A characterization for mixtures of semi-Markov processes. (Q1871360)

A stepped right-continuous random process with a countable space of states \(I\) is considerd. It can be represented by the random sequence \((\sigma_j, \xi_j)_1^\infty\), where \(\sigma_j\) is the \(j\)th jump time, and \(\xi_j\in I\) is the value of the process at time \(\sigma_j\). Let \(\nu_{im}\) be the \(m\)th hitting time of the state \(i\in I\), \(T_{im}= \sigma_{\nu_{im}+1}- \sigma_{\nu_{im}}\), and \(S_{im}= \xi_{\nu_{im}+1}\). The random matrix \((T,S) \equiv (T_{im}, S_{im})_{k\times n}\) \((i\in K\subset I\), \(k\equiv| K|\), \(1\leq m\leq n)\) is called row-wise partially exchangeable if its distribution is invariant with respect to the group of all permutations of the numbers \((1,\dots, n)\) of its columns. The random process is proved to have for any \(K\) and \(n\) a row-wise partially exchangeable matrix \((T,S)\) iff it is representable as a mixture of stepped semi-Markov processes on the space \(I\). The theorem is strengthened for the mixture of stepped Markov processes.