Multidimensional Free Poisson Limits on Free Stochastic Integral Algebras

TL;DR

This paper establishes four-moment theorems for multidimensional free Poisson limits within free stochastic integral algebras, characterizing convergence to free Poisson distributions through moment conditions.

Contribution

It extends four-moment theorems to multidimensional free Poisson limits in free Wigner and free Poisson algebras, including non-free cases with specified parameters.

Findings

Convergence characterized by moments up to order four.

Results apply to free and non-free multidimensional free Poisson limits.

Provides conditions under which stochastic integrals converge to free Poisson distributions.

Abstract

In this paper, we prove four-moment theorems for multidimensional free Poisson limits on free Wigner chaos or the free Poisson algebra. We prove that, under mild technical conditions, a bi-indexed sequence of free stochastic integrals in free Wigner algebra or free Poisson algebra converges to a free sequence of free Poisson random variables if and only if the moments with order not greater than four of the sequence converge to the corresponding moments of the limit sequence of random variables. Similar four-moment theorems hold when the limit sequence is not free, but has a multidimensional free Poisson distribution with parameters $\lambda>0$ and $\alpha=\{\alpha_i: 0\ne \alpha_i\in \mathbb{R}, i=1, 2, \cdots\}$.

Equations

$$f^{\ast }(t_1,t_2,\cdots ,t_q)=\overline{f(t_q,\cdots ,t_2,t_1)}.$$

$$f\stackrel{p}{⌢}g(t_1,\cdots ,t_{m+n-2p})$$

$$=$$

$$f\otimes g(t_1,\cdots ,t_{m+n})=f(t_1,\cdots ,t_m)g(t_{m+1},\cdots ,t_{m+n}).$$

$$f{\star }_p^{p-1}g(t_1,\cdots ,t_{m+n-2p+1})$$

$$=$$

$$\times$$

$$S(0,t)(dx)=\frac{1}{2\pi t}\sqrt{4t-x^2}dx,|x|\le 2\sqrt{t}.$$

$${\int }_{-2\sqrt{t}}^{2\sqrt{t}}{x}^{2m}S(0,t)(dx)=C_m{t}^m,$$

$$\phi (s_{i_1}{s}_{i_2}\cdots s_{i_d})=\sum _{\pi \in N{C}_2(n)}\prod _{(a,b)\in \pi }c(i_a,i_b),$$

$$⟨I^S(f),I^S(g){⟩}_{L^2(\mathcal{A},\phi )}=⟨f,g{⟩}_{L^2({\mathbb{R}}_{+}^n)},$$

$${\kappa }_m(X)=\lambda {\alpha }^m,\forall m\ge 1,$$

$${\kappa }_1(Y(\lambda ))=0,{\kappa }_m(Y(\lambda ))=\lambda ,m\ge 2.$$

$$C_0(x,\lambda )=1,C_1(x,\lambda )=x,x{C}_m(x,\lambda )=C_{m+1}(x,\lambda )+C_m(x,\lambda )+\lambda C_{m-1}(x,\lambda ),m\ge 1.$$

$$I^{\hat{N}}(f)=C_{k_1}(\hat{N}(A_1),\mu (A_1))\cdots C_{k_l}(\hat{N}(A_l),\mu (A_l)).$$

$${\kappa }_n(b_{i(1)},b_{i(2)},\cdots ,b_{i(n)})={\alpha }_{i(1)}{\alpha }_{i(2)}\cdots {\alpha }_{i(n)}\underset{N\to \omega }{\lim }N{\phi }_N(p_N^{(i(1))}{p}_N^{(i(2))}\cdots p_N^{(i(n))}),$$

$${\kappa }_1(a_i)=0,i=1,2,\cdots ;$$

$${\kappa }_n(a_{i(1)},a_{i(2)},\cdots ,a_{i(n)})={\kappa }_n(b_{i(1)},\cdots ,b_{i(n)}),\forall i(1),i(2),\cdots ,i(n),n\in \mathbb{N},n\ge 2.$$

$$\phi (a_{\chi (1)}\cdots a_{\chi (n)})=\sum _{\pi \in NC(n),\pi \le \chi }\prod _{V\in \pi }{\lambda }_{\chi (V)},$$

$$\phi (a_{\chi (1)}\cdots a_{\chi (n)})=$$

$$=$$

$$I(f_n^{(i_1)}\cdots I(f_n^{(i_m)})=\sum _{(r_1,r_2,\cdots ,r_{m-1})\in A_m}I((\cdots (f_n^{(i_1)})\stackrel{r_1}{⌢}{f}_n^{(i_2)})\stackrel{r_2}{⌢}{f}_n^{(i_3)}\cdots \stackrel{r_{m-2}}{⌢}{f}_n^{(i_{m-1})})\stackrel{r_{m-1}}{⌢}{f}_n^{(i_m)}),$$

$$A_m=$$

$$r_3\le 3q-2r_1-2r_2,\cdots ,r_{m-1}\le (m-1)q-2r_1-\cdots -2r_{r_{m-2}}\}.$$

$$\phi (I(f_n^{(i_1)})\cdots I(f_n^{(i_m)}))=\sum _{r_1,r_2,\cdots ,r_{m-1}\in B_m}(\cdots (f_n^{(i_1)}\stackrel{r_1}{⌢}{f}_n^{(i_2)})\stackrel{r_2}{⌢}{f}_n^{(i_3)}\cdots \stackrel{r_{m-2}}{⌢}{f}_n^{(i_{m-1})})\stackrel{r_{m-1}}{⌢}{f}_n^{(i_m)},$$

Full text

∗ The second author was supported by the Project sponsored by the NSFC grant 11431011 and startup funding from Hebei Normal University

Key Words Free Wigner chaos, Free Poisson chaos, Fourth moment theorem, Multidimensional free Poisson distributions.

2010 MSC 46L54

Introduction

A basic question in free probability is to find distributions of random variables in a noncommutative probability space. No one can find distributions of random variables in an arbitrary noncommutative probability space, but mathematicians want to investigate distributions of as many as possible random variables. The initial approach to this question is to study special distributions inspired by the work in classical probability, such as free Gaussian distributions (i.e., semicircle distributions), free Poisson distributions, etc. But not too many distributions in classical probability could be defined and studied in free probability. Mathematicians extended their well-studied research territory by imposing operations such as free addition and free multiplication among the well known distributions. A more complicated and advanced operation is integration: integrating a deterministic (multi-variable) function with respect to a random measure of well known random variables such as semicircle random variables or free Poisson random variables. The resulting random variable is called the free stochastic (multiple) integral of the function with respect to the random measure. The study on the distributions of free stochastic integrals is one of the main topics in free probability. Unfortunately, after integrating a function $f\in L^{2}(\mathbb{R}_{+}^{q})$ ($q\geq 4$) with respect to a random measure of semicircular or free Poisson random variables, one can hardly get any non-trivial semicircle or free Poisson random variables (see Corollary 4.5 in [BP], Proposition 1.5 in [NP], Corollary 1.7 in [KNPS], and Corollary 1.6 in [SB2]). A natural question, then, is to study the convergence of a sequence of free stochastic integrals. On the other hand, the limit theory (e.g., central and Poisson limit theorems) is a key topic in both classical and free probability. Mathematicians, thus, began to study the convergence problem among (free) stochastic integrals more than a decade ago.

Let $\{W_{t}:t\geq 0\}$ be a standard Brownian motion on $\mathbb{R}_{+}$, and let $n\in\mathbb{N}$. Denote by $I^{W}(f)$ the multiple stochastic Wiener-It$\hat{o}$ integral of order $n$ of a function $f\in L^{2}(\mathbb{R}_{+}^{n})$. Denoted by $L^{2}_{s}(\mathbb{R}_{+}^{n})$ the subset of $L^{2}(\mathbb{R}_{+}^{n})$ composed of symmetric functions. The collection of random variables $\{I^{W}(f):f\in L^{2}_{s}(\mathbb{R}_{+}^{n})\}$ is called the $n$th Wiener chaos associated with $W$. In their seminal paper [NuP], Nualart and Peccati proved that, given a sequence of elements with variance one living in a Wiener chaos of a fixed order, the convergence of this sequence to the standard normal distribution is equivalent to the convergence of the fourth moment of the sequence to three. This result is now known as the fourth moment theorem. The above result has led to a wide collection of new results and inspired several new research directions (see [NP1] and the constantly updated web-page: http://sites.google.com/site/malliavinstein/home). The fourth moment theorem was proved to hold as well for sequences of vectors of multiple integrals possibly of different orders in [PT].

Non-commutative counterparts of the results in classical probability have been established in the context of the chaos associated with a free Brownian motion and a free Poisson random measure. The mentioned concepts and notations in this section will be defined in Section 1.

Convergence of multiple integrals with respect to a free Brownian motion. Let $I^{S}(f)$ be the multiple free stochastic integral of a function $f\in L^{2}(\mathbb{R}_{+}^{q})$ ($q\geq 2$) with respect to a free Brownian motion $S$. Kemp, Nourdin, Peccati, and Speicher [KNPS] proved a beautiful fourth moment theorem for the convergence of a sequence of multiple Wigner integrals in law to the standard semicircular distribution (Theorem 1.3 in [KNPS]). A similar result was obtained in [NP] for such a sequence to converge to a centered free Poisson distribution (Theorem 1.4 in [NP]). Nourdin, Peccati, and Speicher [NPS] proved a fourth moment theorem for a sequence of multidimensional free stochastic integral vectors to converge to a multidimensional semicircular limit theorem on the free Wigner chaos (Theorem 1.3 in [NPS]).

