Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution
Let \(Z_ k\), \(-\infty <k<\infty\) be a sequence of iid r.v's with d.f. \(F\in D(\Lambda)\), i.e. \[ \lim_{n\to \infty}F^ n(a_ nx+b_ n)=\Lambda (x),\quad x\in {\mathbb{R}} \] for some \(a_ n>0\), \(b_ n\in {\mathbb{R}}\) (n\(\geq 1)\) and \(\Lambda (x)=\exp \{-e^{-x}\}.\) The main object of this paper is the sequence of point processes based on the moving average process \(X_ n=\sum^{\infty}_{j=-\infty}C_ jZ_{n-j}\), \(n=0,\pm 1,..\). where \(\sum^{\infty}_{j=-\infty}| C_ j|^{1-\delta}<\infty\), \(\delta >0\) and \(\max \{| C_ i|:\) \(i=0,\pm 1,...\}=1\). For a fixed integer m define the vectors \[ X_{n,k}=a_ n^{-1}(Z_ k-b_ n,\quad -Z_ k-b_ n,\quad Z_{k- i},\quad 0<| i| \leq 2m) \] and let \(\epsilon_ x(F)=1\) if \(x\in F\), 0 otherwise; \(x\in E\), \(F\in {\mathcal E}\), E is a subset of compactified Euclidean space, \({\mathcal E}\) is the \(\sigma\)-algebra generated by open sets. Then (under certain conditions) \(\sum^{\infty}_{k=1}\epsilon_{(k_ n^{-1},X_{n,k})}\Rightarrow N\) where N is the sum of 2 independent Poisson random measures with specified mean measures. Numerous applications comprise the convergence of maxima and exceedances. Convergent sequences of point processes based on the max-moving average process \(\{\) \(\bigvee^{\infty}_{j=-\infty}C_ jZ_{n-j}\}\) are also investigated.
scientific article; zbMATH DE number 4074067
scientific article