Heavy traffic limit theorems in fluctuation theory

Abstract

Letbe a family of random walks withFor ε↓0 under certain conditions the random walkU(∊)nconverges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. LetM∊= max {U(∊)n,n≧ 0},v0= min {n:U(∊)n=M∊},v1= max {n:U(∊)n=M∊}. The joint limiting distribution of ∊2σ∊–2v0and ∊σ∊–2M∊is determined. It is the same as for ∊2σ∊–2v1and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊= 0) and lim ∊–1P(M∊<x) are determined. Proofs are by direct comparison of corresponding probabilities forU(∊)nand for a special family of random walks related toMI/M/1 queues, using the central limit theorem.