Moments of transition-additive random variables defined on finite, regenerative random processes (Q1109431)

A finite Markov chain \(\{\xi_ n\}\) is under consideration. Trajectories of the chain are considered until some ``valid'' stopping time \(\tau\). This time is defined in the following way. A set of all pairs of states (i,j) is divided on three subsets: \(S_ 1\), \(S_ 2\), \(S_ 3\). Each pair (i,j) is labelled by a number \(v_{ij}\). Let the present state of the chain be \(\xi_ n=i\) and the next one \(\xi_{n+1}=j\). If \((i,j)\in S_ 1\) then we put \(\tau =n+1\) and do not consider an appropriate trajectory after this time. If \((i,j)\in S_ 2\) then we put \(\tau =\infty\) and also do not consider the trajectory at all. If \((i,j)\in S_ 3\) then \(\tau >n+1\) and the trajectory is developing further. The authors suggest linear algebraic equations for probabilities \(g_ i=p(\tau <\infty | \xi_ 0=i)\). Naturally, they look like well-known equations for calculation of probabilities of absorption in some states. Let \(Z_ i\) be a sum of labels for a ``valid'' trajectory beginning in the state i. The authors also suggest similar equations for the quantities \[ \mu^ p_ i=E(Z^ p_ i| \xi_ 0=i,\tau <\infty),\quad p>1(p-integer). \] An evident algorithm is proposed for calculating these quantities.