Extremes of Markov sequences (Q1890877)

For a stationary Markov sequence \(X_j\), \(j \geq 0\), with transition density \(f(x,y)\), the authors estimate the maximum over \(u_n \leq z \leq \omega\) of \(|P (\max (X_0, X_1, \ldots, X_n) < z) - P^{n + 1} (X_0 < z) |\), where \(\omega = \sup \{u : P(X_0 < u) < 1\}\) and \(u_n\) is defined by \(P(X_0 > u_n) = 1/n\), and \(f(x,y)\) is assumed to satisfy a uniform boundedness property in \(x\). In some particular cases, the estimate is shown to be of the magnitude \((\log n)^{3/2}/n^{1/2}\).