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.