Convergence of multiple integrals with respect to a free Poisson random measure. Let $\widehat{N}$ be a centered free Poisson random measure on $\mathbb{R}_{+}$, and $I^{\widehat{N}}(f)$ be the multiple integral of $f\in L^{2}(\mathbb{R}_{+}^{q})$ ($q\geq 2$). Bourguin and Peccati [BP] proved a fourth moment theorem for convergence to a semicircular distribution on the free Poisson algebra (Theorem 4.3 in [BP]). In [SB2], S. Bourguin gave a fourth moment theorem for convergence to a free centered Poisson distribution on the free Poisson algebra (Theorem 1.5 in [SB2]). He later proved a fourth moment theorem for convergence to vector-valued semicircular limits on the free Poisson algebra (Theorem 1.3 in [SB1]). Recently, Bourguin and Nourdin provided an equivalent condition for the convergence discussed in [SB1] in terms of the fourth moment of the Euclidean norm of the involved random vectors when the kernels of the multiple integrals are fully symmetric (Theorem 6.3 in [BN]).

The work in the present paper. Given a bi-indexed sequence of multiple free stochastic integrals with respect to free Wigner chaos or a free Poisson random measure, the aim of this paper is to investigate the convergence of the sequence to a multidimensional free Poisson distribution defined in [AG]. The rest of this paper is organized as follows. In Section 1, we recall relevant elements in free probability, free Wigner chaos, and free Poisson algebra used in sequel. In Section 2, we study the convergence problem for a bi-indexed sequence of free stochastic integrals of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$ ($q$ is even) with respect to free Bromnian motion. We prove that such a sequence converges in distribution to a free sequence of free Poisson random variables if and only if its joint moments with order less than or equal to four converge to the corresponding moments of the limit sequence (Theorem 2.7). In section 3, a similar theorem is proved when the free stochastic integrals are defined with respect to a centered free Poisson random measure and the kernels are symmetric, bounded, with bounded supports (Theorem 3.3). Finally, in Section 4, we study the convergence problem of sequences studied in Sections 2 and 3, when the limit sequence has a multidimensional free Poisson distribution with parameters $\lambda>0$ and $\alpha=\{\alpha_{1},\alpha_{2},\cdots\}$, a sequence of non-zero real numbers. Using the techniques in Sections 2, 3, [NP], and [SB2], we get four-moment theorems in this case (Theorems 4.2 and 4.4).

1. Preliminaries

In this section, we recall the relevant elements in free probability (especially, free stochastic integration) used in sequel. For details on the subject, the reader is referred to [SB2], [BP], [NS], [NP], and [VDN]. Basics on operator algebras can be found in [KR].

Symmetric functions and contractions. For an integer $q\geq 2$, let $L^{2}(\mathbb{R}_{+}^{q})$ be the space of all complex-valued functions on $\mathbb{R}_{+}^{q}=\{(x_{1},\cdots,x_{q}):x_{i}\geq 0,i=1,2,\cdots,q\}$ that are square-integrable with respect to $\mu$, the Lebesgus measure on $\mathbb{R}^{q}$. Given $f\in L^{2}(\mathbb{R}_{+}^{q})$, we write

[TABLE]

We say that $f\in L^{2}(\mathbb{R}_{+}^{q})$ is symmetric if $f(t_{1},\cdots,t_{q})=f^{*}(t_{1},\cdots,t_{q})$, for all $(t_{1},\cdots,t_{q})\in\mathbb{R}_{+}^{q}$.

Let $m,n$ be natural numbers, $f\in L^{2}(\mathbb{R}_{+}^{m}),g\in L^{2}(\mathbb{R}_{+}^{n})$, and $p\leq\min\{m,n\}$ be a nonnegative integer. The $p$-th arc contraction $f\stackrel{{\scriptstyle p}}{{\smallfrown}}g$ is defined by

[TABLE]

When $p=0$, we define $f\stackrel{{\scriptstyle 0}}{{\smallfrown}}g=f\otimes g$, the $m+n$-variable function defined by

[TABLE]

When $p=m=n$, $f\stackrel{{\scriptstyle p}}{{\smallfrown}}g=\langle g,f^{*}\rangle$.

The star contraction of index $(p,p-1)$ of $f$ and $g$ is defined by

[TABLE]

One can verify the relations

[TABLE]

Free Probability, Free Brownian motion, and free Wigner Chaos. The main references for the contents of this subsection are [BS] and [NP]. Let $(\mathcal{A},\varphi)$ be a tracial $W^{*}$-probability space, that is, $\mathcal{A}$ is a finite von Neumann algebra and $\varphi:\mathcal{A}\rightarrow\mathbb{C}$ is a faithful, normal, and tracial state on $\mathcal{A}$. Self-adjoint operators in $\mathcal{A}$ are referred to random variables. The law (or distribution) of a random variable $X\in\mathcal{A}$ is the unique Borel measure on $\mathbb{R}$ having the same moments as $X$ (see Proposition 3.13 in [NS]). For $1\leq p\leq\infty$, one writes $L^{p}(\mathcal{A},\varphi)$ to indicate the $L^{p}$ space obtained by completing $\mathcal{A}$ with respect to the norm $\|X\|_{p}=\varphi(|X|^{p})^{1/p}$, where $|X|=(X^{*}X)^{1/2}$, for $p<\infty$, and $\|\cdot\|_{\infty}$ stands for the operator norm.

Let $\mathcal{A}_{1},\cdots,\mathcal{A}_{n}$ be unital subalgebras of $\mathcal{A}$. Let $X_{1},\cdots,X_{m}$ be elements chosen among the $\mathcal{A}_{i}$’s such that, for $1\leq j<m$, $X_{j}$ and $X_{j+1}$ do not come from the same $\mathcal{A}_{i}$, and such that $\varphi(X_{j})=0$ for each $j$. The subalgebras $\mathcal{A}_{1},\cdots,\mathcal{A}_{n}$ are said to be free or freely independent if, in this circumstance, $\varphi(X_{1}\cdots X_{m})=0$. Random variables are free if the unital algebras they generate are free.

A partition of $[n]=\{1,2,\cdots,n\}$ is a collection of mutually disjoint nonempty subsets $B_{1},B_{2},\cdots$, $B_{r}$ of $[n]$ such that $B_{1}\cup B_{2}\cup\cdots\cup B_{r}=[n]$. The subsets are called the blocks of the partition. The blocks are ordered by their least elements, i.e. $\min B_{i}<\min B_{j}$ iff $i<j$. The cardinality of $B_{i}$ is denoted by $|B_{i}|$. The set of all partitions of $[n]$ is denoted by $\mathcal{P}(n)$. A partition $\pi\in\mathcal{P}(n)$ has a crossing if there are two distinct blocks $B_{i}$ and $B_{j}$ in $\pi$ with elements $x_{i},y_{i}\in B$ and $x_{j},y_{j}\in B_{j}$ such that $x_{i}<x_{j}<y_{i}<y_{j}$. If a partition $\pi$ has no crossings, we say that $\pi$ is a non-crossing partition. The set of all non-crossing partitions of $[n]$ is denoted by $NC(n)$. It is well known (see Page 144 in [NS]) that the reversed refined order induces a lattice structure on $NC(n)$, where the partial order is defined by $\pi\leq\sigma$, for $\pi,\sigma\in NC(n)$ if every block in $\sigma$ is the union of some blocks in $\pi$.

A set $\{a_{i}:i\in I\}$ of random variables is a free family of random variables (or, we say, $a_{i},i\in I$, are free) if and only if for $1<n\in\mathbb{N}$ and $\chi:\{1,2,\cdots,n\}\rightarrow I$, we have $\kappa_{n}(a_{\chi(1)},a_{\chi(2)},\cdots,a_{\chi(n)})=0$ whenever $\chi$ is not constant (Theorem 11.20 in [NS]).

The (centered) semicircular distribution (or Wigner law) $S(0,t)$ is the probability distribution

[TABLE]

The odd moments of this distribution are [math] and even moments are given by

[TABLE]

where $C_{m}=\frac{1}{m+1}\left(\begin{matrix}2m\\ m\end{matrix}\right)$ is the $m$-th Catalan number (see Lecture 2 in [NS]).

Let $d\geq 2$ be a natural number, and let $c=(c(i,j))_{d\times d}$ be a positive definite symmetric matrix. A $d$-dimensional vector $(s_{1},\cdots,s_{d})$ of random variables in $(\mathcal{A},\varphi)$ is said to be a semicircular family with covariance $c$ if for every $n\geq 2$ and $i_{1},\cdots,i_{n}\in\{1,\cdots,d\}$

[TABLE]

where $NC_{2}(n)=\{\pi\in NC(n):|V|=2,\forall V\in\pi\}$ (see Definition 8.15 in [NS]).

A free Brownian motion $S$ consists of $(i)$ a filtration $\{\mathcal{A}_{t}:t\geq 0\}$, that is, a family of von Neumann subalgebras of $\mathcal{A}$ such that $\mathcal{A}_{s}\subset\mathcal{A}_{t}$ whenever $0\leq s<t$, $(ii)$ a collection $S=\{S_{t}:t\geq 0\}$ of random variables in $\mathcal{A}$ such that: $(a)$ $S_{0}=0$ and $S_{t}\in\mathcal{A}_{t}$, for $t>0$, $(b)$ for every $t$, $S_{t}$ has the distribution $S(0,t)$, and $(c)$ for every $0<s<t$, the operator $S_{t}-S_{s}$ is free from $\mathcal{A}_{s}$, and has the distribution $S(0,t-s)$.

