘˜=0[𝑒+𝑣=π‘˜|||𝑐𝑒𝑑𝑣|||]2πœ”π‘˜!=βˆžξ“π‘˜=0πœ”π‘˜ξ“![𝑒+𝑣=π‘˜1βˆšπœ”π‘’!πœ”π‘£!π‘π‘’π‘žπ‘£βˆšπ‘π‘’πœ”π‘’!|||𝑐𝑒|||βˆšπ‘žπ‘£πœ”π‘£!|||𝑑𝑣|||]2(5.15)From the Cauchy-Bunyakovsky-Schwarz inequality, inequality (5.2), and Newton's binomial formula, we obtain𝐸||||||πœ‘β‹„πœ“2ξ‚„β‰€βˆžξ“π‘˜=0πœ”π‘˜ξ“![𝑒+𝑣=π‘˜1πœ”π‘’!πœ”π‘£!π‘π‘’π‘žπ‘£ξ“]Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=βˆžξ“π‘˜=0[𝑒+𝑣=π‘˜πœ”π‘˜!πœ”π‘’!πœ”π‘£!1𝑝𝑒1π‘žπ‘£ξ“]Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=βˆžξ“π‘˜=0[𝑒+𝑣=π‘˜ξƒ©πœ”π‘˜πœ”π‘’ξƒͺ1𝑝𝑒1π‘žπ‘£ξ“]Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]β‰€βˆžξ“π‘˜=0[𝑒+𝑣=π‘˜π‘€π‘‘π‘˜ξƒ©π‘˜π‘’ξƒͺ1𝑝𝑒1π‘žπ‘£ξ“]Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=π‘€βˆžξ“π‘˜=0π‘‘π‘˜[𝑒+𝑣=π‘˜ξƒ©π‘˜π‘’ξƒͺ1𝑝𝑒1π‘žπ‘£ξ“]Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=π‘€βˆžξ“π‘˜=0π‘‘π‘˜ξ‚ƒ1𝑝+1π‘žξ‚„π‘˜ξ“Γ—[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=π‘€βˆžξ“π‘˜=0π‘‘π‘˜ξ‚ƒ1π‘‘ξ‚„π‘˜[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=π‘€βˆžξ“π‘˜=01[𝑒+𝑣=π‘˜π‘π‘’πœ”π‘’!|||𝑐𝑒|||2π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2]=π‘€βˆžξ“π‘’=0π‘π‘’πœ”π‘’!|||𝑐𝑒|||2βˆžξ“π‘£=0π‘žπ‘£πœ”π‘£!|||𝑑𝑣|||2|||√=𝑀𝐸Γ(|||𝑝𝐼)𝑓2𝐸|||βˆšΞ“(|||π‘žπΌ)𝑔2ξ‚„.(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 πœ”'s become zero. Thus, for example, some complete sums like βˆ‘π‘’+𝑣=π‘˜(π‘˜π‘’)(1/𝑝𝑒)(1/π‘žπ‘£) from Case 1 will become incomplete (that means some of the pais (𝑒, 𝑣), with 𝑒+𝑣=π‘˜, will be missing) in Case 2, and therefore instead of βˆ‘π‘’+𝑣=π‘˜(π‘˜π‘’)(1/𝑝𝑒)(1/π‘žπ‘£)=1/π‘‘π‘˜, we will have βˆ‘π‘’+𝑣=π‘˜(π‘˜π‘’)(1/𝑝𝑒)(1/π‘žπ‘£)<1/π‘‘π‘˜. Therefore, all the inequalities from Case 1 will also remain true in Case 2. Hence, 𝑋 is of class (𝑀, 𝑑)-W-H in this case too.

Corollary 5.6. Let 𝑑 be a number greater than or equal to 1, and 𝛼={𝛼𝑛}𝑛β‰₯2 a sequence of nonnegative numbers producing a convergent series βˆ‘βˆžπ‘›=2𝛼𝑛. Let 𝑋 be a random variable, having finite moments of any order, and let {πœ”π‘›}𝑛β‰₯1 be the principal SzegΓΆ-Jacobi parameters of 𝑋. If the sequence {πœ”π‘›}𝑛β‰₯1 is (𝑑,𝛼)-subadditive, then 𝑋 is of class (𝑀,𝑑)-W-H, where βˆπ‘€βˆΆ=max{1,(1/𝑑)βˆžπ‘›=2(1+𝛼𝑛)}.

