Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

We consider a discrete-time two-dimensional process { ( X 1 , n , X 2 , n ) } on Z + 2 with a supplemental process { J n } on a finite set, where the individual processes { X 1 , n } and { X 2 , n } are both skip-free. We assume that the joint process { Y n } = { ( X 1 , n , X 2 , n , J n ) } is Markovian and that the transition probabilities of the two-dimensional process { ( X 1 , n , X 2 , n ) } are modulated depending on the state of the supplemental process { J n } . This modulation is space homogeneous except for the boundaries of Z + 2 . We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.