Sharp asymptotics of large deviations for general state-space Markov-additive chains in \(\mathbb{R}^d\)
Let \(X_N\) be a Markov chain with state-space \(E\), \(\sigma\)-field \(\mathcal E\), assumed to be irreducible with respect to a maximal irreducibility measure \(\varphi\) on \((E,{\mathcal E})\) and aperiodic. Consider a time-homogeneous Markov-additive chain \((X_N,S_N)\), \(N=0,1,2,\dots\), where \(S_N=\sum^N_{i=0}\xi_i\) while \((X_N,\xi_N)\) is a Markov chain taking values in the space \(E\times\mathbb{R}^d\). Using tilted split-chain distribution, a representation formula for large deviations has been reported. Asymptotic results of the form \[ P(S_N\in N\Gamma; x_N\in A)=N^r e^{-Na}(d_0+o(1)) \] as \(N\to \infty\) are obtained.
scientific article; zbMATH DE number 1448744