Corollary 5.7. If 𝑋 is a random variable, having finite moments of all orders, such that its principal SzegΓΆ-Jacobi sequence {πœ”π‘›}𝑛β‰₯1 is 𝑑-subadditive, then 𝑋 is of class 𝑑-W-H.

Corollary 5.8. If 𝑋 is a random variable, having finite moments of all orders, such that its principal SzegΓΆ-Jacobi sequence {πœ”π‘›}𝑛β‰₯1 is exp-𝑑-subadditive, then 𝑋 is of class 2π‘‘βˆ’1-W-H. In particular, by taking 𝑝=π‘ž=πœ†=2𝑑, it is concluded that if β„‹ denotes the chaos space generated by 𝑋, and if πœ‘(𝑋), πœ“(𝑋)βˆˆβ„‹, such that Ξ“(2𝑑/2𝐼)πœ‘(𝑋)βˆˆβ„‹ and Ξ“(2𝑑/2𝐼)πœ“(𝑋)βˆˆβ„‹, then πœ‘(𝑋)β‹„πœ“(𝑋)βˆˆβ„‹ and the following inequality holds: 𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||≀𝐸Γ(2𝑑/2|||𝐼)πœ‘(𝑋)2𝐸|||Ξ“(2𝑑/2|||𝐼)πœ“(𝑋)2ξ‚„.(5.17)

Observation 5. If 𝑝=π‘ž=2𝑑, then to prove that the inequality,
𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||βˆšβ‰€πΈΞ“(|||2𝑑𝐼)πœ‘(𝑋)2𝐸|||βˆšΞ“(|||2𝑑𝐼)πœ“(𝑋)2ξ‚„,(5.18)holds whenever both expectations from the right-hand side are finite, we do not need the condition that each πœ”-binomial coefficient (πœ”π‘˜πœ”π‘Ÿ) is less than or equal to π‘€π‘‘π‘˜(π‘˜π‘Ÿ), but it is enough to assume that for each π‘˜β‰₯0, the sum of all binomial coefficients, having πœ”π‘˜ on the top, is less than or equal to π‘€π‘‘π‘˜ times the sum of all classic binomial coefficients having π‘˜ on the top, that meansπ‘˜ξ“π‘Ÿ=0ξƒ©πœ”π‘˜πœ”π‘Ÿξƒͺβ‰€π‘€π‘‘π‘˜β‹…2π‘˜(5.19)for all π‘˜β‰₯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 (𝑒, 𝑣) such that 𝑒+𝑣 is equal to a fixed number π‘˜ since 𝑝=π‘ž=2𝑑, the product π‘π‘’π‘žπ‘£ is always the same (independent of the pair) and equal to (2𝑑)π‘˜.

The following proposition will be useful in showing later that πœ†=2𝑑 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 β„‹ be the chaos space generated by 𝑋 and {πœ”π‘›}𝑛β‰₯1 the principal SzegΓΆ-Jacobi sequence of 𝑋. We define π‘†π‘›βˆ‘βˆΆ=π‘›π‘˜=0(πœ”π‘›πœ”π‘˜) for all 𝑛β‰₯0, and πœ†0∢=limsupπ‘›β†’βˆžπ‘†π‘›1/𝑛. Then,
(1) if there exists a positive number πœ† such that the inequality,𝐸||||||πœ‘(𝑋)β‹„πœ“(𝑋)2|||βˆšβ‰€πΈΞ“(|||πœ†πΌ)πœ‘(𝑋)2𝐸|||βˆšΞ“(|||πœ†πΌ)πœ“(𝑋)2ξ‚„,(5.20)holds for all πœ‘βˆˆβ„‹ and πœ“βˆˆβ„‹, such that βˆšΞ“(πœ†πΌ)πœ‘βˆˆβ„‹ and βˆšΞ“(πœ†πΌ)πœ“βˆˆβ„‹, then πœ†β‰₯πœ†0;(2) if πœ†0 satisfies (5.20), then πœ†0 is optimal (i.e., the smallest among all positive πœ†'s satisfying this inequality);(3) if π‘†π‘›β‰€πœ†π‘›0 for all 𝑛β‰₯0, then πœ†0 is optimal.

