Limit laws for a sequence between the maximum and the sum of independent exponentials

Consider a stochastic process {Xn}, n = 0, 1, 2, ... with initial value X0 and a sequence of independent, random variables, {Yi}, i [epsilon] N with exponential distribution with parameter one, where Xn+1 = max(Xn, [alpha]Xn + Yn+1), 0 < [alpha] < 1. In this paper, we show that this sequence behaves like the sequence of maxima as far as record values are concerned, that {Xn - [Log(n)]/[1 - [alpha]]} converges weakly to a nondegenerate random variable Z and, finally, we show that .