We consider the number of overlappingoccurrences up to a fixed time
of one or several “rare ”patterns in a stationary finite-state
Markov chain. We derive a bound for the total variation distance between the
distribution of this quantity and a compound Poisson distribution, using
general results on compound Poisson approximation for Markov chains by
Erhardsson. If the state space is $\{0, 1\}$ and the pattern is a head
run ($111 \dots 111$), the bound is completely explicit and improves on an
earlier bound given by Geske, Godbole, Schaffner, Skolnick and Wallstrom. In
general, the bound can be computed by solving five linear equation systems of
dimension at most the number of states plus the sum of the lengths of the
patterns. We also give approximations with error bounds for the distributions
of the first occurrence time of a head run of fixed length and the longest head
run occurringup to a fixed time. Finally, we consider the sojourn time in an
“extreme” subset of the state space by a stationary
birth–death chain and derive a bound for the total variation distance
between the distribution of this quantity and a compound Poisson
distribution.