Related papers: Weak convergence of self-normalized partial sums p…
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$ with the finite Choquet expectation, upper mean…
We obtain strong moment invariance principles for normalized multiple iterated sums and integrals of the form $\mathbb{S}^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...<k_\nu\leq Nt}\xi(k_1)\otimes\cdots\otimes\xi(k_\nu)$, $t\in[0,T]$ and…
In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions…
The principal results of this contribution are the weak and strong limits of maxima of contracted stationary Gaussian random sequences. Due to the random contraction we introduce a modified Berman condition which is sufficient for the weak…
We consider an Ornstein-Uhleneck (OU) process associated to self-normalised sums in i.i.d. symmetric random variables from the domain of attraction of $N(0, 1)$ distribution. We proved the self-normalised sums converge to the OU process (in…
We find a sufficient condition under which a central limit theorem for a stationary linear process is quenched. We find a stationary linear process szatisfying the Maxwell-Woodroofe condition for which the variances of partial sums are…
Limit distributions for the greatest convex minorant and its derivative are considered for a general class of stochastic processes including partial sum processes and empirical processes, for independent, weakly dependent and long range…
The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we…
In nature or societies, the power-law is present ubiquitously, and then it is important to investigate the mathematical characteristics of power-laws in the recent era of big data. In this paper we prove the superposition of non-identical…
Given $n$ independent random marked $d$-vectors $X_i$ with a common density, define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near…
Let $S$ be a Scott set, or even an $\omega$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if…
A family of random matrices is said to converge strongly to a limiting family of operators if the operator norm of every noncommutative polynomial of the matrices converges to that of the limiting operators. Recent developments surrounding…
The Cram\'er-Wold device characterises weak convergence of probability measures on $\mathbb{R}^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for…
For normalized sums $Z_n$ of i.i.d. random variables, we explore necessary and sufficient conditions which guarantee the normal approximation with respect to the R\'enyi divergence of infinite order. In terms of densities $p_n$ of $Z_n$,…
In previous papers, we studied the asymptotic behaviour of $S_N(A,X)=(2N+1)^{-d/2}\sum_{n \in A_N} X_n,$ where $X$ is a centered, stationary and weakly dependent random field, and $A_N=A \cap [-N,N]^d$, $A \subset \mathbb{Z}^d$. This leads…
Let $\{S_n,n\geq 0\} $ be a random walk whose increments belong without centering to the domain of attraction of an $\alpha$-stable law $\{Y_t,t\geq 0\}$, i.e. $S_{nt}/a_n\Rightarrow Y_t,t\geq 0,$ for some scaling constants $a_n$. Assuming…
In this paper, we quantitative convergence in $W_2$ for a family of Langevin-like stochastic processes that includes stochastic gradient descent and related gradient-based algorithms. Under certain regularity assumptions, we show that the…
Let $r=r(n)$ be a sequence of integers such that $r\leq n$ and let $X_1,\ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $\mathbb{R}^n$. Limit theorems for the…
Consider the set of Borel probability measures on $\mathbf{R}^k$ and endow it with the topology of weak convergence. We show that the subset of all probability measures which belong to the domain of attraction of some multivariate extreme…
In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.