Renewal theory with a trend
type: Item
instance of: scholarly article
title: Renewal theory with a trend
zbMATH Open document ID: 1261.60037
author: Allan Gut
published in: Statistics \& Probability Letters
publication date: 2011-07-26T00:00:00Z
review text: Let \(Y, Y_1, \dots \) be i.i.d. random variables with finite mean 0 and set \(X_k=Y_k+k^\gamma \mu\) for \(k\geq 1\), and some \(\mu>0\) and \(0<\gamma<1\). Set \[ T_n=\sum_{k=1}^nY_k,\;S_n=\sum_{k=1}^nX_k,\quad n\geq 1. \] In the paper, some analogs of results from renewal theory for random walks (strong law, Marcinkiewich-Zygmund strong law, central limit theorem) are proved for the family of first passage times \[ \tau(t):=\min\{n: S_n>t\},\quad t\geq 0. \]
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60G40
zbMATH DE Number: 5932092
zbMATH Keywords: renewal theory
zbMATH Keywords: first passage time
zbMATH Keywords: strong law
zbMATH Keywords: Marcinkiewich-Zygmund strong law
zbMATH Keywords: central limit theorem
zbMATH Keywords: stopping time
MaRDI profile type: Publication
full work available at URL: j.spl.2011.03.031
OpenAlex ID: W2010917276
cites work: Q5812603
cites work: On the moments and limit distributions of some first passage times
cites work: Probability: A Graduate Course
cites work: Stopped Random Walks
cites work: Q4095644
DOI: 10.1016/J.SPL.2011.03.031
description: scientific article; zbMATH DE number 5932092