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Renewal Theory for Random Walks with Positive Drift

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Stopped Random Walks

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Abstract

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.

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  1. Department of Mathematics, Uppsala University, SE-751 06, Uppsala, Sweden

    Allan Gut

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Gut, A. (2009). Renewal Theory for Random Walks with Positive Drift. In: Stopped Random Walks. Springer Series in Operations Research and Financial Engineering. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87835-5_3

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