  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.

Probability