Related papers: Weak convergence of multivariate partial maxima pr…

This article deals with the limit distribution for a stochastic differential equation driven by a non-symmetric cylindrical $\alpha$-stable process. Under suitable conditions, it is proved that the solution of this equation converges weakly…

Statistical inference for non-stationary data is hindered by the failure of classical central limit theorems (CLTs), not least because there is no fixed Gaussian limit to converge to. To resolve this, we introduce relative weak convergence,…

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability $1/2$, respectively. The theorem entails that the law of every one-dimensional regular…

For It\^o stochastic equations in $\mathbb{R}^{d}$ with drift in $L_{d}$ several results are discussed such as the existence of weak solutions, the existence of the corresponding Markov process, Aleksandrov type estimates of their Green's…

This paper proves several weak limit theorems for the joint version of extreme order statistics and partial sums of independently and identically distributed random variables. The results are also extended to almost sure limit version.

The aim of this paper is to show that the discrete maximal function $$\mathcal{M}_{h}f(x)=\sup_{N\in\mathbb{N}}\frac{1}{|\mathbf{N}_{h}\cap[1, N]|}\Big|\sum_{n\in \mathbf{N}_{h}\cap[1, N]}f(x-n)\Big|,\ \ \mbox{for $x\in\mathbb{Z}$},$$ is of…

In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation…

It is well known and readily seen that the maximum of $n$ independent and uniformly on $[0,1]$ distributed random variables, suitably standardised, converges in total variation distance, as $n$ increases, to the standard negative…

For certain subordinators $(X_t)_{t\ge 0}$ it is shown that the process $(-t\log X_{ts})_{s>0}$ tends to an extremal process $(\hat\eta_s)_{s>0}$ in the sense of convergence of the finite dimensional distributions. Additionally it is also…

The functional autoregressive model is a Markov model taylored for data of functional nature. It revealed fruitful when attempting to model samples of dependent random curves and has been widely studied along the past few years. This…

Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling features commonly found in observed financial returns. The extremal properties of these processes are of considerable interest for…

We consider a stationary regularly varying time series which can be expressedas a function of a geometrically ergodic Markov chain. We obtain practical conditionsfor the weak convergence of the tail array sums and feasible estimators…

In this paper we prove a criterion of convergence in distribution in Skorokhod space. We apply this criterion to some special Levy processes and obtain almost-sure versions of limit theorems for these processes.

We investigate branching processes in nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely, therefore, we condition on non-extinction or…

Let $(X_1, \xi_1), (X_2,\xi_2),\ldots$ be i.i.d.~copies of a pair $(X,\xi)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $\xi$ is a positive random variable. Define $S_k := \xi_1+\ldots+\xi_k$, $k \in…

Many practical optimization problems lack strong convexity. Fortunately, recent studies have revealed that first-order algorithms also enjoy linear convergences under various weaker regularity conditions. While the relationship among…

We investigate the properties of a continuous time GARCH process as the solution to a L\'evy driven stochastic functional integral equation. This process occurs as a weak limit of a sequence of discrete time GARCH processes as the time…

In this paper the sufficient conditions for convergence in Skorokhod space $D[0,1]$ of sequence of random processes with random time substitution are obtained.

We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration. The theory provides a differential structure which describes the infinitesimal evolution of Wiener functionals at very small…

Let T be an ergodic automorphism of the d-dimensional torus T^d, and f be a continuous function from T^d to R^l. On the probability space T^d equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical…