A Boundary Problem for Sums of Lattice Random Variables Defined on a Regular Finite Markov Chain

This paper deals with two-dimensional Markov process $\{\xi_n,k_n\}$ the first coordinateof which $\{\xi_n\}$ may be considered as a sequence of sums of lattice random variables defined on a regular finite Markov chain $\{k_n\}$. Some identities are obtained that establish a certain relation between generating functions of various distributions connected with this process. Some properties of the components of these identities are investigated. With the help of these properties we study the asymptotical behaviour of the joint distribution of random variables $\max\limits_{0<m<n}\zeta_m,\zeta_n,k_n$ when some conditions are satisfied.