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Abstract:We study the joint convergence in distribution of a sequence $X_N = I_p(f_N)$ of multiple Wiener--Itô integrals of order $p\geq 2$ 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.

Submission history

From: Ciprian Tudor [view email] [via CCSD proxy]
[v1] Fri, 12 Jun 2026 08:22:47 UTC (37 KB)