Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs

A discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process), $\{{\boldsymbol{Y}}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$, is a two-dimensional skip-free random walk $\{(X_{1,n},X_{2,n})\}$ on $\mathbb{Z}_+^2$ with a supplemental process $\{J_n\}$ on a finite set $S_0$. The supplemental process $\{J_n\}$ is called a phase process. The 2d-QBD process $\{{\boldsymbol{Y}}_n\}$ is a Markov chain in which the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ vary according to the state of the phase process $\{J_n\}$. This modulation is assumed to be space homogeneous except for the boundaries of $\mathbb{Z}_+^2$. Under certain conditions, the directional exact asymptotic formulae of the stationary distribution of the 2d-QBD process have been obtained in "T. Ozawa and M. Kobayashi, Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process, Queueing Systems (2018) DOI:10.1007/s11134-018-9586-x." In this paper, we give an example of 2d-QBD process and proofs of some lemmas and propositions appeared in that paper.