Maxima of recurrent random sequences. The case of heavy tails (Q1851548)

The paper deals with random sequences \(\{X_n\}\), \(n\geq 0\), of the form \(X_n = a_nX_{n-1}+b_n\), where \(\{a_n\}\) and \(\{b_n\}\), \(n\geq 1\), are sequences of independent nonnegative random values with distributions \(A\) and \(B\), respectively, \(X_0\geq 0\), \(Ma_n<1\), \(Mb_n<\infty\), \(MX_0<\infty\), \(X_0\), \(\{a_n\}\) and \(\{b_n\}\) are independent in totality. The author studies the behavior of maxima \(M_n = \max\{X_0,\dots,X_n\}\) as \(n\to\infty\). As application a result is obtained for the maxima of shot noise processes with subexponential distribution of amplitudes and response function majorized by the exponential one.