arXiv subjects

Ciprian A Tudor

Publications and source records attributed to Ciprian A Tudor.

6 recordsLinked to original sources

arXiv2026

Joint convergence in Wiener chaos via transport hierarchy and Malliavin covariances

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 $Z\sim N(0,σ^2)$, 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 $Y\in L^2(Ω)$, 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 $ρ_a = \lim_N \mathbf{E}[Y_{a,N}]$. 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.

math.PR↗

arXiv2025

Quartic variation of the solution to the semilinear stochastic heat equation: limit behavior and asymptotic independence with respect to the data

This work concerns the limit behavior of the quartic variation (i.e., the power variation of order four) with respect to the time variable of the solution to the semilinear stochastic heat equation with space-time white noise. In a first step, we prove that this sequence satisfies a Central Limit Theorem and we deduce a similar result for the viscosity parameter estimator associated with the quartic variation. Then, by using a recent variant of the Stein-Malliavin calculus, we analyze the asymptotic independence between the quartic variation (as well as the associated viscosity parameter estimator) and the data used to construct it.

math.PR↗

arXiv2024

The spatial average of solutions to SPDEs is asymptotically independent of the solution

Let $\left(u(t,x), t\geq 0, x\in \mathbb{R}^d\right)$ be the solution to the stochastic heat or wave equation driven by a Gaussian noise which is white in time and white or correlated with respect to the spatial variable. We consider the spatial average of the solution $F_{R}(t)= \frac{1}{σ_R}\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx$, where $σ^2_R= \mathbf{E} \left(\int_{\vert x\vert \leq R} \left( u(t,x)-1\right) dx\right)^2$. It is known that, when $R$ goes to infinity, $F_R(t)$ converges in law to a standard Gaussian random variable $Z$. We show that the spatial average $F_R(t)$ is actually asymptotic independent by the solution itself, at any time and at any point in space, meaning that the random vector $(F_R(t), u(t, x_0))$ converges in distribution, as $R\to \infty$, to $(Z, u(t, x_0))$, where $Z$ is a standard normal random variable independent of $u(t, x_0)$. By using the Stein-Malliavin calculus, we also obtain the rate of convergence, under the Wasserstein distance, for this limit theorem.

math.PR↗

arXiv2023

Multidimensional Stein method and quantitative asymptotic independence

If $\mathbb{Y}$ is a random vector in $\mathbb{R}^{d}$, we denote by $P_{\mathbb{Y}}$ its probability distribution. Consider a random variable $X$ and a $d$-dimensional random vector $\mathbb{Y}$. Inspired by \cite{Pi}, we develop a multidimensional Stein-Malliavin calculus which allows to measure the Wasserstein distance between the law $P_{ (X, \mathbb{Y})}$ and the probability distribution $P_{Z}\otimes P_{\mathbb{Y}}$, where $Z$ is a Gaussian random variable. That is, we give estimates, in terms of the Malliavin operators, for the distance between the law of the random vector $(X, \mathbb{Y})$ and the law of the vector $(Z, \mathbb{Y})$, where $Z$ is Gaussian and independent of $\mathbb{Y}$. Then we focus on the particular case of random vectors in Wiener chaos and we give an asymptotic version of this result. In this situation, this variant of the Stein-Malliavin calculus has strong and unexpected consequences. Let $(X_{k}, k\geq 1)$ be a sequence of random variables in the $p$th Wiener chaos ($p\geq 2$), which converges in law, as $k\to \infty$, to the Gaussian distribution $N(0, σ^2)$. Also consider $(\mathbb{Y}_{k}, k\geq 1)$ a $d$-dimensional random sequence converging in $L^{2}(Ω)$, as $k\to \infty$, to an arbitrary random vector $\mathbb{U}$ in $\mathbb{R}^{d}$ and assume that the two sequences are asymptotically uncorrelated. We prove that, under very light assumptions on $\mathbb{Y}_{k}$, we have the joint convergence of $(X_{k}, \mathbb{Y}_{k}), k\geq 1)$ to $(Z, \mathbb{U})$ where $Z\sim N(0, σ^{2})$ is indeendent of $\mathbb{U}$. These assumptions are automatically satisfied when the components of the vector $\mathbb{Y}_{k}$ belong to a finite sum of Wiener chaoses or when $\mathbb{Y}_{k}=Y$ for every $k\geq 1$, where $\mathbb{Y}$ belongs to the Sobolev-Malliavin space $\mathbb{D}^{1,2}$.

math.PR↗

arXiv2023

Multidimensional Stein's method for Gamma approximation

Let F ($ν$) be the centered Gamma law with parameter $ν$ > 0 and let us denote by P Y the probability distribution of a random vector Y. We develop a multidimensional variant of the Stein's method for Gamma approximation that allows to obtain bounds for the second Wasserstein distance between the probability distribution of an arbitrary random vector (X, Y) in R x R n and the probability distribution F ($ν$) $\otimes$ P Y. In the case of random vectors with components in Wiener chaos, these bounds lead to some interesting criteria for the joint convergence of a sequence ((X n , Y n), n $\ge$ 1) to F ($ν$) $\otimes$ P Y , by assuming that (X n , n $\ge$ 1) converges in law, as n $\rightarrow$ $\infty$, to F ($ν$) and (Y n , n $\ge$ 1) converges in law, as n $\rightarrow$ $\infty$, to an arbitrary random vector Y. We illustrate our criteria by two concrete examples.

math.PR↗

arXiv2023

Asymptotic normality for a modified quadratic variation of the Hermite process

We consider a modified quadratic variation of the Hermite process based on some well-chosen increments of this process. These special increments have the very useful property to be independent and identically distributed up to asymptotically negligible remainders. We prove that this modified quadratic variation satisfies a Central Limit Theorem and we derive its rate of convergence under the Wasserstein distance via Stein-Malliavin calculus. As a consequence, we construct, for the first time in the literature related to Hermite processes, a strongly consistent and asymptotically normal estimator for the Hurst parameter.

math.PR↗