Extreme values in stationary sequences (Q676666)

Let \(\{X_j,\;j\geq 1\}\) be a strictly stationary sequence\ of random variables satisfying the \(\varphi\)-mixing condition and let \(M_n=\max\{X_i:1\leq i\leq n\}\). The author studies the asymptotic behaviour of \(P(M_n\leq y_n)\), where \(\{y_n\}\) is any sequence of real constants. In particular, he shows that the condition \(P(X_{j+1}>y_n\mid X_1>y_n)\to0\) \((\forall j\in N)\) as \(n\to \infty\), is sufficient for the convergence \(P(M_n\leq y_n)-[P(X_1\leq y_n)]^n\to 0\) as \(n\to \infty\). Furthermore, the author estimates rates of convergence of the distribution of the maximum \(M_n\) and presents a method of computing the extremal index of the sequence \(\{X_j\}\).