Let be a stationary Harris recurrent Markov chain on a Polish
state space , with stationary distribution .
Let be the number of visits
to by , where is "rare" in the sense
that is "small". We want to find an approximating
*compound Poisson* distribution for , such
that the approximation error, measured using the *total
variation* distance, can be *explicitly* bounded with a bound of
order not much larger than . This is motivated by the
observation that approximating *Poisson* distributions often
give larger approximation errors when the visits to by
tend to occur in *clumps*, and also by the compound Poisson
*limit theorems* of classical *extreme value theory*.

We here propose an approximating compound Poisson distribution which
in a natural way takes into account the *regenerative*
properties of Harris recurrent Markov chains. A total variation
distance error bound for this approximation is derived, using the
compound Poisson *Stein equation* of
Barbour, Chen and Loh
(1992),
and certain *couplings*. When the chain has an
*atom* (e.g., a singleton), the bound is particularly simple,
and depends only on much studied quantities like *hitting
probabilities* and *expected hitting times*, which satisfy
*Poisson's equation* and can in some cases be bounded using
*Lyapunov* functions. Adding a few terms to the bound gives
*upper and lower* bounds for the error in the approximation with
*arbitrary* compound Poisson, or *normal*, distributions.
Applications of the above results are given, to the number of
*overlapping occurrences* of fixed sequences in finite-state
Markov chains (in particular, the sequence 111...111 in a Markov
chain on ), and to the visits by *birth-death* chains
and arbitrary *finite-state* Markov chains to "rare" sets.

The most important application concerns the *Johnson-Mehl*
crystallization model of stochastic geometry. In this model, for each
point of a Poisson point process on ,
an interval starts to grow in from with constant speed in
both directions at time . Let be the number of
components of the *uncovered set* (= the complement of the union
of growing random intervals) which intersect , at time .
can be interpreted as the number of visits to a "rare" set
by a Markov chain. We first give an approximating Poisson
distribution for , and a total variation
distance error bound, using the *Stein-Chen method* and the
existence of *monotone increasing* couplings. Then, using the
general results described above and suitable couplings, we give an
approximating compound Poisson distribution, and an error bound for
this approximation. Under a mild condition, the latter bound is of
considerably smaller order than the former. These results are
illustrated numerically.

**Keywords:** Compound Poisson approximation,
error bound, stationary Harris chain, "rare" set, Stein equation,
coupling, expected hitting times, overlapping occurrences of
patterns, Johnson-Mehl model, uncovered set.

**AMS 1991 subject classification:** Primary
60E15, 60J05. Secondary 60D05, 60G70.