Statistica Sinica 7(1997), 149-156

ON SPITZER'S FORMULA FOR THE MOMENT

OF LADDER VARIABLES

Yuan S. Chow

Academia Sinica

Abstract: Let X,X₁,X₂,... be i.i.d. random variables,EX=0,   S_n=X₁+...+X_n and ρ(ε) be the first n>=1 such that S_n(ε)<=0, where S_n(ε)=S_n-εn; for ε>=0. We prove that if E(|X^-|²)<∞, E|S_ρ(0)| is the limit of E|S_ρ(ε)(ε)| as ε→ 0. When E(X²)<∞; , this limit is evaulated by a probabilistic method. Thus we have a new proof of the Spitzer's formula for the moment of ladder variable S_ρ(0).

Key words and phrases: Euler constant, ladder variable, moment, random walks, Spitzer's formula, uniformly integrable.