Compound Poisson approximation of the number distribution for monotone strings of fixed length in a random sequence

We study the number distribution for monotone strings of a length $s$ in a sequence of $n$ random independent variables uniformly distributed on the set $\{0,\dots,N-1\}$ where $N$ is a constant. By means of the Stein method we construct an estimate of the variation distance between this distribution and a compound Poisson distribution. As a corollary of this result we prove the limit theorem as $n,s\to\infty$ for the number of monotone strings. The approximating distribution is the distribution of the sum of Poisson number of independent random variables with geometric distribution.