  The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$
is a sum of 1-dependent non-negative integer-valued random variables (rvs),
satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a
statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is
statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the
approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter
depending on $x$.
