Joint convergence in Wiener chaos via transport hierarchy and Malliavin covariances

Abstract

We study the joint convergence in distribution of a sequence $X_N=I_p(f_N)$ of multiple Wiener--Itô integrals of order $p\ge 2$ that converges to a Gaussian limit $Z\sim N(0,{\sigma }^2)$, together with another sequence $Y_N=I_q(g_N)$ converging in law. The central finding is that the joint convergence of $(X_N,Y_N)$ is completely governed by the asymptotic behavior of the iterated Malliavin covariances $Y_{r+1,N}=⟨D{X}_N,D{Y}_{r,N}{⟩}_H$, $r\ge 0$: joint convergence holds as soon as these covariances converge jointly with $Y_N$, and the structure of the limiting distribution is then explicitly determined by their limits. Moreover, the convergence of the Malliavin covariances is necessary for joint convergence, as shown by a counterexample. When $q<p$, the sequence $X_N$ is asymptotically independent of any $Y\in L^2(\Omega )$, a result which strengthens the stable convergence results in [12] and extends the multidimensional Fourth Moment Theorem [9]. When $q\ge p$, genuine asymptotic dependence appears and its structure depends critically on the ratio $q/p$. Writing $q=ap+r^{\prime }$ with $0\le r^{\prime }<p$, the iterated Malliavin covariances form a transport hierarchy of depth $a$ that terminates in both the non-critical regime $ap<q<(a+1)p$ and the critical regime $q=ap$, but with different structures: the hierarchy is nilpotent in the non-critical case and recurrent in the critical one, due to the non-vanishing limit ${\rho }_a={\lim }_N\mathbf{E}[Y_{a,N}]$. In both cases, the limiting characteristic function admits an explicit series representation whose coefficients are determined by a simple recursion. Under exponential moment assumptions, the series closes in closed form, and the two regimes differ by exactly one additional factor that appears only in the critical case.