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Ciprian A. Tudor

Publications and source records attributed to Ciprian A. Tudor.

At least 19 recordsLinked to original sources
arXiv2026

Multidimensional Stein's method for asymptotic independence with invariant measures of diffusion

We derive a multidimensional Stein's method for asymptotic independence in the case of a general target $μ$ with a density, being invariant measure of a diffusion process. It allows us to give a general bound in Wasserstein distance between the law of a couple $(X, Y)$, where $X$ is a random variable, and $Y$ a random vector and $μ\otimes \mathrm{Law}(Y)$. We focus in particular in the case where $X$ and $Y$ are differentiable in the Malliavin sense, by being function of a finite number of stochastic Wiener integrals.

math.PR↗
arXiv2024

On explosion time in stochastic differential equations driven by fractional Brownian motion

In this article, we study the explosion time of the solution to autonomous stochastic differential equations driven by the fractional Brownian motion with Hurst parameter $H>1/2$. With the help of the Lamperti transformation, we are able to tackle the case of non-constant diffusion coefficients not covered in the literature. In addition, we provide an adaptive Euler-type numerical scheme for approximating the explosion time.

math.PR↗
arXiv2024

Asymptotic expansion of the drift estimator for the fractional Ornstein-Uhlenbeck process

We present an asymptotic expansion formula of an estimator for the drift coefficient of the fractional Ornstein-Uhlenbeck process. As the machinery, we apply the general expansion scheme for Wiener functionals recently developed by the authors [26]. The central limit theorem in the principal part of the expansion has the classical scaling T^{1/2}. However, the asymptotic expansion formula is a complex in that the order of the correction term becomes the classical T^{-1/2} for H in (1/2,5/8), but T^{4H-3} for H in [5/8, 3/4).

math.PR↗
arXiv2024

Modified wavelet variation for the Hermite processes

We define an asymptotically normal wavelet-based strongly consistent estimator for the Hurst parameter of any Hermite processes. This estimator is obtained by considering a modified wavelet variation in which coefficients are wisely chosen to be, up to negligeable remainders, independent. We use Stein-Malliavin calculus to prove that this wavelet variation satisfies a multidimensional Central Limit Theorem, with an explicit bound for the Wasserstein distance.

math.ST↗
arXiv2020

Limiting behavior of large correlated Wishart matrices with chaotic entries

We study the fluctuations, as $d,n\to \infty$, of the Wishart matrix $\mathcal{W}_{n,d}= \frac{1}{d} \mathcal{X}_{n,d} \mathcal{X}_{n,d}^{T} $ associated to a $n\times d$ random matrix $\mathcal{X}_{n,d}$ with non-Gaussian entries. We analyze the limiting behavior in distribution of $\mathcal{W}_{n,d}$ in two situations: when the entries of $\mathcal{X}_{n,d}$ are independent elements of a Wiener chaos of arbitrary order and when the entries are partially correlated and belong to the second Wiener chaos. In the first case, we show that the (suitably normalized) Wishart matrix converges in distribution to a Gaussian matrix while in the correlated case, we obtain its convergence in law to a diagonal non-Gaussian matrix. In both cases, we derive the rate of convergence in the Wasserstein distance via Malliavin calculus and analysis on Wiener space.

math.PR↗
arXiv2020

Quantitative normal approximations for the stochastic fractional heat equation

In this article we present a {\it quantitative} central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space-time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius $R$ converges, as $R$ tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

math.PR↗
arXiv2019

Limit behavior of the Rosenblatt Ornstein-Uhlenbeck process with respect to the Hurst index

We study the convergence in distribution, as $H\to \frac{1}{2}$ and as $H\to 1$, of the integral $\int_{\mathbb{R}} f(u) dZ^{H}(u) $, where $Z ^{H}$ is a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right) $ and $f$ is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.

math.PR↗
arXiv2019

Behavior with respect to the Hurst index of the Wiener Hermite integrals and application to SPDEs

We consider the Wiener integral with respect to a $d$-parameter Hermite process with Hurst multi-index ${\bf H}= (H_{1},\ldots, H_{d}) \in \left( \frac{1}{2}, 1\right) ^{d}$ and we analyze the limit behavior in distribution of this object when the components of ${\bf H}$ tend to $1$ and/or $\frac{1}{2}$. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process.

math.PR↗
arXiv2019

Integration-by-Parts Characterizations of Gaussian Processes

The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes.