For a natural number $n$, and $f\in L^{2}(\mathbb{R}_{+}^{n})$, the (free stochastic) multiple integral $I^{S}(f)$ is defined as follows. $(a)$ Define $I^{S}(f)=(S_{b_{1}}-S_{a_{1}})\cdots(S_{b_{n}}-S_{a_{n}})$ for a function

[TABLE]

where intervals $(a_{i},b_{i}),i=1,\cdots,n$, are pairwise disjoint; $(b)$ extend the above $I^{S}(f)$ linearly to the integrals of simple functions vanishing on diagonals, that is, linear combinations of functions of type $(1.1)$; $(c)$ exploit the isometric relation ((3.3) in [NP])

[TABLE]

where $f$ and $g$ are simple functions vanishing on diagonals, and use a density argument to define $I^{S}(f)\in L^{2}(\mathcal{A},\varphi)$ for $f\in L^{2}(\mathbb{R}_{+}^{n})$. We define $I^{S}(\alpha)=\alpha$, for a complex number $\alpha$. The set of all random variables $I^{S}(f),f\in L^{2}(\mathbb{R}_{+}^{n})$, is called the $n$th free Wigner chaos associated with $S$. For $m,n\in\mathbb{N}$, and $f\in L^{2}(\mathbb{R}_{+}^{m}),g\in L^{2}(\mathbb{R}_{+}^{n})$, we have the following orthogonal and isometric formula (Proposition 3.5 in [BP])

[TABLE]

where $\delta_{m,n}=1$, if $m=n$; $\delta_{m,n}=0$, if $m\neq n$.

Free Wigner chaos is defined as the linear space of all multiple integrals $I^{S}(f)$, for $f\in L^{2}(\mathbb{R}_{+}^{n})$, $n\geq 0$, where $L^{2}(\mathbb{R}_{+}^{0})=\mathbb{C}$. Free Wigner chaos becomes a unital $*$-algebra by imposing the following operations

[TABLE]

(see Remark 3.4 in [BP] and (3.4) in [NP]). The multiplication unit of the free Wigner Chaos is $1\in L^{2}(\mathbb{R}^{0})=\mathbb{C}$, since, $I^{S}(1)$ is defined as $1$.

The free Poisson algebra. The main reference for this section is [BP]. A free Poisson distribution $N(\lambda,\alpha)$ with parameters $\lambda>0$ and $\alpha\in\mathbb{R}$ with $\alpha\neq 0$ is a probability distribution on $\mathbb{R}$ defined as follows. A random variable $X\in\mathcal{A}$ has a distribution $N(\lambda,\alpha)$ if and only if

[TABLE]

where $\kappa_{m}$ is the $m$th free cumulant of $(\mathcal{A},\varphi)$ (see Proposition 12.11 in [NS]). When $\alpha=1$, we denote $N(\lambda,\alpha)$ by $N(\lambda)$. If $X(\lambda)$ has a distribution $N(\lambda)$, we denote the distribution of $Y(\lambda):=X(\lambda)-\lambda\mathrm{1}$ by $\widehat{N}(\lambda)$, which is called the centered free Poisson distribution of parameter $\lambda$, which is characterized by

[TABLE]

Let $(Z,\mathcal{Z})$ be a Polish space with $\mathcal{Z}$ the associated Borel $\sigma$-field, and let $\mu$ be a positive $\sigma$-finite measure over $(Z,\mathcal{Z})$ without atoms. We denote by $\mathcal{Z}_{\mu}$ the class of those $A\in\mathcal{Z}$ such that $\mu(A)<\infty$. Let $(\mathcal{A},\varphi)$ be a tracial $W^{*}$-probability space and $\mathcal{A}_{+}$ be the cone of positive operators in $\mathcal{A}$. Then, a free Poisson random measure with control $\mu$ on $(Z,\mathcal{Z})$ and values in $(\mathcal{A},\varphi)$ is a mapping $N:\mathcal{Z}_{\mu}\rightarrow\mathcal{A}_{+}$ with the following properties.

1. (1)

For $A\in\mathcal{Z}_{\mu}$, $N(A)$ is a free Poisson random variable with parameter $\mu(A)$. 2. (2)

If $2\leq r\in\mathbb{N}$, and $A_{1},\cdots,A_{r}\in\mathcal{Z}_{\mu}$ are disjoint, then $N(A_{1}),\cdots,N(A_{r})$ are free, and $N(\cup_{i=1}^{r}A_{i})=\sum_{i=1}^{r}N(A_{i})$.

The existence of free Poisson measures on an appropriate $W^{*}$-probability space is guaranteed by Theorem 3.3 in [BnT] and Theorem 5.1 in [BV]. If $N$ is a free Poisson random measure with control $\mu$, we will denote by $\widehat{N}$ the mapping from $\mathcal{Z}_{\mu}$ into $\mathcal{A}$ defined by $\widehat{N}(A)=N(A)-\mu(A)\mathrm{1},A\in\mathcal{Z}_{\mu}$, where $\mathrm{1}$ is the unit of $\mathcal{A}$. Then $\widehat{N}(A)$ has the centered free Poisson distribution $\widehat{N}(\mu(A))$. We call $\widehat{N}$ a free centered Poisson measure. In this paper, we let $(Z,\mathcal{Z},\mu)=(\mathbb{R}_{+},\mathcal{B}(\mathbb{R}_{+}),\mu)$, where $\mu$ is the Lebesgue measure on $\mathbb{R}_{+}$. Then $\mathcal{Z}_{\mu}=\mathcal{B}_{\mu}(\mathbb{R}_{+})$ is the set of all Borel subsets $B$ of $\mathbb{R}_{+}$ with finite Lebesgue measure ($\mu(B)<\infty$).

Centered free Charlier polynomials are defined by the following recurrence relations

[TABLE]

Let $q\geq 2$ be an integer, and let $\mathcal{E}_{q}$ be the linear subspace of $L^{2}(\mathbb{R}_{+}^{q})$ generated by functions of the type

[TABLE]

where $k_{1}+\cdots+k_{l}=q$, each $A_{j}$ is bounded, and $A_{j}\cap A_{j+1}=\emptyset$, for $j=1,2,\cdots,l-1$. For a function of the type (1.4), the multiple free stochastic integral of $f$ with respect to the centered free Poisson random measure $\widehat{N}$ is defined by

[TABLE]

We extend $I^{\widehat{N}}(f)$ to general $f\in\mathcal{E}_{q}$ by linearity.

For $q,q^{\prime}\geq 0$, and $f\in\mathcal{E}_{q}$ and $g\in\mathcal{E}_{q^{\prime}}$, one has the following orthogonal and isometric property

[TABLE]

(see Proposition 3.5 in [BP]). Then, for a $f\in L^{2}(\mathbb{R}_{+}^{q})$, one can define $I^{\widehat{N}}(f)\in L^{2}(\mathcal{A},\varphi)$ by the density of $\mathcal{E}_{q}$ in $L^{2}(\mathbb{R}_{+}^{q})$ and the above isometry property. The space $L^{2}(\mathcal{X}(\widehat{N}),\varphi)=\{I^{\widehat{N}}(f):f\in L^{2}(\mathbb{R}_{+}^{q}),q\geq 0\}$ is a unital $*$-algebra with the following operations. For $f\in L^{2}(\mathbb{R}_{+}^{p}),g\in L^{2}(\mathbb{R}_{+}^{q})$, $p,q\geq 1$,

[TABLE]

(see (3) of Page 2145 in [SB2]) The algebra $L^{2}(\mathcal{X}(\widehat{N},\varphi))$ is called the free Poisson (multiple integral) algebra.

2. Multidimensional free Poisson limits on the free Wigner chaos

In this section, we study the convergence of a sequence in the free Wigner chaos to a multidimensional free Poisson distribution, aiming at giving a fourth moment type theorem for the convergence.

Multidimensional (finite dimensional) free Poisson distributions were studied in [RS]. More general (infinite dimensional) multidimensional free Poisson distributions were defined in [AG] through a multidimensional free Poisson limit theorem.

Definition 2.1** (Definition 2.7 in [AG]).**

Let $\{\alpha_{i}:i=1,2,\cdots\}$ be a sequence of real numbers, $\{\lambda_{i}>0:i=1,2,\cdots\}$ with $\lambda=\sup\{\lambda_{i}:i\geq 1\}<\infty$, and for each $N\in\mathbb{N},N\geq\lambda$, $(\mathcal{A},\varphi_{N})$ be a $C^{*}$-probability space. A sequence $\{b_{i}:i=1,2,\cdots\}$ of random variables in a non-commutative probability space $(\mathcal{B},\phi)$ has a multidimensional free Poisson distribution, if there is a family $\{p_{N}^{(i)}:i\in\mathbb{N}\}$ of projections in $\mathcal{A}_{N}$ and an ultrafilter $\omega\in\beta\mathbb{N}\setminus\mathbb{N}$ such that $\varphi_{N}(p_{N}^{(i)})=\frac{\lambda_{i}}{N}$, and

[TABLE]

for all $(i(1),i(2),\cdots,i(n))\in\mathbb{N}^{n}.$

If we choose $\{p_{N}^{(i)}:i=1,2,\cdots\}$ to be an orthogonal sequence of projections for each $N$, then we get $\kappa_{n}(b_{\epsilon(1)},\cdots,b_{\epsilon(n)})=0,$ if $\epsilon:\{1,2,\cdots n\}\rightarrow\mathbb{N}$ is not constant, and $n\geq 2$. It follows that the sequence $\{b_{n}:n=1,2,\cdots\}$ is a free family of free Poisson random variables. In other words, a free sequence $\{b_{n}:n=1,2,\cdots\}$ of free Poisson random variables has a multidimensional free Poisson distribution in sense of Definition 2.1. Let $a_{i}=b_{i}-\varphi(b_{i})1,\forall i\in\mathbb{N}$, for a sequence $\{b_{n}:n=1,2,\cdots\}$ having a multidimensional free Poisson distribution. We then have

[TABLE]

[TABLE]

We say that $\{a_{i}:i=1,2,\cdots\}$ has a centered multidimensional free Poisson distribution.

