New moments criteria for convergence towards normal product/tetilla laws

Abstract

In the framework of classical probability, we consider the normal product distribution $F_{\infty }\sim N_1\times N_2$ where $N_1,N_2$ are two independent standard normal random variable, and in the setting of free probability, $F_{\infty }\sim \left(S_1{S}_2+S_2{S}_1\right)/\sqrt{2}$ known as {\it tetilla law} \cite{d-n}, where $S_1,S_2$ are freely independent normalized semicircular random variables. We provide novel characterization of $F_{\infty }$ within the second Wiener (Wigner) chaos. More precisely, we show that for any generic element $F$ in the second Wiener (Wigner) chaos with variance one the laws of $F$ and $F_{\infty }$ match if and only if $μ_4 (F)= 9 \, (\mbox{resp. }φ(F^4)=2.5)$, and $μ_{2r}(F)= ((2r-1)!!)^2 \, (\mbox{resp. }φ(F^{2r})=φ(F^{2r}_\infty ))$ for some $r\ge 3$, where ${\mu }_r(F)$ stands for the $r$th moment of the random variable $F$, and $\phi$ is the relevant tracial state. We use our moments characterization to study the non central limit theorems within the second Wiener (Wigner) chaos and the target random variable $F_{\infty }$. Our results generalize the findings in Nourdin \& Poly \cite{n-p-2w}, Azmoodeh, et. al \cite{a-p-p} in the classical probability, and of Deya \& Nourdin \cite{d-n} in the free probability setting.

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References in corpus (4)

• Central limit theorems for sequences of multiple stochastic integrals
• Noncentral convergence of multiple integrals
• Chaos of a Markov operator and the fourth moment condition
• A bound on the 2-Wasserstein distance between linear combinations of independent random variables

Cited by in corpus (1)

• Malliavin-Stein Method: a Survey of Recent Developments