REMARKS OF THE SOJOURN TIMES OF A SEMI-MARKOV PROCESS

Let {X(n)}0infinity (X0 < X1 < ... ) be a homogeneous Markov chain and {T(n)}0infinity a sequence of non-negative integer-valued r.v.'s conditionally independent given {X(n)}0infinity. Under certain conditions {(X(n), T(n))}0infinity is a Markov renewal process. The semi-Markov process {xi(n)}0infinity associated with {(X(n), T(n))}0infinity is non-decreasing with {T(n)}0infinity as its sojourn times. In this paper we determine the marginal distributions of {xi(n)}0infinity. Under certain assumptions on P{T(n) = i parallel-to X(n)} the {T(n)}oinfinity is asymptotically i.i.d. and possesses a mixing property. This is used to show that Var(SIGMA1(n) T(n+i)) = nL(n), where {L(n)} is a slowly varying sequence. We also show that {T(n)}0infinity obeys the strong law of large numbers. Finally, under some suitable moment restrictions we prove that {T(n)}0infinity has the central limit property.