The asymptotic joint distribution of the state occupancy times in an alternating renewal process

Abstract

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function Fi, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let Ui(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (Ui(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.