Clustering of large deviations in moving average processes: the short memory regime (Q6591595)

The authors discuss clustering of large deviations events in a light-tailed system with short memory regime (for long memory case see the article by the same authors [Stochastic Processes Appl. 163, 387--423 (2023; Zbl 1520.60018)]). More precisely, let \({Z_n, n\in\mathbb{Z}}\) be a sequence of i.i.d. nondegenerate random variables (the noise) with distribution \(F_Z\) satisfying \N\[\N\int_{\mathbb{R}}e^{tz}F_Z(dz)<\infty\ \ \text{for all}\ \ t\in\mathbb{R}\ \ \text{and}\ \ \int_{\mathbb{R}}zF_Z(dz)=0. \N\]\NFor infinite moving average processes \N\[\N X_n=\sum_{i=-\infty}^\infty a_iZ_{n-i},\ \ n\in\mathbb{Z} \N\]\Nwith real sequence \({a=\{a_i\}_{i\in\mathbb{Z}}\in\ell^2},\) there is considered the following events \N\[\NE_j(n,\varepsilon)=\left\{\frac{1}{n}\sum_{i=j}^{n+j-1}X_i\geq\varepsilon\right\},\ \ j\geq0,\ \varepsilon>0. \N\]\NIn the short memory regime (i.e., \({\sum_{j=-\infty}^\infty |a_j|<\infty}\) and \({\sum_{j=-\infty}^\infty a_j\neq0}\)) the authors describe the limiting distribution of the large deviation cluster caused by the rare event \({E_0(n, \varepsilon)}\) as well as the behaviour of the size of that cluster as \(\varepsilon\) becomes small. It turns out that the size of the cluster is of the order \(\varepsilon^{-2}.\)