arXiv subjects

Salim Boukfal

Publications and source records attributed to Salim Boukfal.

4 recordsLinked to original sources

arXiv2025

Weak approximation for Gaussian processes from renewal processes

In previous works, Bardina and Rovira (2023) constructed a family of processes that converge strongly towards Brownian motion, defined from renewal processes, are constructed. In this paper we prove that some of these processes can be utilized to build approximations of Gaussian processes such as fractional Brownian motion or multiple Stratonovich integrals and we provide sufficient conditions on renewal processes to ensure that the convergence holds. An illustrative example of such a Gaussian process is the fractional Brownian motion with any Hurst parameter.

math.PR↗

arXiv2025

Kac-Stroock type approximations for the Brownian motion

In the present paper we show that the processes $X_n = \{X_n(t) \colon t \in [0,1]\}$, $n \in \mathbb{N}$, defined by $X_n(t) = \sqrt{n}C\int_0^t (-1)^{L(nu)} du$, where $L = \{L(t) \colon t \geq 0\}$ is a renewal processes whose inter-arrival times satisfy some integrability conditions and $C > 0$ is some normalizing constant, weakly converge, in the space of continuous functions over $[0,1]$, $\mathcal{C}([0,1])$, to the Brownian motion as $n$ approaches infinity. Thus, generalizing the result of D. W. Stroock (1982), where $L$ is taken to be a standard Poisson process. In particular, we see that these results are a mere consequence of Donsker's invariance principle.

math.PR↗

arXiv2025

Weak convergence of stochastic integrals with applications to SPDEs

In this paper we provide sufficient conditions for sequences of random fields of the form $\int_{D} f(x,y) \theta_n(y) dy$ to weakly converge, in the space of continuous functions over $D$, to integrals with respect to the Brownian sheet, $\int_{D} f(x,y)W(dy)$, where $D \subset \mathbb{R}^d$ is a rectangular domain, $x \in D$, $f$ is a function satisfying some integrability conditions and $\{\theta_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,x]}\theta_n(y)dy$ converge in law to the Brownian sheet (in the sense of the finite dimensional distribution convergence). We then apply these results to stablish the weak convergence of solutions of the stochastic Poisson equation.

math.PR↗

arXiv2025

Weak convergence of stochastic integrals

In this paper we provide sufficient conditions for sequences of stochastic processes of the form $\int_{[0,t]} f_n(u) \theta_n(u) du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, $\int_{[0,t]} f(u)W(du)$, where $\{f_n\}_n$ is a sequence satisfying some integrability conditions converging to $f$ and $\{\theta_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,t]}\theta_n(u)du$ converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

math.PR↗