On the moments of ladder epochs for driftless random walks
Copyright policy )On the moments of ladder epochs for driftless random walks
Let X, X1, X2, … be i.i.d. Sn =Σ1 n Xj , E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X 2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ 1/2) =∞. In this paper, we prove that if E((X– ) 2 ) <∞, then E(τ 1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ 1–1/p ) <∞, p > 1, is Σn–1–1p E|Sn| <∞.
Stopping times; optimal stopping problems; gambling theory, Sums of independent random variables; random walks, ladder epoch, moments, random walks, Optimal stopping in statistics
Stopping times; optimal stopping problems; gambling theory, Sums of independent random variables; random walks, ladder epoch, moments, random walks, Optimal stopping in statistics
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