Theorem 2.2**.**

Let $\{a_{n}:n=1,2,\cdots\}$ be a free sequence of free Poisson random variables with parameters $\lambda_{i},i=1,2,\cdots$, in a non-commutative probability space $(\mathcal{A},\varphi)$. Then for $2\leq n\in\mathbb{N}$ and $\chi:\{1,2,\cdots,n\}\rightarrow\mathbb{N}$, we have

[TABLE]

where $\chi$ in the sum subscript of the right hand side is the partition of $\{1,2,\cdots,n\}$ defined by $l_{1}\sim_{\chi}l_{2}$ if and only if $\chi(l_{1})=\chi(l_{2})$, for $1\leq l_{1},l_{2}\leq n$, $\chi(V)$, the subscript of $\lambda$, is the common value of $\chi$ when restricted to $V\in\pi$.

Proof.

The proof is a simple application of the moments and free cumulants conversion formula ($(11.8)$ in [NS]). Under the assumptions of this theorem, we have

[TABLE]

∎

As the discussions in the proof of Theorem 1.4 in [NP], iterative applications of the product formula $(1.3)$ yields the following formula

[TABLE]

for all $i_{1},\cdots,i_{m},m\in\mathbb{N},m\geq 2$, where

[TABLE]

Then, we have

[TABLE]

where $B_{m}=\{(r_{1},\cdots,r_{m-1})\in A_{m}:2r_{1}+2r_{2}+\cdots+2r_{m-1}=mq\}$.

We need several technical results to prove the main result of this section.

Lemma 2.3**.**

Given an even number $q\geq 2$ and $\lambda>0$, consider a bi-indexed sequence $\{f_{n}^{(i)}:n,i\in\mathbb{N}\}$ of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$ such that $\lim_{n\rightarrow\infty}\langle f_{n}^{(i)},f_{n}^{(j)}\rangle_{L^{2}(\mathbb{R}_{+}^{q})}=\delta_{i,j}\lambda_{i}$, for all $i,j\in\mathbb{N}$. As $n\rightarrow\infty$, one has that, for $i,j\in\mathbb{N}$,

[TABLE]

if and only if $\|f_{n}^{(i)}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(i)}-f_{n}^{(i)}\|_{L^{2}(\mathbb{R}_{+}^{q})}\rightarrow 0$ and $\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(i)}\|_{L^{2}(\mathbb{R}_{+}^{q})}\rightarrow 0$, for all $r=1,2,\cdots,q-1$, $r\neq q/2$, and $i=1,2,\cdots$; and $\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(j)}\|_{L^{2}(\mathbb{R}_{+}^{q})}\rightarrow 0$, for all $r=1,2,\cdots,q-1$, if $i\neq j$, $i,j=1,2,\cdots$.

Proof.

When $i=j$, this lemma is Lemma 5.1 in [NP].

Now we consider $i\neq j$. In this case, by the product formula (1.3), we have

[TABLE]

By (1.2) and (1.0), we get

[TABLE]

On the other hand,

[TABLE]

We thus get that $\lim_{n\rightarrow\infty}(\varphi(I(f_{n}^{(i)})I(f_{n}^{(j)})^{2}I(f_{n}^{(i)})-2I(f_{n}^{(i)})I(f_{n}^{(j)})^{2}))=\lambda_{i}\lambda_{j}$ if and only if

[TABLE]

Let $\overline{s}=(s_{1},\cdots,s_{q/2}),\overline{s}^{r}=(s_{q/2},\cdots,s_{1}),d\overline{s}=ds_{1}\cdots ds_{q/2}$, $\overline{t_{1}}=(t_{1},\cdots,t_{q/2}),d\overline{t_{1}}=dt_{1}\cdots dt_{q/2}$, and $\overline{t_{2}}=(t_{q/2+1},\cdots,t_{q}),d\overline{t_{2}}=dt_{q/2+1}\cdots dt_{q}$. We have

[TABLE]

where second and third equalities hold because $f_{n}^{(j)}$ and $f^{(j)}_{n}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(j)}$ are symmetric, respectively.

Now suppose that $(2.1)$ holds, by the conclusion of the case that $i=j$ (see also Lemma 5.1 in [NP]), $\lim_{n\rightarrow\infty}\|f_{n}^{(j)}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(j)}-f_{n}^{(j)}\|=0$. It follows that

[TABLE]

Combining the above discussion with $(2.2)$, we get

[TABLE]

for $r=1,2,\cdots,q-1$.

Conversely, if $(2.3)$ hold, we get $(2.2)$. Hence, $(2.1)$ holds for $i\neq j$. ∎

Lemma 2.4**.**

Let $q\geq 2$ be an even integer, and consider a bi-indexed sequence $\{f_{n}^{(i)}:n,i\in\mathbb{N}\}$ of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$ such that $\lim_{n\rightarrow\infty}\langle f_{n}^{(i)},f_{n}^{(j)}\rangle_{L^{2}(\mathbb{R}_{+}^{q})}=\delta_{ij}\lambda_{i}$, $\lambda_{i}>0$ for all $i,j\in\mathbb{N}$. If $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(i)}-f_{n}^{(i)}\|_{L^{2}(\mathbb{R}_{+}^{q})}=0$, $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(i)}\|=0$, for all $i=1,2,\cdots$, and $1\leq r\leq q-1$ and $r\neq q/2$, and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(j)}\|=0$, for $r=1,2,\cdots,q-1$, all $i\neq j$, then for $1<m\in\mathbb{N}$ and $\chi:\{1,2,\cdots,m\}\rightarrow\mathbb{N}$, we have

[TABLE]

where $D_{m}=B_{m}\cap\{0,\frac{q}{2},q\}^{(m-1)}$, and $\{a_{i}:i=1,2,\cdots\}$ is a free sequence of free Poisson random variables $a_{i}$ with parameters $\lambda_{i}>0$, $i=1,2,\cdots$.

Proof.

We follow the ideas of the proof of Lemma 5.2 in [NP]. For $r_{1},\cdots r_{j}\in\{0,q/2,q\}$, let

[TABLE]

We show that $G_{j}$ is either a scalar or a multiple of an object of the type $H_{1}\otimes\cdots\otimes H_{l}$, $l\geq 1$, where $H_{i}=f_{n}^{(l)}$, or $H_{i}=f_{n}^{(s_{1})}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}\cdots\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(s_{k})}$. We prove it by induction on $j$. When $j=1$, $G_{j}=f_{n}^{(\chi(1))}\otimes f_{n}^{(\chi(2))}$, if $r_{1}=0$; $G_{1}=f_{n}^{(\chi(1))}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(\chi(2))}$, if $r_{1}=q/2$; $G_{1}=\langle f_{n}^{(\chi(1))},f_{n}^{(\chi(2))}\rangle$, is a scalar, if $r_{1}=q$. Suppose that $G_{j-1}$ has the desired form. We prove that so does $G_{j}$. Let

[TABLE]

where $G_{j-1}$ is a scalar, when $l=0$. If $r_{j}=0$, $G_{j}=H_{1}\otimes\cdots\otimes H_{l}\otimes f_{n}^{(\chi(j+1))}$. If $r_{j}=q/2$, $G_{j}=H_{1}\otimes\cdots\otimes(H_{l}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(\chi(j+1))})$. If $r_{j}=q$, then $G_{j}=\langle H_{l},f_{n}^{(\chi(j+1))}\rangle H_{1}\otimes\cdots\otimes H_{l-1}$. It follows that $G_{j}$ has the desired form in all cases.

Now we prove that

[TABLE]

as $n\rightarrow\infty$. We need a Cauchy-Schwartz type inequality. For $f\in L^{2}(\mathbb{R}_{+}^{p})$, $g\in L^{2}(\mathbb{R}_{+}^{q})$, and $0\leq r\leq\min\{p,q\}$, we prove that

[TABLE]

It is obvious that we need only to prove the non-trivial case $0<r<\min\{p,q\}$. We use some vector notations to simplify the expressions. Let $\overline{x}_{1}=(x_{1},x_{2},\cdots,x_{p-r}),\overline{t}^{i}=(t_{r},t_{r-1},\cdots,t_{1}),\overline{t}=(t_{1},t_{2},\cdots,t_{r}),\overline{x}_{2}=(x_{q-r+1},\cdots,x_{p+q-2r})$. We then have

[TABLE]

When $i_{1}=i_{2}=\cdots=i_{k+1}$, $(2.4)$ is $(5.2)$ in Lemma 5.2 in [NP]. If $i_{1}=\cdots=i_{k}\neq i_{k+1}$, we have

[TABLE]

When $|\{i_{1},\cdots,i_{k}\}|>1$, let $i_{p}$ be the first index in the ordered sequence $\{i_{1},i_{2},\cdots,i_{k}\}$ such that $i_{1}\neq i_{p}$. By $(2.5)$ and the above discussion, we get

[TABLE]

We’ve proved $(2.4)$.

