Remarks of the sojourn times of a semi-Markov process

Let {Xn}[infinity]0(X0<X1<...) be a homogeneous Markov chain and {Tn}[infinity]0 a sequence of non-negative integer-valued r.v.'s conditionally independent given {Xn}[infinity]0. Under certain conditions {(Xn,Tn)}[infinity]0 is a Markov renewal process. The semi-Markov process {[xi]n}[infinity]0 associated with {(Xn,Tn)}[infinity]0 is non-decreasing with {Tn}[infinity]0 as its sojourn times. In this paper we determine the marginal distributions of {[xi]n}[infinity]0. Under certain assumptions on P{Tn=iXn} the {Tn}[infinity]0 is asymptotically i.i.d. and possesses a mixing property. This is used to show that Var([summation operator]n1Tn+i)=nL(n), where {L(n)} is a slowly varying sequence. We also show that {Tn}[infinity]0 obeys the strong law of large numbers. Finally, under some suitable moment restrictions we prove that {Tn}[infinity]0 has the central limit property.