Remarks of the sojourn times of a semi-Markov process

Abstract

Let { X n } ∞ 0( X 0< X 1<⋯) be a homogeneous Markov chain and { T n } ∞ 0 a sequence of non-negative integer-valued r.v.'s conditionally independent given { X n } ∞ 0. Under certain conditions {( X n , T n )} ∞ 0 is a Markov renewal process. The semi-Markov process { ξ n } ∞ 0 associated with {( X n , T n )} ∞ 0 is non-decreasing with { T n } ∞ 0 as its sojourn times. In this paper we determine the marginal distributions of { ξ n } ∞ 0. Under certain assumptions on P{ T n = i|| X n } the { T n } ∞ 0 is asymptotically i.i.d. and possesses a mixing property. This is used to show that Var(∑ n 1 T n+ i )= nL( n), where { L( n)} is a slowly varying sequence. We also show that { T n } ∞ 0 obeys the strong law of large numbers. Finally, under some suitable moment restrictions we prove that { T n } ∞ 0 has the central limit property.