For $1<m\in\mathbb{N}$, and $\chi:\{1,2,\cdots,m\}\rightarrow\mathbb{N}$, we try to find $\lim_{n\rightarrow\infty}G_{m-1}$, where

[TABLE]

$(r_{1},r_{2},\cdots,r_{m-1})\in D_{m}$. Since $r_{1},\cdots,r_{m-1}\in D_{m}$, we have $r_{m-1}=q$, by the definitions of $A_{m}$ and $D_{m}$. Let $\{r_{j(1)},r_{j(2)},\cdots,r_{j(k-1)},r_{m-1}\}$ be the set of all $q$’s in the sequence $r_{1},r_{2},\cdots,r_{m-1}$, where $j(1)<j(2)<\cdots<j(k-1)<m-1$. Let $l_{1}$ be the number such that $r_{j(1)-l_{1}}=0$, and $r_{j(1)-l_{1}+1}r_{j(1)-l_{1}+2}\cdots r_{j(1)}\neq 0$. Then

[TABLE]

if $l_{1}>1$; $G_{m-1}=G_{j(1)-l_{1}}\otimes f_{n}^{\chi(j(1))}\stackrel{{\scriptstyle q}}{{\smallfrown}}f_{n}^{(\chi(j(1)+1))}\cdots$, if $l_{1}=1$, where

[TABLE]

[TABLE]

are constants. Let $V_{1}=\{j(1)-l_{1}+1,j(1)-l_{1}+2,\cdots,j(1)+1\}$. Removing $\alpha_{1}$ or $\beta_{1}$ from $G_{m-1}$, we get a new expression $\widetilde{G}_{m-l_{1}}$, which has the same form as $G_{m-1}$. We continue the above procedure until we remove all factors in $G_{m-1}$. We then get a non-crossing partition $\pi=\{V_{1},V_{2},\cdots,V_{k}\}$ in $NC_{\geq 2}(m)=\{\pi\in NC(m):|V|>1,\forall V\in\pi\}$. By the hypothesis and $(2.4)$, $\lim_{n\rightarrow\infty}\alpha_{1}\beta_{1}\neq 0$, if and only if

[TABLE]

In this case, $\lim_{n\rightarrow\infty}\alpha_{1}=\lim_{n\rightarrow\infty}\beta_{1}=\lambda_{\chi(j(1))}$. It implies that $\lim_{n\rightarrow\infty}G_{m-1}\neq 0$, if and only if $\pi\leq\chi$, that is, $\chi|_{V_{i}}$ must be a constant, for $i=1,2,\cdots,k$. Moreover,

[TABLE]

if $\pi\leq\chi$. For $\overline{r}=(r_{1},r_{2},\cdots,r_{m-1})\in D_{m}$, the above process gives rise to a partition $\pi\in NC_{\geq 2}(m)$ denoted by $\pi(\overline{r})$. We denote the above $G_{m-1}$ by $G_{m-1}(\overline{r})$. We have proved that

[TABLE]

if and only if $\pi(\overline{r})\leq\chi$.

Conversely, if $\pi\in NC_{\geq 2}(m)$ and $\pi\leq\chi$, then we can arrange the blocks of $\pi$ as $V_{1},V_{2},\cdots,V_{k}$ so that the minimal element of $V_{i}$ is less than that of $V_{j}$, if $i<j$. Since $\pi$ is non-crossing, there are interval blocks in $\pi$. Let $V_{i}$ be the interval block with minimum index. Let $V_{i}=\{p+1,p+2,\cdots,p+j\}$. Define $r_{p+1}=r_{p+2}=\cdots=r_{p+j-2}=q/2,r_{p+j-1}=p$,if $j>2$; $r_{p+1}=q$, if $k=2$. Since $\pi\setminus\{V_{i}\}$ is still non-crossing, and less than $\chi|_{\{1,2,\cdots,m\}\setminus\{p+1,\cdots,p+j\}}$, we can continue the process till we take all blocks of $\pi$. Each time when choosing $r\in\{q/2,q\}$ to link $i$ and $j$ ($i<j$) in a block, we define $r_{j-1}=r$. We have defined $r_{i}$’s connecting the ascending ordered set of numbers in each block of $\pi$. For the smallest number $m_{i}$ of block $V_{i}$, we define $r_{m_{i}-1}=0$, if $m_{i}\neq 1$, for $i=1,2,\cdots,k$. We thus get a $r_{1},\cdots,r_{m-1}\in\{0,q/2,q\}$. We now show that

[TABLE]

In fact, for an interval $V_{i}$, we have $2r_{p+1}+\cdots 2r_{p+j-1}=2(|V_{i}|-2)\frac{q}{2}+2q=|V_{i}|q$. Note that every block must be an interval block in some step of the above process of definition $r_{i}$’s. Moreover, every non-zero $r_{i}$’ must link two elements in a block, when the block is treated as an interval block. It follows that $2(r_{1}+\cdots+r_{m-1})=\sum_{V\in\pi}|V|q=mq$. We need to prove that $(r_{1},r_{2},\cdots,r_{m-1})\in A_{m}$. It is obvious that $(r_{i})_{i\in V}\in A_{|V|}$, for a block $V\in\pi$. For each $r_{l}$, let $V_{1},\cdots,V_{p}$ be the blocks of $\pi$ such that $V_{i}^{\prime}:=V_{i}\cap\{1,2,\cdots,l-1\}\neq\emptyset$, $i=1,2,\cdots,p$. Then

[TABLE]

Therefore, $\overline{r}:=(r_{1},r_{2},\cdots,r_{m-1})\in D_{m}$. Moreover, the construction of $\overline{r}$ shows that $\pi(\overline{r})=\pi$. Since $\pi\leq\chi$, by the previous proof, the express $G_{m-1}$ corresponding to $(r_{1},r_{2},\cdots,r_{m-1})$ has a limit of $\prod_{V\in\pi}\lambda_{\chi(V)}$.

We have established a one to one mapping from

[TABLE]

onto $\mathfrak{P}:=\{\pi\in NC_{\geq 2}(m):\pi\leq\chi\}$. Note that

[TABLE]

if and only if $\mathfrak{F}\neq\emptyset$. Moreover, by Theorem 2.2, we have $\varphi(a_{\chi(1)}\cdots a_{\chi(m)})\neq 0$ if and only if $\mathfrak{P}\neq\emptyset$. Hence,

[TABLE]

if and only if $\varphi(a_{\chi(1)}\cdots a_{\chi(m)})\neq 0$. If $\varphi(a_{\chi(1)}\cdots a_{\chi(m)})\neq 0$, we have

[TABLE]

∎

Remark 2.5**.**

We want to give an example to illustrate the process of constructing

[TABLE]

from $\pi\in NC_{\geq 2}(m)$. Consider a partition

[TABLE]

where $V_{2}$ is an interval block. Define $r_{2}=10/2=5,r_{3}=10$, since $|V_{2}|=3>2$. Removing $V_{2}$, we get a new non-crossing partition $\pi_{1}=\{V_{1},V_{3}\}$ of $\{1,5,6,7,8,9,10\}$. $V_{3}$ is an interval block of $\pi_{1}$. Define $r_{7}=10$, since $|V_{3}|=2$. Removing $V_{3}$ from $\pi_{1}$, we get $\pi_{2}=\{1,5,6,9,10\}$, which is partition $1_{5}$ of $\{1,5,6,9,10\}$. Define $r_{4}=5,r_{5}=5,r_{8}=5,r_{9}=10$. Finally, define $r_{1}=r_{6}=0$. It is easy to verify that $(r_{1},r_{2},\cdots,r_{9})\in D_{10}$.

Lemma 2.6**.**

Under the assumptions of Lemma 2.4, we have

[TABLE]

as $n\rightarrow\infty$, for $(r_{1},r_{2},\cdots,r_{m-1})\in E_{m}:=B_{m}\setminus D_{m}$, $\chi:\{1,2,\cdots,m\}\rightarrow\mathbb{N}$, and $m>1$.

Proof.

Let $l$ be the minimal index, $1\leq l\leq m-1$, such that $r_{l}\neq q/2$, and $1\leq r_{l}<q$. Let $G_{0}:=f_{n}^{(\chi(1))}$, $G_{1}=f_{n}^{(\chi(1))}\stackrel{{\scriptstyle r_{1}}}{{\smallfrown}}f_{n}^{(\chi(2))}$, $\cdots,G_{m-1}=G_{m-2}\stackrel{{\scriptstyle r_{m-1}}}{{\smallfrown}}f_{n}^{(\chi(m))}$. Since $r_{i}\in\{0,q/2,q\}$, for $i=1,2,\cdots,l-1$, and $r_{l}\geq 1$, by the proof of Lemma 2.4, $G_{l-1}$ is a multiple of an object of the type $H_{1}\otimes\cdots\otimes H_{k}$, where $k\geq 1$, each $H_{i}$ is either equal to $f_{n}^{(j)}$ or to an iterated contraction of the type

[TABLE]

for some $s\geq 1$. let $G_{l-1}=cH_{1}\otimes\cdots\otimes H_{k}$.

