## On some limit properties of the reward from a Markov replacement process.(English)Zbl 0542.60089

The finite state Markov processes with replacements are given in connection with the point processes. The marked point process with rewards is considered with marks $$\ell =1,...,q$$ such that $$\ell \sim(i,k)$$ means the transition from state i into state k, $$\ell \sim(i,+k)$$ means the replacement (the instantaneous shift of the trajectory from state i into state k) and $$\ell \sim(i,k,+k')$$ means the transition $$i\to k$$ and in the same moment the replacement $$k\to k'.$$
A compensator (integral from the intensity) $$\{$$ $$\tilde A_ t,t\geq 0\}=\{^ 1\tilde A_ t,...,^ q\tilde A_ t\}$$ of the process is investigated if the process is under such a stationary replacement policy of destination f, under which one class of recurrent states exists only. Next there is considered the compensator $$\{A_ t,t\geq 0\}$$ of the process under a common replacement policy F. Let $$R_ t$$ be the reward from the process up to the time t and $$\theta$$ the mean reward per time unit.
Theorem 1: If $$\lim_{t\to \infty}(^{\ell}A_ t-^{\ell}\tilde A_ t)=0$$ F-almost sure holds for all $$\ell =1,...,q$$ then $$\lim_{t\to \infty}t^{-1}R_ t=\theta$$ F-almost sure. In Theorem 2 there are proved conditions under which $$t^{-1/2}(R_ t-\theta t)$$ for $$t\to \infty$$ has asymptotically normal distribution N(0,$$\xi)$$, where $$\xi$$ is a certain constant.

### MSC:

 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.) 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

### Keywords:

marked point process; replacement policy
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### References:

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