Proof. (1) Let πœ†>0 such that (5.20) holds whenever both expectations from its right-hand side are finite. For every 0β‰€π‘Ÿβ‰€π‘˜, let us choose πœ‘βˆΆ=π‘“π‘Ÿ(𝑋) and πœ“βˆΆ=π‘“π‘˜βˆ’π‘Ÿ(𝑋). The inequality 𝐸[|πœ‘β‹„πœ“|2√]≀𝐸[|Ξ“(πœ†πΌ)πœ‘|2√]𝐸[|Ξ“(πœ†πΌ)πœ“|2] is equivalent to πœ”π‘˜!β‰€πœ†π‘Ÿπœ”π‘Ÿ!πœ†π‘˜βˆ’π‘Ÿπœ”π‘˜βˆ’π‘Ÿ!. This implies that (πœ”π‘˜πœ”π‘Ÿ)β‰€πœ†π‘˜. Summing up from π‘Ÿ=0 to π‘Ÿ=π‘˜, we obtain π‘†π‘˜β‰€(π‘˜+1)πœ†π‘˜ for all π‘˜β‰₯0. Taking the π‘˜th root in both sides of this inequality and then passing to the superior limit as π‘˜β†’βˆž, we get limsupπ‘˜β†’βˆžπ‘˜βˆšπ‘†β‰€πœ†. Thus, πœ†0β‰€πœ†.
(2) It follows from 1.(3) It follows from letting π‘€βˆΆ=1 and π‘‘βˆΆ=πœ†0/2 in Observation 5.

6. Examples

Example 6.1. Let 𝑠β‰₯1 be a fixed real number and let 𝑋 be a random variable, having finite moments of all orders, and the principal SzegΓΆ-Jacobi parameters πœ”π‘›=𝑛𝑠>0 for all 𝑛β‰₯1. It is clear that {πœ”π‘›}𝑛β‰₯1 is an exp-𝑠-subadditive (in fact exp-𝑠-additive) sequence. Therefore, {πœ”π‘›}𝑛β‰₯1 is also 2π‘ βˆ’1-subadditive. Thus, 𝑋 is of class 2π‘ βˆ’1-W-H. Moreover, the following lemma holds.

Lemma 6.2. For all 𝑠β‰₯1, it holds that [π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛≀2𝑠,(6.1)limπ‘›β†’βˆž[π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛=2𝑠.(6.2)0<𝑠<1Formula (6.2) holds even for 𝑠β‰₯1.

