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--It\^o integrals of order $p\geq 2$ 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 $Y_{r+1,N} = \langle DX_N, DY_{r,N}\rangle_H$, $r\geq 0$: 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 $q \geq p$, genuine asymptotic dependence appears and its structure depends critically on the ratio $q/p$. Writing $q = ap + r'$ with $0\leq r' < p$, 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.