On the maximum and minimum of partial sums of random variables

Peggy Strait

Vol. 52 (1974), No. 2, 585–593

DOI: 10.2140/pjm.1974.52.585

Abstract

Let X₁,X₂,⋯,X_n be independent identically distributed random variable with S_k = X₁ + X₂ + ⋯ + X_k and S_k⁺ = max[0,S_k]. We shall derive formulas for the computation of E[min_1≦k≦nS_k⁺],E[max_1≦k≦nS_k], and E[min_1≦k≦nS_k]. The formulas are then applied to the case of standard normal random variables.

Mathematical Subject Classification 2000

Primary: 60G50

Milestones

Received: 21 February 1973

Revised: 6 June 1973

Published: 1 June 1974