Proof. If [π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛≀{[π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ]𝑠}1/𝑛=2𝑛𝑠/𝑛=2𝑠.(6.3), then we have𝑠β‰₯1 Since β„ŽβˆΆ(0, the function ∞)→ℝ, β„Ž(π‘₯)=π‘₯𝑠, [π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛=𝑛1/𝑛[π‘›ξ“π‘˜=01π‘›ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛β‰₯𝑛1/𝑛{[π‘›ξ“π‘˜=01π‘›ξƒ©π‘›π‘˜ξƒͺ]𝑠}1/𝑛=𝑛1/𝑛2𝑛𝑛𝑠/𝑛=1ξ‚€π‘›βˆšπ‘›ξ‚π‘ βˆ’12𝑠.(6.4) is convex, and thus1ξ‚€π‘›βˆšπ‘›ξ‚π‘ βˆ’12𝑠≀[π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛≀2𝑠.(6.5) Hence,π‘›βˆšπ‘›β†’1Since π‘›β†’βˆž, as limπ‘›β†’βˆž[βˆ‘π‘›π‘˜=0(π‘›π‘˜)𝑠]1/𝑛, we conclude that 2𝑠 exists and is equal to 0<𝑠<1.
If 1ξ‚€π‘›βˆšπ‘›ξ‚π‘ βˆ’12𝑠β‰₯[π‘›ξ“π‘˜=0ξƒ©π‘›π‘˜ξƒͺ𝑠]1/𝑛β‰₯2𝑠,(6.6), then in a similar way, we can show thatlimπ‘›β†’βˆž[βˆ‘π‘›π‘˜=0(π‘›π‘˜)𝑠]1/𝑛=2𝑠and 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 πœ”π‘›=𝑛𝑠 be a random variable, having finite moments of any order whose principal SzegΓΆ-Jacobi parameters are 𝑛β‰₯1 for all 𝑠, where 𝑠β‰₯1 is a fixed real number, such that β„‹. Let 𝑋 be the chaos space generated by 𝑝.
(1) If π‘ž and (1/𝑝)+(1/π‘ž)≀(1/2π‘ βˆ’1) are positive numbers, such that πœ‘βˆˆβ„‹, then for all πœ“βˆˆβ„‹ and βˆšΞ“(𝑝𝐼)πœ‘βˆˆβ„‹, such that βˆšΞ“(π‘žπΌ)πœ“βˆˆβ„‹ and 𝑓(𝑋)⋄𝑔(𝑋)βˆˆβ„‹, there exists 𝐸||||||πœ‘β‹„πœ“2|||βˆšβ‰€πΈΞ“(|||𝑝𝐼)πœ‘2𝐸|||βˆšΞ“(|||π‘žπΌ)πœ“2ξ‚„.(6.7) and the following inequality holds:πœ†0∢=2𝑠(2)πœ† is the smallest among all positive numbers 𝐸||||||πœ‘β‹„πœ“2|||βˆšβ‰€πΈΞ“(|||πœ†πΌ)πœ‘2𝐸|||βˆšΞ“(|||πœ†πΌ)πœ“2ξ‚„,(6.8), for which the inequality,πœ‘βˆˆβ„‹holds for all πœ“βˆˆβ„‹ and βˆšΞ“(πœ†πΌ)πœ‘βˆˆβ„‹, such that βˆšΞ“(πœ†πΌ)πœ“βˆˆβ„‹ and 𝑠.

Example 6.4. This is a modification of the previous example. Let 𝑠>2 be a fixed real number, such that π‘‹π‘Ÿ. Let πœ”π‘›=𝑛𝑠+π‘Ÿπ‘› be a random variable whose principal SzegΓΆ-Jacobi parameters are 𝑛β‰₯0 for all π‘Ÿ>βˆ’1, where π‘Ÿ>βˆ’1. The condition πœ”1>0 follows from the inequality πœ”π‘›>0, and ensures that 𝑛β‰₯2 for all 𝑠>2. The reason why we have chosen π‘Ÿβ‰₯0 will become more transparent later.

Claim 1. If {πœ”π‘›}𝑛β‰₯1, then the sequence 2π‘ βˆ’1 is π‘š-subadditive.

Indeed, for all 𝑛 and (π‘š+𝑛)𝑠≀2π‘ βˆ’1ξ‚€π‘šπ‘ +𝑛𝑠,π‘š+𝑛<2π‘ βˆ’1ξ€·ξ€Έ.π‘š+𝑛(6.9) natural numbers, we haveπ‘ŸMultiplying the second inequality by πœ”π‘š+𝑛≀2π‘ βˆ’1ξ‚€πœ”π‘š+πœ”π‘›ξ‚.(6.10) and adding the resulting inequality to the first one, we getβˆ’1<π‘Ÿ<0

Claim 2. If {πœ”π‘›}𝑛β‰₯1, then the sequence 2π‘ βˆ’1 is not π‘Ÿξ…žβˆΆ=βˆ’π‘Ÿ>0-subadditive.

Indeed, if we define 𝑛β‰₯1, then for all πœ”2𝑛=(2𝑛)π‘ βˆ’2π‘›π‘Ÿξ…ž=2π‘ βˆ’1ξ‚€π‘›π‘ βˆ’π‘Ÿξ…žπ‘›+π‘›π‘ βˆ’π‘Ÿξ…žπ‘›ξ‚ξ‚€2+2π‘›π‘ βˆ’1ξ‚π‘Ÿβˆ’1ξ…ž=2π‘ βˆ’1ξ‚€πœ”π‘›+πœ”π‘›ξ‚ξ‚€2+2π‘›π‘ βˆ’1ξ‚π‘Ÿβˆ’1ξ…ž>2π‘ βˆ’1ξ‚€πœ”π‘›+πœ”π‘›ξ‚.(6.11), we haveβˆ’1<π‘Ÿ<0

Claim 3. If 𝛼={𝛼𝑛}𝑛β‰₯2, then there exists a sequence βˆ‘βˆžπ‘›=2𝛼𝑛 of positive numbers, such that {πœ”π‘›}𝑛β‰₯1 is convergent, and the sequence 2π‘ βˆ’1 is (𝛼, π‘Ÿξ…ž=βˆ’π‘Ÿβˆˆ(0)-subadditive.

