Signed Poisson approximations for Markov chains (Q1613613)

The authors consider a sum of Markov dependent random variables. They prove that, with the aid of ``signed Poisson approximation'', better results can be achieved to compare with the usual normal approximation. Let \(E_a\) denote the distribution concentrated at a point \(a\in {\mathbb R}\), \(E\equiv E_0\). Let \(\lambda \in {\mathbb R}\), and let \(F\) be a distribution. Then \(\exp \{\lambda (F-E)\}\) is called a signed compound Poisson measure. In particular, \(\exp \{\lambda (E_1-E)\}\) is called a signed Poisson measure. Let \(\xi _0,\xi _1,\ldots \) be an \(s\)-state Markov chain with the (irreducible and aperiodic) transition matrix \(P = (p_{ij})_{i,j=1,\ldots ,s}\) and initial distribution \(\Pi \). For a fixed integer-valued function \(f\) we denote by \(F_{n_0}\) the distribution of the sum \( S_{n_0} = \sum _{j=0}^n f(\xi _j). \) Then the main result of the paper states \[ \|F_{n_0}-G_0^n\|= {\mathcal O}(n^{-1/2})\quad \text{and}\quad \|F_{n_0}-G_0^n(E+G_{01})\|= o(n^{-1/2}) \] where \[ G_{0} = \exp \left \{{\sigma ^2+\mu \over 2} (E_1-E) + {\sigma ^2-\mu \over 2}(E_{-1}-E) \right \},\quad G_{01} =K(E_1-E) + n(L-\mu) (E_1-E)^3/6, \] \(\mu \) is the mean value, \(\sigma ^2\) is the (asymptotic) variance, and \(K, L\) are parameters depending on the distribution of the Markov chain. The main improvement to compare with the usual normal approximation is the total variation norm in the above result. The paper contains also a couple of further results based on the above concept.