Case I. $H_{k}=f_{n}^{(j)}$. We then have

[TABLE]

By (2.5) and the fact that $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\|=\sqrt{\lambda}$, for $i\in\mathbb{N}$, we get

[TABLE]

Case II. $H_{k}=f_{n}^{(j_{1})}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}\cdots\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(j_{s})}$. The proof is similar to Case I. The only difference is that, in this case, we need to prove that

[TABLE]

as $n\rightarrow\infty$. When $j_{1}=j_{2}=\cdots=j_{s}$, by the assumptions of this lemma, (2.5), and the proof of $(2.4)$, we have

[TABLE]

Finally, suppose that $j_{t}$ be the first number in the ordered sequence $j_{1},j_{2},\cdots,j_{s}$ such that $j_{t}\neq j_{1}$, that is, $j_{1}=j_{2}=\cdots=j_{t-1}\neq j_{t}$. By the above computations and $(2.5)$, we have

[TABLE]

∎

We are able to give a four-moment theorem for the multidimensional free Poisson limit over the free Wigner chaos.

Theorem 2.7**.**

Let $q\geq 2$ be an even integer, $\{\alpha_{i}:\alpha_{i}\in\mathbb{R},\alpha_{i}\neq 0,i=1,2,\cdots\}$ and $\{\lambda_{i}:\lambda_{i}>0,\lambda_{i}\in\mathbb{R}\}$ be two sequences of numbers, and $\{a_{i}:i=1,2,\cdots\}$ be a free sequence of centered free Poisson random variables $a_{i}$ with parameters $\lambda_{i}$ and $\alpha_{i}$ (that is, $\kappa_{n}(a_{i})=\lambda_{i}\alpha_{i}^{n},n\geq 2$, $\kappa_{1}(a_{i})=0$) in a non-commutative probability space $(\mathcal{A},\varphi)$. Let $\{I(f_{n}^{(i)}):i,n\in\mathbb{N}\}$ be a bi-indexed sequence of multiple integrals of order $q$ with respect to the free Brownian motion $S$, where $f_{n}^{(i)}$ is a symmetric function in $L^{2}(\mathbb{R}_{+}^{q})$. Then the following two statements are equivalent.

1. (1)

$\{I(f_{n}^{(i)}):i=1,2,\cdots\}$* converges in joint distribution to $\{a_{i}:i=1,2,\cdots\}$, as $n\rightarrow\infty$;* 2. (2)

The following equations

[TABLE]

[TABLE]

and

[TABLE]

hold, for all $i,j\in\mathbb{N}$.

Proof.

Replacing $a_{i}$ by $\frac{a_{i}}{\alpha_{i}}$, and $f_{n}^{(i)}$ by $\frac{f_{n}^{(i)}}{\alpha_{i}}$, we can assume that $\alpha_{1}=\alpha_{2}=\cdots=1$.

$(1)\Rightarrow(2)$. Note that $\lim_{n\rightarrow\infty}\langle f_{n}^{(i)},f_{n}^{(j)}\rangle=\lim_{n\rightarrow\infty}\varphi(I(f_{n}^{(i)}f_{n}^{(j)}))=\varphi(a_{i}a_{j})=\delta_{ij}\lambda_{i}$, for all $i,j=1,2,\cdots$. Moreover, for $i,j\in\mathbb{N}$, we have

[TABLE]

[TABLE]

Converse, for $m>1,m\in\mathbb{N}$ and $\chi:\{1,2,\cdots,m\}\rightarrow\mathbb{N}$, the condition (2), Theorem 2.2, Lemmas 2.3, 2,4, and 2.6 imply that

[TABLE]

∎

3. Multidimensional free Poisson limits on the free Poisson algebra

This section is devoted to proving a four-moment theorem for the convergence of a multiple integral sequence of functions in $L^{2}(\mathbb{R}_{+}^{q})$ ($q\geq 2$) with respect to a centered free Poisson random measure $\widehat{N}$. We still use $I(f)$ to denote the free stochastic integral of a function $f\in L^{2}(\mathbb{R}_{+}^{q})$ with respect to $\widehat{N}$. Let’s introduce some notations adopted from Section 3 of [SB2].

Let $i$ and $m$ be two non-negative integers such that $0\leq i\leq m-1$. We define a multiset $M_{i}^{m}=\{1,\cdots,1,0,\cdots,0\}$, where the element $1$ has multiplicity of $i$ and the element [math] has multiplicity of $m-1-i$. Let $\mathfrak{S}^{m}_{i}$ be the permutation group of the multiset $M_{i}^{m}$. For $\sigma\in\mathfrak{S}_{i}^{m}$, and a sequence $\{f_{n}:n\in\mathbb{N}\}$ of functions in $L^{2}(\mathbb{R}_{+}^{q})$, we define the following

[TABLE]

When $q>2$ is odd, we define

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In order to prove the main result in this section, we need some technical results.

Lemma 3.1**.**

Let $q>1$ and $m\geq 2$ be integers. Let $\{f_{n}:1\leq n\leq m\}$ be a (finite) sequence of functions in $L^{2}(\mathbb{R}_{+}^{q})$. Then

[TABLE]

and

[TABLE]

where $\pi(n)=0$, if $n$ is even; $\pi(n)=1$, if $n$ is odd.

The proof of the above lemma is exactly the same as those of Lemma 4.1 and Lemma 4.2 in [SB2], because the proofs of Lemma 4.1 and Lemma 4.2 in [SB2] are processes of manipulating the numbers $r_{i},r_{i}-\sigma(i)$, and the sets $\mathfrak{A}_{m}^{\sigma}$, $\mathfrak{B}_{m}^{\sigma}$, and $\mathfrak{S}_{i}^{m}$, and independent of the function $f$.

Lemma 3.2**.**

Let $q>1$ be an integer, and $\{f_{n}^{(i)}:n,i\in\mathbb{N}\}$ be a bi-indexed sequence of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$ such that $\lim_{n\rightarrow\infty}\langle f_{n}^{(i)},f_{n}^{(j)}\rangle=\delta_{ij}\lambda_{i}$, for all $i,j\in\mathbb{N}$. Then $(2.1)$ hold for all $i,j\in\mathbb{N}$, where free stochastic integrals are defined with respect to the centered free Poisson random measure, if and only if the following conditions are satisfied.

1. (1)

The case that $q$ is even. We have $\lim_{n\rightarrow\infty}(\|f_{n}^{(i)}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(i)}-f_{n}^{(i)}\|+\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(i)}\|)=0$, for all $1\leq r<q$ and $r\neq q/2$, and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{r}^{r-1}f_{n}^{(i)}\|=0$, for all $1\leq r\leq q$, for $i=1,2,\cdots$; and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(j)}\|=0$, for all $1\leq r<q$, and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{r}^{r-1}f_{n}^{(j)}\|=0$, for all $1\leq r\leq q$, if $i\neq j$. 2. (2)

The case that $q$ is odd. We have $\lim_{n\rightarrow\infty}(\|f_{n}^{(i)}\star_{(q+1)/2}^{(q-1)/2}f_{n}^{(i)}-f_{n}^{(i)}\|+\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(i)}\|)=0$, for all $1\leq r\leq q-1$, and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{r}^{r-1}f_{n}^{(i)}\|=0$, for all $1\leq r\leq q$ and $r\neq(q+1)/2$, $i=1,2,\cdots$; $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle r}}{{\smallfrown}}f_{n}^{(j)}\|=0$, for all $1\leq r\leq q-1$, and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{r}^{r-1}f_{n}^{(j)}\|=0$, for all $1\leq r\leq q$, if $i\neq j$.

Proof.

When $i=j$, this lemma is Lemma 4.3 in [SB2].

Now we prove this lemma when $i\neq j$. The proof is very similar to that of Lemma 2.4. But, in the present case, we use the production formula (1.6). When $q$ is even, the proof is almost as same as that of Lemma 2.4.

When $q$ is odd, we have

[TABLE]

Using the fact that multiple free Poisson integrals of different orders are orthogonal in $L^{2}(\mathcal{A},\varphi)$ (see (1.5)), we get

[TABLE]

On the other hand,

[TABLE]

We thus get a statement similar to that involving $(2.2)$:

[TABLE]

if and only if

[TABLE]

[TABLE]

As in the proof of Lemma 2.4, we can prove (with the same methods as those in the corresponding proofs in Lemma 2.4) that $\langle f\star_{(q+1)/2}^{(q-1)/2}g,g\rangle=\langle f,g\star_{(q+1)/2}^{(q-1)/2}g\rangle$, for $f,g\in L^{2}(\mathbb{R}_{+}^{q})$, and therefore,

[TABLE]

if $(2.1)$ holds in the present situation. It implies that if $(2.1)$ holds, then

[TABLE]

Conversely, $(3.4)$ implies $(3.3)$. Therefore, $(2.1)$ holds in the case that $q$ is odd and $i\neq j$. ∎

Now we are ready to give the main result of this section.

Theorem 3.3**.**

Let $q\geq 2$ be an even integer, $\{\alpha_{i}:\alpha_{i}\in\mathbb{R},\alpha_{i}\neq 0,i=1,2,\cdots\}$ and $\{\lambda_{i}:\lambda_{i}>0,\lambda_{i}\in\mathbb{R}\}$ be two sequences of numbers, and $\{a_{i}:i=1,2,\cdots\}$ be a free sequence of centered free Poisson random variables $a_{i}$ with parameters $\lambda_{i}$ and $\alpha_{i}$ in a non-commutative probability space $(\mathcal{A},\varphi)$. Let $\{I(f_{n}^{(i)}):i,n\in\mathbb{N}\}$ be a bi-indexed sequence of multiple integrals of order $q$ with respect to the centred free Poisson random measure $\widehat{N}$, where $f_{n}^{(i)}$ is a symmetric function in $L^{2}(\mathbb{R}_{+}^{q})$ such that there are numbers $M_{i}$ and $L_{i}$, and a $q$-dimensional square $S_{n,i}$ such that

[TABLE]

and

[TABLE]

for $i\in\mathbb{N}$. Then the following two statements are equivalent.

