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

Abstract: We consider a discrete-time two-dimensional process ${(L_{1,n},L_{2,n})}$ on $\mathbb{Z}+^2$ with a supplemental process ${J_n}$ on a finite set, where individual processes ${L{1,n}}$ and ${L_{2,n}}$ are both skip free. We assume that the joint process ${Y_n}={(L_{1,n},L_{2,n},J_n)}$ is Markovian and that the transition probabilities of the two-dimensional process ${(L_{1,n},L_{2,n})}$ are modulated depending on the state of the background process ${J_n}$. This modulation is space homogeneous except for the boundaries of $\mathbb{Z}_+^2$. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process and, under several conditions, obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.