Related papers: Functional Limit Theorems for Marked Hawkes Point …

We consider a classical risk process with arrival of claims following a non-stationary Hawkes process. We study the asymptotic regime when the premium rate and the baseline intensity of the claims arrival process are large, and claim size…

The purpose of this paper is to establish the convergence in law of the sequence of "midpoint" Riemann sums for a stochastic process of the form f'(W), where W is a Gaussian process whose covariance function satisfies some technical…

We prove a functional central limit theorem for partial sums of symmetric stationary long range dependent heavy tailed infinitely divisible processes with a certain type of negative dependence. Previously only positive dependence could be…

In this paper, we prove maximal inequalities and study the functional central limit theorem for the partial sums of linear processes generated by dependent innovations. Due to the general weights, these processes can exhibit long-range…

We study the asymptotic shape of the trajectory of the stochastic gradient descent algorithm applied to a convex objective function. Under mild regularity assumptions, we prove a functional central limit theorem for the properly rescaled…

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting…

In order to obtain functional limit theorems for heavy tailed stationary processes arising from dynamical systems, one needs to understand the clustering patterns of the tail observations of the process. These patterns are well described by…

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and…

We give an overview of the recent asymptotic results on the geometry of excursion sets of stationary random fields. Namely, we cover a number of limit theorems of central type for the volume of excursions of stationary (quasi--, positively…

We establish diffusion and fractional Brownian motion approximations for motions in a Markovian Gaussian random field with a nonzero mean.

We consider the stochastic heat equation on $\mathbb R^d$ with multiplicative space-time white noise noise smoothed in space. For $d\geq 3$ and small noise intensity, the solution is known to converge to a strictly positive random variable…

In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…

In this article, we fill a gap in the literature on Hawkes processes. In particular, we derive a CLT for a non linear compound marked Hawkes process. We also provide an upper bound on the convergence rate using the functional 1-Wasserstein…

Consider the following local empirical process indexed by $K\in \mathcal{G}$, for fixed $h>0$ and $z\in \mathbb{R}^d$: $$G_n(K,h,z):=\sum_{i=1}^n K \Bigl(\frac{Z_i-z}{h^{1/d}}\Big) - \mathbbE \Bigl(K \Bigl(\frac{Z_i-z}{h^{1/d}}\Big)\Big),$$…

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct new examples of processes which exhibit both divergent…

We review and present some known results for non-linear functionals of Gaussian variables in the context of discrete Gaussian fields defined on the $d$ dimensional lattice. Our main result is a Central Limit Theorem in the spirit of the…

We present the reduction of generalized Langevin equations to a coordinate-only stochastic model, which in its exact form, involves a forcing term with memory and a general Gaussian noise. It will be shown that a similar…

The Courtade-Kumar conjecture posits that dictatorship functions maximize the mutual information between the function's output and a noisy version of its input over the Boolean hypercube. We present two significant advancements related to…

Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent $H\in [0,1]$, generalising standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical…

We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph $\mathbb G$. The process is constructed as the solution to a system of Poisson driven stochastic…