1. (1)

$\{I(f_{n}^{(i)}):i=1,2,\cdots\}$* converges in distribution to $\{a_{i}:i=1,2,\cdots\}$, as $n\rightarrow\infty$;* 2. (2)

The equations $(2.6)$, $(2.7)$, and $(2.8)$ hold, for all $i,j\in\mathbb{N}$, where the free integrals in $(2.7)$ and $(2.8)$ are defined with respect to the centered free Poisson random measure.

Proof.

As the proof of Theorem 2.7, replacing $a_{i}$ by $\frac{a_{i}}{\alpha_{i}}$, and $f_{n}^{(i)}$ by $\frac{f_{n}^{(i)}}{\alpha_{i}}$, we can assume that $\alpha_{1}=\alpha_{2}=\cdots=1$.

The implication of $(1)\Rightarrow(2)$ is same as the proof of $(1)\Rightarrow(2)$ of Theorem 2.7.

Conversely, for $\chi(1),\cdots,\chi(m),1<m\in\mathbb{N}$, by $(3.2)$, we have

[TABLE]

By the proofs of Lemmas 2.4 and 2.6, the first sum on the right-hand side of the above equation converges to $\varphi(a_{\chi(1)}a_{\chi(2)}\cdots a_{\chi(m)})$, as $n\rightarrow\infty$. Note also that the second sum in the above equation disappears when $m=2$. Therefore, it remains to prove

[TABLE]

We need a Cauchy-Schwartz type inequality for $f\star_{r}^{r-1}g$, where $1<r\leq q$, $f,g\in L^{2}(\mathbb{R}_{+}^{q})$ satisfy the bounded conditions presented in this theorem. Assume that

[TABLE]

[TABLE]

where $supp\{f\}$ is the support of $f$, $S(f)$ and $S(g)$ are $q$-dimensional squares, and $\mu(S(f))$ is the volume of $S(f)$. We have

[TABLE]

We, therefore, get the following inequality

[TABLE]

Let

[TABLE]

Then we have

[TABLE]

where $\sigma=0$ or $1$, and $f,g\in L^{2}(\mathbb{R}_{+}^{q})$ satisfying the above bounded conditions. For two sequences $\{f_{n}:n=1,2,\cdots\}$ and $\{g_{n}:n=1,2,\cdots\}$ with uniform (square) supports

[TABLE]

we have

[TABLE]

whenever $\lim_{n\rightarrow\infty}\|f_{n}\|=0$ and $\sup\{\|g_{n}\|:n=1,2,\cdots\}<\infty$. Moreover, the above calculation also shows that

[TABLE]

for all $(t_{1},\cdots,t_{2q-2r+\sigma})\in\mathbb{R}_{+}^{2q-2r+\sigma}$, and

[TABLE]

Let $G_{0,n}=f_{n}^{(\chi(1))}$,

[TABLE]

It implies that

[TABLE]

whenever $\lim_{n\rightarrow\infty}\|G_{i,n}\|=0$, for some $1\leq i\leq m-2$. Also $(3.6)$ implies $(3.5)$.

Therefore, it is sufficient to prove $\lim_{n\rightarrow\infty}\|G_{i,n}\|=0$, for some $1\leq i\leq m-2$.

There exist $k$ less that $m$ and $l$ less that $m-1$, such that $\chi(1)=\cdots=\chi(k)\neq\chi(k+1)$, and $\sigma(1)=\cdots=\sigma(l-1)=0,\sigma(l)=1$.

Case I. $l\leq k$. If there is an item $r_{l_{1}}$ in the sequence $r_{1},r_{2},\cdots r_{l-1}$ such that $r_{1},\cdots,r_{l_{1}-1}\in\{0,q/2,q\}$ and $r_{l_{1}}$ is not in $\{0,q/2,q\}$, then by the proof of Lemma 2.4, $G_{l_{1}-1,n}=cH_{1}\otimes\cdots\otimes H_{v}$, where $c$ is a constant, $H_{v}=f_{n}^{(s_{1})}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}\cdots\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(s_{t})}$, $H_{v}=f_{n}^{(s_{1})}$ if $t=1$, other $H_{j}$’s have the same form as $H_{v}$. By Lemma 3.2 and the proof of Lemma 2.4, we get

[TABLE]

since $\lim_{n\rightarrow\infty}\|f_{n}^{(s_{t})}\stackrel{{\scriptstyle r_{l_{1}}}}{{\smallfrown}}f_{n}^{(\chi(l_{1}+1))}\|=0$, by Lemma 3.2. Otherwise, if $r_{1},\cdots,r_{l-1}\in\{0,q/2,q\}$, the above calculation shows that

[TABLE]

since $\lim_{n\rightarrow\infty}\|f_{n}^{(s^{\prime}_{t^{\prime}})}\star_{r_{l}}^{r_{l}-1}f_{n}^{(\chi(l+1))}\|=0$ by Lemma 3.2, no matter whether $s_{t^{\prime}}^{\prime}=\chi(l+1)$ or not, where we assume $G_{l-1,n}=c^{\prime}H_{1}^{\prime}\otimes\cdots\otimes H_{v^{\prime}}^{\prime}$, and $H_{v^{\prime}}^{\prime}=f_{n}^{(s_{1}^{\prime})}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}\cdots\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(s_{t^{\prime}}^{\prime})}$.

Case II. $k<l$. We show that

[TABLE]

By the proof of Lemma 2.4, $G_{k,n}=f_{n}^{(\chi(1))}\stackrel{{\scriptstyle r_{1}}}{{\smallfrown}}f_{n}^{(\chi(2))}\cdots\stackrel{{\scriptstyle r_{k}}}{{\smallfrown}}f_{n}^{(\chi(k+1))}=H_{1}\otimes\cdots\otimes H_{p}$, where each factor $H_{i}$ has a similar form to $G_{k,n}$, but all $r_{j}$’s are nonzero. Let $f_{n}^{(\chi(k+1))}$ be a factor of $H_{p}$. Let $H_{p}=f_{n}^{(j_{1})}\stackrel{{\scriptstyle s_{1}}}{{\smallfrown}}f_{n}^{(j_{2})}\cdots\stackrel{{\scriptstyle s_{t}}}{{\smallfrown}}f_{n}^{(\chi(k+1))}$. If there is a number $s_{u}$ in $\{s_{1},\cdots,s_{t-1}\}$, which is neither $q/2$ nor $q$, but, $s_{1},\cdots,s_{u-1}\in\{q/2,q\}$, then by the proof of Case I of this theorem, we have

[TABLE]

If $s_{1},\cdots,s_{t}\in\{q/2,q\}$, we get $\lim_{n\rightarrow\infty}\|H_{p}\|\leq C_{1}\lim_{n\rightarrow\infty}\|f_{n}^{(j_{t})}\stackrel{{\scriptstyle s_{t}}}{{\smallfrown}}f_{n}^{(\chi(k+1))}\|=0$, by Lemma 3.2, since $\chi(k+1)\neq j_{t}$, where $C_{1}$ is a constant. Equation $(3.7)$ follows now. ∎

4. Another case of infinite dimensional free Poisson distribution limits

When $\lambda_{1}=\lambda_{2}=\cdots=\lambda>0$, and $p_{N}^{(1)}=p_{N}^{(2)}=\cdots=p_{N}$ with $\varphi_{N}(p_{N})=\lambda/N$, for $N\in\mathbb{N},N>\lambda$, in Definition 2.1, we shall get a special kind of infinite dimensional free Poisson distributions as follows.

Definition 4.1**.**

A sequence $\{b_{i}:i=1,2,\cdots\}$ of random variables in a non-commutative probability space $(\mathcal{A},\varphi)$ has an infinite dimensional free Poisson distribution with parameters $\lambda$ and $\alpha:=(\alpha_{1},\alpha_{2},\cdots,)$, a sequence of non-zero real numbers, if

[TABLE]

for $n\in\mathbb{N}$ and $\chi:\{1,2,\cdots,n\}\rightarrow\mathbb{N}$. Let $a_{i}=b_{i}-\varphi(b_{i})$, for $i=1,2,\cdots$. Then

[TABLE]

We say that $\{a_{n}:n=1,2,\cdots\}$ has a centered infinite dimensional free Poisson distribution denoted by $Z(\lambda,\alpha)$.

In this section, we shall give four-moment theorems for convergence to a centered infinite dimensional free Poisson distribution $Z(\lambda,\alpha)$ of a bi-index $\{I(f_{n}^{(i)}):i,n=1,2,\cdots\}$ of free stochastic integrals of $f_{n}^{(i)}\in L^{2}(\mathbb{R}_{+}^{q})$ with respect to free Brownian motion or centered free Poisson random measure ($q>1$).

After coefficient adjusting, we can assume $\alpha_{1}=\alpha_{2}=\cdots=1$. In this case, $Z(\lambda,\alpha)$ is denoted by $Z(\lambda)$. If a sequence $\{a_{i}:i=1,2,\cdots\}$ of random variables has a distribution $Z(\lambda)$, then its distribution is same as the distribution of a single free Poisson random variable. A necessary condition for a bi-indexed sequence $\{I(f_{n}^{(i)}):n,i=1,2,\cdots\}$ to converge to $\{a_{i}:i=1,2,\cdots\}$ is the following asymptotically linear dependence $(4.1)$, where $f_{n}^{(i)},i,n\in\mathbb{N}$, are symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$, and the free stochastic integrals $I(f_{n}^{(i)})$ are defined with respect to the free Brownian motion or a centered free random measure. In certain sense, $(4.1)$ indicates that limit behavior of the bi-indexed sequence is like the limit behavior of a single-indexed sequence. Following the strategy for the proofs of main results in previous two sections, we find that the present situation is, in certain sense, like a special case $i=j$ in Theorems 2.7 and 3.3. Therefore, we can adopt most of the ideas in the proofs in Sections 2 and 3, and in the four-moment theorem results for single-indexed sequences of free stochastic integrals ( Theorem 1.4 in [NP] and Theorem 1.5 in [SB2]).

