Strong limit of processes constructed from a renewal process
type: Item
instance of: scholarly article
title: Strong limit of processes constructed from a renewal process
zbMATH Open document ID: 1548.60068
author: Xavier Bardina
author: Carles Rovira
published in: Open Mathematics
publication date: 2024-09-26T00:00:00Z
review text: The authors establish strong convergence to Brownian motion for an extension of the uniform transport process based on a renewal-reward process. Let \(U_1,U_2,\ldots\) be independent and identically distributed random variables, and \(S_k=U_1+\cdots+U_k\). For a positive function \(\beta\) with \(\sum_{n\geq1}\beta(n)<\infty\), define\N\[\NT_{\beta(n)}(t)=\eta_0+\sum_{k=1}^\infty\eta_k\mathbf{1}_{[0,t]}(\beta(n)S_k)\,,\N\]\Nwhere \(\eta_0,\eta_1,\ldots\) are independent and identically distributed Bernoulli random variables with mean \(1/2\), which are also independent of the \(U_i\). The authors study the processes\N\[\Nx_n(t)=\left(\beta(n)\frac{\mathbf{E}[U_1^2]}{\mathbf{E}[U_1]}\right)^{-1/2} \int_0^t(-1)^{T_{\beta(n)}(u)}\,\text{d}u\,,\N\]\Nshowing strong convergence of these processes to Brownian motion as \(n\to\infty\). In the special case where the \(U_i\) have a uniform distribution on \((0,1)\) and \(\beta(n)=n^{-k}\) for \(k>1\), the authors further establish the corresponding rate of convergence, showing that for each \(q>0\) there exists a constant \(\alpha=\alpha(q)\) such that\N\[\N\mathbf{P}\left(\max_{0\leq t\leq1}|x_n(t)-x(t)|>\alpha n^{-k/4}(\log n)^{3/2}\right)=o(n^{-q})\N\]\Nas \(n\to\infty\), where \(\{x(t):t\geq0\}\) is a standard Brownian motion.
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 7919380
zbMATH Keywords: strong convergence
zbMATH Keywords: renewal process
zbMATH Keywords: Brownian motion
MaRDI profile type: Publication
cites work: A stochastic model related to the telegrapher's equation
cites work: Almost Sure Convergence of Uniform Transport Processes to Brownian Motion
cites work: Rate of convergence of uniform transport processes to brownian motion and application to stochastic integrals
cites work: Rate of convergence of transport processes with an application to stochastic differential equations
cites work: An explicit solution to the Skorokhod embedding problem for double exponential increments
cites work: Strong approximation of diffusion processes by transport processes
cites work: The complex Brownian motion as a strong limit of processes constructed from a Poisson process
cites work: A strong uniform approximation of fractional Brownian motion by means of transport processes
cites work: A Strong Approximation of Subfractional Brownian Motion by Means of Transport Processes
cites work: Approximations of fractional stochastic differential equations by means of transport processes
cites work: A strong convergence to the Rosenblatt process
cites work: Q5524051
DOI: 10.1515/MATH-2023-0112
description: scientific article; zbMATH DE number 7919380