As before, let 1), 𝑛β‰₯2. Let 𝛼𝑛 be a fixed natural number. Let us find a positive number πœ”π‘›β‰€ξ‚€1+𝛼𝑛2π‘ βˆ’1ξ‚€πœ”π‘˜+πœ”π‘›βˆ’π‘˜ξ‚(6.12), such that1β‰€π‘˜<𝑛for all π‘›π‘ βˆ’π‘Ÿξ…žξ‚€π‘›β‰€1+𝛼𝑛2π‘ βˆ’1ξ‚ƒπ‘˜π‘ +(π‘›βˆ’π‘˜)π‘ ξ‚„βˆ’ξ‚€1+𝛼𝑛2π‘ βˆ’1π‘Ÿξ…žπ‘›(6.13). This inequality is equivalent to1β‰€π‘˜<𝑛for all 𝑛𝑠≀2π‘ βˆ’1[π‘˜π‘ +(π‘›βˆ’π‘˜)𝑠]. Since 𝛼𝑛, if we choose π‘›π‘ βˆ’π‘Ÿξ…žξ‚€π‘›=1+π›Όπ‘›ξ‚π‘›π‘ βˆ’ξ‚€1+𝛼𝑛2π‘ βˆ’1π‘Ÿξ…žπ‘›,(6.14) such that1β‰€π‘˜<𝑛then the last inequality holds, for all 𝑛. In fact, for π‘˜=π‘›βˆ’π‘˜ even, if we choose π‘˜=𝑛/2 (that means 𝛼𝑛), then we can see that 𝛼𝑛 cannot be chosen smaller than the value of the solution of (6.14). Solving (6.14) for 𝛼𝑛=π‘Ÿξ…žξ‚€2π‘ βˆ’1ξ‚βˆ’1π‘›π‘ βˆ’1βˆ’2π‘ βˆ’1π‘Ÿξ…ž.(6.15), we get𝑛β‰₯2Since 0<π‘Ÿξ…ž<1 and 𝛼𝑛>0, we can see that 𝑛β‰₯2 for all 𝑠>2. Moreover, since we have chosen βˆ‘βˆžπ‘›=2𝛼𝑛, the series βˆ‘βˆžπ‘›=21/π‘›π‘ βˆ’1 has the same nature as the series π‘Ÿβ‰₯0, which is convergent.

Therefore, according to Theorem 5.8 for π‘‹π‘Ÿ, 2π‘‘βˆ’1 is of class βˆ’1<π‘Ÿ<0-W-H, while for π‘‹π‘Ÿ, (𝑀,2π‘‘βˆ’1) is of class 𝑀β‰₯1-W-H for some 𝑀 (π‘Ÿ depends on {π‘‹π‘Ÿ}π‘Ÿ>βˆ’1). Moreover, the family 𝑀 is not uniformly Wick-HΓΆlderian (that means we cannot find the same π‘Ÿ>βˆ’1 for all limπ‘Ÿβ†’(βˆ’1)+πœ”1=0). This follows from the observation that limπ‘Ÿβ†’(βˆ’1)+πœ”2=2π‘ βˆ’2>0, while 𝑝. If we assume the existence of two positive numbers π‘ž and 𝐸[|𝑓1(π‘‹π‘Ÿ)⋄𝑓1(π‘‹π‘Ÿ)|2√]≀𝐸[|Ξ“(𝑝𝐼)𝑓1(π‘‹π‘Ÿ)|2√]𝐸[|Ξ“(π‘žπΌ)𝑓1(π‘‹π‘Ÿ)|2], such that the inequality: π‘Ÿ>βˆ’1 holds for all 𝑓1(π‘‹π‘Ÿ)⋄𝑓1(π‘‹π‘Ÿ)=𝑓2(π‘‹π‘Ÿ) since 𝑓1, where 𝑓2 and π‘‹π‘Ÿ, are the orthogonal polynomials of degree 1 and 2, respectively, generated by πœ”2!β‰€π‘πœ”1!β‹…π‘žπœ”1!.(6.16), then we would conclude thatπœ”2β‰€π‘π‘žπœ”1This inequality reduces to π‘Ÿ>βˆ’1 for all π‘Ÿ, which is impossible since the left-hand side converges to a positive number, while the right-hand side tends to zero as βˆ’1 goes to 𝑠=2.

