Local limit theorem for the maximum of a random walk

2014/03/28 by Johannes Kugler, Kugler, Johannes
Mathematics · #60G50 (Primary) #60G70 #60K05 (Secondary) #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60G50 #msc:60G70 #msc:60K05

paper · pdf · doi:10.48550/arxiv.1403.7372

19 pages

arxiv created 2014/03/31 · arxiv updated 2014/04/01

Abstract

Consider a family of Δ-latticed aperiodic random walks \S^(a),0≤ a≤ a₀\ with increments X_i^(a) and non-positive drift -a. Suppose that sup_{a≤ a0}E[(X^(a))²]<∞ and sup_{a≤ a0}E[max\0,X^(a)\^2+ε]<∞ for some ε>0. Assume that X^(a)\xrightarrow[]w X⁽⁰⁾ as a→ 0 and denote by M^(a)=max_{k≥ 0} S_k^(a) the maximum of the random walk S^(a). In this paper we provide the asymptotics of P(M^(a)=yΔ) as a→ 0 in the case, when y→ ∞ and ay=O(1). This asymptotics follows from a representation of P(M^(a)=yΔ) via a geometric sum and a uniform renewal theorem, which is also proved in this paper.

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