Compound Poisson approximations for sums of Markov dependent random variables (Q6622505)

A compound Poisson random variable (CP) is a random sum of i.i.d. variables where the number of summands is a Poisson variable independent of the summands. The authors first review in detail known results on CP-approximations of Markov chains $\{ \xi_n \}_{n\geq 0}$, in particular of those with binomial distribution, i.e., with $P(\xi_0 =1) = p_0$, $P(\xi_0 = 0 ) = 1 - p_0$, $p_0\in [0,1]$, $P(\xi_i =1\mid \xi_{i-1}=1) = p$, $P(\xi_i =0\mid \xi_{i-1}=1) = 1-p$, $P(\xi_i =1\mid \xi_{i-1}=0) = q$, $P(\xi_i =0\mid \xi_{i-1}=0) = 1-q$, $p, q \in (0,1)$. \N\NThey then present their new improvements, culminating in a CP-approxi-mation of order $o(n^{-1/2})$ in the total-variation norm, $d_{TV}(x,y):= \sup_{A\text{ Borel}} |P(x\in A) - P(y\in A)|$, for a general Markov chain with finite number of states. This is a first attempt to prove an analogue of the first Kolmogorov theorem for Markov chains with more than three states. The proof is based on the characteristic function method.