Limit Theorems for a Storage Process with a Random Release Rule

Abstract

We consider a discrete time Storage Process X n with a simple random walk input S n and a random release rule given by a family { U x , x ≥ 0} of random variables whose probability laws { U x , x ≥ 0} form a convolution semigroup of measures, that is, μ x × μ y = μ x + y The process X n obeys the equation: X 0 = 0, U 0 = 0, X n = S n － U S n , n ≥ 1. Under mild assumptions, we prove that the processes and are simple random walks and derive a SLLN and a CLT for each of them.