math.PR↗
arXiv2019

Hurst index estimation in stochastic differential equations driven by fractional Brownian motion

We consider the problem of Hurst index estimation for solutions of stochastic differential equations driven by an additive fractional Brownian motion. Using techniques of the Malliavin calculus, we analyze the asymptotic behavior of the quadratic variations of the solution, defined via higher order increments. Then we apply our results to construct and study estimators for the Hurst index.

math.PR↗
arXiv2019

Generalized $k$-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

We analyze the generalized $k$-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian with Hurst parameter $H>\frac{1}{2}$ in time and which is white in space. The $k$-variations are defined along {\it filters} of any order $p\geq 1$ and of any length. We show that the sequence of generalized $k$-variation satisfies a Central Limit Theorem when $p> H+\frac{1}{4}$ and we estimate the rate of convergence for it via the Stein-Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for $H$ and these estimators are analyzed theoretically and numerically.

math.PR↗
arXiv2019

Parameter estimation for the Rosenblatt Ornstein-Uhlenbeck process with periodic mean

We study the least squares estimator for the drift parameter of the Langevin stochastic equation driven by the Rosenblatt process. Using the techniques of the Malliavin calculus and the stochastic integration with respect to the Rosenblatt process, we analyze the consistency and the asymptotic distribution of this estimator. We also introduce alternative estimators, which can be simulated, and we study their asymptotic properties.

math.PR↗
arXiv2016

Wiener integrals with respect to the Hermite random field and applications to the wave equation

The Hermite random field has been introduced as a limit of some weighted Hermite variations of the fractional Brownian sheet. In this work we define it as a multiple integral with respect to the standard Brownian sheet and introduce Wiener integrals with respect to it. As an application we study the wave equation driven by the Hermite sheet. We prove the existence of the solution and we study the regularity of its sample paths, the existence of the density and of its local times. 2000 AMS Classification Numbers: 60F05, 60H05, 60G18.

math.PR↗
arXiv2015

Sample Paths of the Solution to the Fractional-colored Stochastic Heat Equation

Let u = {u(t, x), t $\in$ [0, T ], x $\in$ R d } be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process u with respect to the time and space variable, respectively. In particular, we derive their exact uniform and local moduli of continuity and Chung-type laws of the iterated logarithm.

math.PR↗
arXiv2014

Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

Hermite processes are self--similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order $1$ is fractional Brownian motion and the Hermite process of order $2$ is the Rosenblatt process. We consider here the sum of two Hermite processes of order $q\geq 1$ and $q+1$ and of different Hurst parameters. We then study its quadratic variations at different scales. This is akin to a wavelet decomposition. We study both the cases where the Hermite processes are dependent and where they are independent. In the dependent case, we show that the quadratic variation, suitably normalized, converges either to a normal or to a Rosenblatt distribution, whatever the order of the original Hermite processes.

math.PR↗
arXiv2013

High order chaotic limits of wavelet scalograms under long--range dependence

Let $G$ be a non--linear function of a Gaussian process $\{X_t\}_{t\in\mathbb{Z}}$ with long--range dependence. The resulting process $\{G(X_t)\}_{t\in\mathbb{Z}}$ is not Gaussian when $G$ is not linear. We consider random wavelet coefficients associated with $\{G(X_t)\}_{t\in\mathbb{Z}}$ and the corresponding wavelet scalogram which is the average of squares of wavelet coefficients over locations. We obtain the asymptotic behavior of the scalogram as the number of observations and scales tend to infinity. It is known that when $G$ is a Hermite polynomial of any order, then the limit is either the Gaussian or the Rosenblatt distribution, that is, the limit can be represented by a multiple Wiener-Itô integral of order one or two. We show, however, that there are large classes of functions $G$ which yield a higher order Hermite distribution, that is, the limit can be represented by a a multiple Wiener-Itô integral of order greater than two.

math.PR↗
arXiv2013

Wavelet estimation of the long memory parameter for Hermite polynomial of Gaussian processes

We consider stationary processes with long memory which are non-Gaussian and represented as Hermite polynomials of a Gaussian process. We focus on the corresponding wavelet coefficients and study the asymptotic behavior of the sum of their squares since this sum is often used for estimating the long-memory parameter. We show that the limit is not Gaussian but can be expressed using the non-Gaussian Rosenblatt process defined as a Wiener Itô integral of order 2. This happens even if the original process is defined through a Hermite polynomial of order higher than 2.

math.ST↗