Example 6.5. Let us take now π‘‹π‘Ÿ, and consider the family of random variables πœ”π‘›βˆΆ=𝑛2+π‘Ÿπ‘› whose principal SzegΓΆ-Jacobi parameters are 𝑛β‰₯1 for all π‘Ÿ>βˆ’1, where π›Όπ‘›βˆΆ=𝛼𝑛. If we take the other SzegΓΆ-Jacobi parameters to be 𝑛β‰₯0 for all 𝛼, where {π‘‹π‘Ÿ}π‘Ÿ>βˆ’1 is a fixed real number, then we can see that π‘Ÿβ‰₯0 are exactly the centered and rescaled Meixner random variables.

We can see exactly as in Example 6.5 that for {πœ”π‘›}𝑛β‰₯1, 22βˆ’1 is βˆ’1<π‘Ÿ<0-subadditive, while for 2, it is not π‘Ÿξ…žβˆΆ=βˆ’π‘Ÿ-suadditive. If 𝛼𝑛=π‘Ÿξ…ž(22βˆ’1βˆ’1)/(𝑛2βˆ’1βˆ’22βˆ’1π‘Ÿξ…ž)=π‘Ÿξ…ž/(π‘›βˆ’2π‘Ÿξ…ž), unfortunately, for βˆ‘βˆžπ‘›=2𝛼𝑛, the series 𝑛 is not convergent. Moreover, as we saw in the previous example, since (6.14) cannot be avoided for {πœ”π‘›}𝑛β‰₯1 even, we can see that the sequence 2 is not (𝛽, 𝛽={𝛽𝑛}𝑛β‰₯2)-subadditive, for any nonnegative sequence βˆ‘βˆžπ‘›=2𝛽𝑛<∞, such that βˆ’1/2β‰€π‘Ÿ<0.

Claim 1. If 𝑛β‰₯π‘˜β‰₯0, then for all ξƒ©πœ”π‘›πœ”π‘˜ξƒͺ≀2π‘›ξƒ©π‘›π‘˜ξƒͺ.(6.17), we haveξ‚€ξƒ©πœ”π‘›πœ”π‘˜ξƒͺ=ξ‚€ξƒ©πœ”π‘›πœ”π‘›βˆ’π‘˜ξƒͺSince (ξƒ©π‘›π‘˜ξƒͺ)=(π‘›π‘›βˆ’π‘˜) and π‘˜β‰€π‘›/2, we may assume that π‘˜β‰₯1.

We fix 𝑛, and prove by induction on 𝑛β‰₯2π‘˜ that for all ξ‚€ξƒ©πœ”π‘›πœ”π‘˜ξƒͺ≀2𝑛(π‘›π‘˜), 𝑛=2π‘˜.

Let us prove first this inequality for πœ”π‘š=π‘š(π‘šβˆ’π‘Ÿξ…ž). Since π‘šβ‰₯1 for all ξƒ©πœ”2π‘˜πœ”π‘˜ξƒͺ=ξƒ©π‘˜ξƒͺ2π‘˜2π‘˜βˆ’π‘Ÿξ…žπ‘˜βˆ’π‘Ÿξ…žβ‹…2π‘˜βˆ’1βˆ’π‘Ÿξ…žπ‘˜βˆ’π‘Ÿξ…žξ‚„ξ‚ƒ2π‘˜βˆ’2βˆ’π‘Ÿξ…žπ‘˜βˆ’1βˆ’π‘Ÿξ…žβ‹…2π‘˜βˆ’3βˆ’π‘Ÿξ…žπ‘˜βˆ’1βˆ’π‘Ÿξ…žξ‚„β‹―Γ—ξ‚ƒ2βˆ’π‘Ÿξ…ž1βˆ’π‘Ÿ