Extremes of integer-valued moving average models with exponential type tails (Q2488437)

An integer moving average time series of the form \(X_n=\sum_{i=-\infty}^{+\infty}\beta_i\circ Z_{n-i}\) is considered, where \(Z_i\) is an i.i.d. sequence of nonnegative integer r.v.s and \(\circ\) denotes the binomial thinning operation (i.e. \(\beta\circ Z\) is a binomial r.v. with the probability of success \(\beta\) and the number of trials \(Z\)). It is assumed that \( {\mathbf P}\{Z>n\}\sim Kn^\xi(1+\lambda)^{-n} \) as \(n\to\infty\). Then under some additional assumptions the distribution of \(X_i\) has analogous asymptotics (with different \(K\), \(\xi\) and \(\lambda\)). Asymptotics of \(\sup_{1\leq i\leq n} X_i\) is investigated. INAR(1) model \(X_n=\beta\circ X_{n-1}+Z_n\) with negative binomial and geometric mixture margins is considered as example.