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.


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.)
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