Davis, Richard; Resnick, Sidney Limit theory for moving averages of random variables with regularly varying tail probabilities. (English) Zbl 0562.60026 Ann. Probab. 13, 179-195 (1985). Let \(\{Z_ k,-\infty <k<\infty \}\) be i.i.d. random variables with \(P(| Z_ 1| >x)=x^{-\alpha}L(x)\), where \(\alpha >0\) and L(x) is slowly varying at \(\infty\) and with \(P(Z_ 1>x)/P(| Z_ 1| >x)\to p\) as \(x\to \infty\), \(0\leq p\leq 1\). Let \(\sum | c_ j|^{\delta}<\infty\) for \(\delta <\alpha\), \(\delta\leq 1\). Then the stationary process \(X_ n=\sum c_ jZ_{n-j}\) exists. The main result of the paper concerns a weak limit behaviour of a point process based on \(\{X_ n\}.\) Its first part is the following: If \(a_ n\) are such that \(nP(| Z_ 1| >a_ nx)\to x^{-\alpha}\) for all \(x>0\), then the sequence \(\{\) \(\sum^{\infty}_{k=1}\epsilon_{(k/n,X_ n/a_ n)}\}\) converges weakly as \(n\to \infty\) in the space of point measures on (0,\(\infty)\times (R\setminus \{0\})\) to \(\sum^{\infty}_{i=0}\sum^{\infty}_{k=1}\epsilon_{(t_ k,j_ kc_ i)}\), where \((t_ k,j_ k)\) are the points of Poisson random measure with mean measure \(\mu (dt,dx)=dt\times \lambda (dx)\), \(\lambda (dx)=\alpha px^{-\alpha -1}1_{(0,\infty)}(x)dx+\alpha q(-x)^{- \alpha -1}1_{(-\infty,0)}(x)dx.\) Here \(\epsilon_ x\) denote the probability measure concentrated in the point x. In the second part of the main theorem a multidimensional version of the first part is obtained. As an application of the main result a variety of weak limit theorems are obtained: for extremes of \(\{X_ 1,...X_ n\}\), the first passages, exceedances, sums and sample covariance functions. Reviewer: T.Inglot Cited in 143 Documents MSC: 60F05 Central limit and other weak theorems 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F17 Functional limit theorems; invariance principles 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:weak limit behaviour of a point process; Poisson random measure × Cite Format Result Cite Review PDF Full Text: DOI