Theorem 4.2**.**

Let $\lambda>0$ and $\alpha:=\{\alpha_{1},\alpha_{2},\cdots\}$ be a sequence of non-zero real numbers, and $\{f_{n}^{(i)}:n,i\in\mathbb{N}\}$ be a bi-index sequence of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$, where $q\in\mathbb{N}$ is even. Let $I(f)$ be the free stochastic integral of $f\in L^{2}(\mathbb{R}_{+}^{q})$ with respect to the free Brownian motion. Then the following two statements are equivalent.

1. (1)

The sequence $\{\{I(f_{n}^{(i)}):i=1,2,\cdots\};n=1,2,\cdots\}$ converges in distribution to $\{a_{i}:i=1,2,\cdots\}$ with distribution $Z(\lambda,\alpha)$. 2. (2)

The following equations hold

[TABLE]

[TABLE]

for all $i,j\in\mathbb{N}$.

Proof.

As the proofs of Theorem 2.7 and 3.3, we only need to prove $(2)\Rightarrow(1)$ when $\alpha_{1}=\alpha_{2}=\cdots=1$. Let $1<m\in\mathbb{N}$ and $\chi:\{1,2,\cdots,m\}\rightarrow\mathbb{N}$. By the proof Lemma 2.3 (or See Lemma 5.1 in [NP]), Condition $(2)$ implies that

[TABLE]

for $r=1,2,\cdots,q-1$ and $r\neq q/2$. Thus, by the proof of $(2.4)$, we have

[TABLE]

since $\lim_{n\rightarrow\infty}\langle f_{n}^{(i)},f_{n}^{(j)}\rangle=\lambda$ and $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\stackrel{{\scriptstyle q/2}}{{\smallfrown}}f_{n}^{(j)}-f_{n}^{(j)}\|=0$. We thus get the following equation ($(5.2)$ in [NP]) from $(4.3)$

[TABLE]

where $(r_{1},\cdots,r_{m-1})\in D_{m}$, and $j$ is the number of $q$’s in the sequence $r_{1},\cdots,r_{m-1}$. It implies from the proof of Lemma 5.2 in [NP] that

[TABLE]

By the proof of Lemma 2.6, we have

[TABLE]

for every $(r_{1},\cdots,r_{m-1})\in E_{m}:=B_{m}\setminus D_{m}$. It follows that

[TABLE]

∎

Lemma 4.3**.**

Let $q>1$ be an odd number and $m\geq 2$ be an integer. Let $\{f_{n,i}:i,n\in\mathbb{N}\}$ be a bi-indexed sequence of symmetric functions in $L^{2}(\mathbb{R}_{+}^{q})$ such that $\lim_{n\rightarrow\infty}\langle f_{n,i},f_{n,j}\rangle=\lambda>0$, for all $i,j\in\mathbb{N}$. There exist numbers $M_{i}$ and $L_{i}$, and a $q$-dimensional square $S_{n,i}$ such that

[TABLE]

[TABLE]

for $i\in\mathbb{N}$. If $\lim_{n\rightarrow\infty}\|f_{n,i}\star_{(q+1)/2}^{(q-1)/2}f_{n,j}-f_{n,j}\|=0$, for all $i,j\in\mathbb{N}$, then

[TABLE]

as $n\rightarrow\infty$, where $\pi(n)=0$, if $n$ is even; $\pi(n)=1$ if $n$ is odd, $\{a_{i}:i=1,2,\cdots\}$ is a sequence of random variables with distribution $Z(\lambda)$.

Proof.

The proof is similar to that of Lemma 2.4. We first prove that

[TABLE]

as $n\rightarrow\infty$, where $(r_{1},\cdots,r_{m-2})\in\mathfrak{D}_{m-1}^{\sigma}$, and $j$ is the number of $q$’s appearing $(r_{1},\cdots,r_{m-2})$. Since $(r_{1},\cdots,r_{m-2})\in\mathfrak{D}_{m-1}^{\sigma}$ and $G_{m-1}$ is a number, $G_{m-1}$ must be a product of $j$ scalar products having either $\langle f_{n,i},f_{n,j}\rangle$ or $(\cdots((f_{n,j_{1}}\star_{(q+1)/2}^{(q-1)/2}f_{n,j_{2}})\star_{(q+1)/2}^{(q-1)/2})\cdots\star_{(q+1)/2}^{(q-1)/2}f_{n,j_{k}})\stackrel{{\scriptstyle q}}{{\smallfrown}}f_{n,j_{k+1}}$. Now we show that

[TABLE]

By the hypotheses of this lemma and the proof of (2.4), we have

[TABLE]

We have proved (4.5), which implies $(4.4)$. Then following the proof of Lemma 4.4 in [SB2] (and Lemma 5.2 in [NP]), we get

[TABLE]

as $n\rightarrow\infty$, where $R_{m,j}$ is the number of non-crossing partitions in $NC_{\geq 2}(m)$ with exactly $j$ blocks. On the other hand,

[TABLE]

∎

Theorem 4.4**.**

Let $q\geq 2$ be an integer, $\lambda>0$ and $\alpha:=\{\alpha_{1},\alpha_{2},\cdots\}$ be a sequence of non-zero real numbers, and $\{a_{i}:i=1,2,\cdots\}$ be a sequence of random variables in a non-commutative probability space $(\mathcal{A},\varphi)$ with distribution $Z(\lambda,\alpha)$. Let $\{I(f_{n}^{(i)}):i,n\in\mathbb{N}\}$ be a bi-indexed sequence of multiple integrals of order $q$ with respect to the centred free Poisson random measure $\widehat{N}$, where $f_{n}^{(i)}$ is a symmetric function in $L^{2}(\mathbb{R}_{+}^{q})$ such that there are numbers $M_{i}$ and $L_{i}$, and a $q$-dimensional square $S_{n,i}$ such that

[TABLE]

[TABLE]

for $i\in\mathbb{N}$. Then the following two statements are equivalent.

1. (1)

$\{I(f_{n}^{(i)}):i=1,2,\cdots\}$* converges in distribution to $\{a_{i}:i=1,2,\cdots\}$ , as $n\rightarrow\infty$;* 2. (2)

The equations $(4.1)$ and $(4.2)$ hold, for all $i,j\in\mathbb{N}$, where the free integrals in $(4.2)$ are defined with respect to the centered free Poisson random measure.

Proof.

As in the previous theorems, we only need to prove $(2)\Rightarrow(1)$ in the case that $\alpha_{1}=\alpha_{2}=\cdots=1$. For $1<m\in\mathbb{N}$ and $\chi:\{1,2,\cdots\}\rightarrow\mathbb{N}$. It is sufficient to prove

[TABLE]

Case I. $q$ is even. Equation $(4.2)$ implies that

[TABLE]

It follows from (4.7), the proof of Lemma 4.3 in [SB2] that

[TABLE]

for $1\leq r<q$, $r\neq q/2$, and $1\leq s\leq q$. Then, exactly following the proof of $(2)\Rightarrow(1)$ in Theorem 3.3, we get (4.6).

Case II. $q$ is odd. By $(3.2)$, we have

[TABLE]

Condition $(2)$, Lemmas 3.2 and 4.3 implies that the first sum in the right-hand side of the above equation approaches $\varphi(a_{\chi(1)}\cdots a_{\chi(m)})$, as $n\rightarrow\infty$. It remains to show that

[TABLE]

as $n\rightarrow\infty$, for all $(r_{1},\cdots,r_{m-2})\in\mathfrak{E}_{m-1}^{\sigma}$. Let $r_{l}$ be the first item in the sequence $r_{1},r_{2},\cdots,r_{m-2}$ which breaks the rules in $\mathfrak{D}_{m-2}^{\sigma}$, that is, for $j<l$, $r_{j}\in\{0,\frac{q+1}{2},q\}$ and $r_{j}\in\{0,q\}$ if and only if $\sigma(j)=0$. We shall prove $(4.8)$ in the all possible cases as follows.

$(i)$. The number $r_{l}$ satisfies $1\leq r_{l}<q$ and $r_{l}\neq(q+1)/2$. In this case, the equation $(4.8)$ follows from the equation $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{r}^{r-\sigma}f_{n}^{(j)}\|=0$, where $\sigma=0$ or $1$, $1\leq r<q,r\neq(q+1)/2$ (see Lemma 3.2 (2)), $(3.6)$, and the proof of Theorem 3.3.

$(ii)$. The number $r_{l}=q$ and $\sigma(l)=1$. In this case, the equation $(4.8)$ follows from the equation $\lim_{n\rightarrow\infty}\|f_{n}^{(i)}\star_{q}^{q-1}f_{n}^{(j)}\|=0$ (see Lemma 3.2 (2)), $(3.6)$, and the proof of Theorem 3.3.

$(iii)$. The number $r_{l}=(q+1)/2$ and $\sigma(l)=0$. In this case, we have

[TABLE]

by Lemma 3.2 (2), since $(q+1)/2\neq q$. The equation $(4.8)$ follows by (4.9), (3.6) and the proof of Theorem 3.3. ∎