  A transition matrix $[U_{i,j}]_{i,j\geq 0}$ on $\mathbb{N}$ is said to be
almost upper triangular if $U_{i,j}\geq 0\Rightarrow j\geq i-1$, so that the
increments of the corresponding Markov chains are at least $-1$; a transition
matrix $[L_{i,j}]_{i,j\geq 0}$ is said to be almost lower triangular if
$L_{i,j}\geq 0\Rightarrow j\leq i+1$, and then, the increments of the
corresponding Markov chains are at most $+1$. In the present paper, we
characterize the recurrence, positive recurrence and invariant distribution for
the class of almost triangular transition matrices. The upper case appears to
be the simplest in many ways, with existence and uniqueness of invariant
measures, when in the lower case, existence as well as uniqueness are not
guaranteed. We present the time-reversal connection between upper and lower
almost triangular transition matrices, which provides classes of integrable
lower triangular transition matrices. These results encompass the case of birth
and death processes (BDP) that are famous Markov chains (or processes) taking
their values in $\mathbb{N}$, which are simultaneously almost upper and almost
lower triangular, and whose study has been initiated by Karlin & McGregor in
the 1950's. They found invariant measures, criteria for recurrence, null
recurrence, among others; their approach relies on some profound connections
they discovered between the theory of BDP, the spectral properties of their
transition matrices, the moment problem, and the theory of orthogonal
polynomials. Our approach is mainly combinatorial and uses elementary algebraic
methods; it is somehow more direct and does not use the same tools.
