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.
Monte Carlo estimation of total variation distance of Markov chains on large spaces, with application to phylogenetics
