Let X1, …, Xn be a sequence of r.v.s
produced by a stationary Markov chain with state space an alphabet Ω
= {ω1, …, ωq}, q [ges ] 2. We consider a set of words
{A1, …, Ar}, r [ges ] 2,
with letters from the alphabet Ω. We allow the words to have self-overlaps as well as
overlaps between them. Let [Escr ] denote the event of the appearance of a word from the set
{A1, …, Ar} at a given position.
Moreover, define by N the number of non-overlapping (competing renewal) appearances of
[Escr ] in the sequence X1, …, Xn. We derive a bound
on the total variation distance between the distribution of N and a Poisson distribution
with parameter [ ]N. The Stein–Chen method and combinatorial arguments concerning the
structure of words are employed. As a corollary, we obtain an analogous result for the i.i.d.
case. Furthermore, we prove that, under quite general conditions, the r.v. N converges in
distribution to a Poisson r.v. A numerical example is presented to illustrate the performance
of the bound in the Markov case.