Joint Distributions of the Numbers of Visits for Finite-State Markov Chains

For a discrete-time Markov chain with finite state space {1, ..., r} we consider the joint distribution of the numbers of visits in states 1, ..., r-1 during the firstNsteps or before theNth visit tor. From the explicit expressions for the corresponding generating functions we obtain the limiting multivariate distributions asN-->[infinity] when staterbecomes asymptotically absorbing and forj=1, ..., r-1 the probability of a transition fromrtojis of order 1/N.