Two-boundary Problems for a Random Walk with Negative Geometric Jumps

Two-boundary problems for a random walk with negative geometric jumps are considered, and the corresponding results for a usual semicontinuous random walk are generalized for them. The following results are obtained: the probability distribution of ruin is found and expressed in terms of the lower and upper boundaries; formulas are given for the joint distribution of the infimum, supremum, and the walk itself at an arbitrary time instance; the transient probabilities and ergodic distribution are evaluated for the process describing the evolution of the random walk with two boundaries. Two-boundary problems for random walks and stochastic processes have several applications in the queue theory, storage and inventory theories, reliability theory, and in many other fields. Two-boundary problems have been studied for semicontinuous random walks and for semicontinuous stochastic processes. Several methods are known for solving those problems, namely combinatorial [1], resolvent [2]–[6], factorization [7], and renewal [8] methods. In this paper we solve two-boundary problems for random walks with negative geometric jumps. This model is a generalization of a usual model of semicontinuous random walks. 1. Main notation Let α ∈ {0, 1, . . .} be a nonnegative integer-valued random variable, E[θ] = ∞ ∑ i=0 aiθ , ai = P[α = i], P[α > 1] > 0, |θ| ≤ 1. Consider the random variable ξ = α− β, ξ ∈ {0,±1, . . .} = Z, where β ∈ {1, 2, . . .} is a positive integer-valued random variable distributed geometrically with parameter b ∈ [0, 1), that is, P[β = n] = (1 − b)bn−1, n > 0. It is clear that E[θ] = E[θ] (1− b)/θ 1− b/θ def = ∞ ∑ i=−∞ piθ , |θ| = 1. An easy evaluation yields the distribution of the random variable ξ: (1) pi = P[ξ = i] = (1− b)b−i−1 E[b;α > i], i ∈ Z. 2000 Mathematics Subject Classification. Primary 60G50, 60J50. c ©2004 American Mathematical Society 55 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use