Renewal Theory for Random Walks with Positive Drift

Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)


Throughout this chapter (Ω, \(\mathfrak{F}\), P) is a probability space on which everything is defined, {S n , n ≥ 0} is a random walk with positive drift, that is, S o = 0, \(S_n=\sum\nolimits^n_{n-1}X_k, \,\, n \geq 1\), where {X k k ≥ 1} is a sequence of i.i.d. random variables, and \(S_n \xrightarrow{{\textrm{a.s.}}} + \infty \,\, {\textrm{as}}\,\, n \rightarrow \infty \). We assume throughout this chapter, unless stated otherwise, that 0 < EX 1 = μ < ∞ (recall Theorems 2.8.2 and 2.8.3). In this section, however, no such assumption is necessary.


Random Walk Central Limit Theorem Passage Time Renewal Process Renewal Theory 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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