Strong limit theorems for large and small increments of \(\ell^ p\)- valued Gaussian processes
type: Item
instance of: scholarly article
title: Strong limit theorems for large and small increments of \(\ell^ p\)- valued Gaussian processes
zbMATH Open document ID: 0791.60028
author: Miklós Csörgo
author: Q. M. Shao
published in: The Annals of Probability
publication date: 1994-07-07T00:00:00Z
review text: Let \(\{Y(t), t \geq 0\}=\{X_ k(t), t \geq 0\}^ \infty_{k=1}\) be a sequence of independent Gaussian processes with \(\mathbb{E} X_ k(t)=0\) and stationary increments \(\sigma^ 2_ k (h)=\mathbb{E}(X_ k (t+h)-X_ k(t))^ 2\). The authors are interested in the limit behaviour of the increments of the process \(Y(t)\). In fact they consider \[ I(T)=\sup_{0 \leq t \leq T} \sup_{0 \leq s \leq a_ T} \| Y(t+s)-Y(t) \|_{\ell^ p}, \] where \(a_ T\) is a positive continuous function. The main results claim that with a suitable normalizing factor \(\gamma_ T\) the almost sure limit of the process \(\gamma_ TI(T)\) is 1 provided that some regularity conditions hold true.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 528628
zbMATH Keywords: \(\ell^ p\)-valued Gaussian processes
zbMATH Keywords: large increments
zbMATH Keywords: moduli of continuity
zbMATH Keywords: stationary increments
zbMATH Keywords: regularity conditions
reviewed by: Pál Révész
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989007
OpenAlex ID: W2078169579
DOI: 10.1214/AOP/1176989007
description: scientific article; zbMATH DE number 528628