Inequalities on the Overshoot beyond a Boundary for Independent Summands with Differing Distributions

摘要

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 𝔼Rbk.