Inequalities on the overshoot beyond a boundary for independent summands with differing distributions

Abstract

Let { Sn : n ≥ 0 }(S0 = 0) denote the successive sums of independent non-negative random variates, of possibly differing distributions. Define: (1) the number N (b) = inf { n ≥ 0 : Sn > b } of sums in the interval [0, b]; and (2) the overshoot Rb = SN (b) - b. This paper bounds the tail P { Rb > c } and the moments E Rbk.