Exact asymptotics of the stationary tail probabilities in an arbitrary direction in a two-dimensional discrete-time QBD process

We deal with a discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process for short) on $\mathbb{Z}_+^2\times S_0$, where $S_0$ is a finite set, and give a complete expression for the asymptotic decay function of the stationary tail probabilities in an arbitrary direction. The 2d-QBD process is a kind of random walk in the quarter plane with a background process. In our previous paper (Queueing Systems, vol. 102, pp. 227-267, 2022), we have obtained the asymptotic decay rate of the stationary tail probabilities in an arbitrary direction and clarified that if the asymptotic decay rate $\xi_{\boldsymbol{c}}$, where $\boldsymbol{c}$ is a direction vector in $\mathbb{N}^2$, is less than a certain value $\theta_{\boldsymbol{c}}^{max}$, the sequence of the stationary tail probabilities in the direction $\boldsymbol{c}$ geometrically decays without power terms, asymptotically. In this paper, we give the function according to which the sequence asymptotically decays, including the case where $\xi_{\boldsymbol{c}}=\theta_{\boldsymbol{c}}^{max}$. When $\xi_{\boldsymbol{c}}=\theta_{\boldsymbol{c}}^{max}$, the function is given by an exponential function with power term $k^{-\frac{1}{2}}$ except for two boundary cases, where it is given by just an exponential function without power terms. This result coincides with the existing result for a random walk in the quarter plane without background processes, obtained by Malyshev (Siberian Math. J., vol. 12, ,pp. 109-118, 1973).