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

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.