Upper bounds for the maximum of a random walk with negative drift

2011/07/27 by Johannes Kugler, Kugler, Johannes, Vitali Wachtel +1
Business, Management and Accounting · Decision Sciences · Mathematics · #60G50 #60G52 #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1107.5400

openalex publication_date 2011/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a random walk S_n=∑_i=0ⁿ X_i with negative drift. This paper deals with upper bounds for the maximum M=max_{n≥ 1}S_n of this random walk in different settings of power moment existences. As it is usual for deriving upper bounds, we truncate summands. Therefore we use an approach of splitting the time axis by stopping times into intervals of random but finite length and then choose a level of truncation on each interval. Hereby we can reduce the problem of finding upper bounds for M to the problem of finding upper bounds for M_τ=max_{n≤ τ}S_n. In addition we test our inequalities in the heavy traffic regime in the case of regularly varying tails.

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