Characteristic of exit moments and models of enlargement of states for finite Markov chains in terms of global memory functionals

A notion of local- and global-memory functionals for discrete-type distribu tions is introduced by analogy with the notion of the memory for continuous -type distributions introduced in Muth's papers. In the class of PH-distrib utions (i.e., of distributions of the time of Markov chain exit from? a sub set of states) the necessary and sufficient conditions nle obtained for the case where the exit time has an exponential (continous-time) or a geometri c (discrete time) distribution. A new notion of a global memory functional for decomposition of the state space of a finite Markov chain is int Its pr operties as a measure of quality of decomposition and enlargement of a stat e space are studied. The asymptotic optimality is proved.