Long strange segments in a long-range-dependent moving average.
Let \(\{Z_ t\}\) be i.i.d. random variables with zero mean and \(L_ 1(\lambda ), L_ 2(\lambda )\) slowly varying functions at infinity. Assume that \(P(| Z_ 0| >\lambda ) = L_ 1(\lambda )\lambda ^ {-\alpha }\), \(\lim _ {\lambda \to \infty } \{P(Z_ 0>\lambda )/P(| Z_ 0| >\lambda )\}=p\), \(\lim _ {\lambda \to \infty } \{P(Z_ 0<-\lambda )/P(| Z_ 0| >\lambda )\}=q\) for some \(\alpha >1\) and \(0<p=1-q\leq 1\). Define \(X_ n=\mu +\sum _ {j=-\infty }^ {\infty } \varphi _ {n-j} Z_ j\). Further assume that there exists a function \(\varphi (t)=L_ 2(t)t^ {-h}\geq 0\) with \(h>\max \{\alpha ^ {-1}, 0.5\}\) such that \(\lim _ {j\to \infty } \{\varphi _ j/\varphi (j)\} =\xi _ +\), \(\lim _ {j\to \infty } \{\varphi _ {-j}/\varphi (j)\} =\xi _ -\) for some \(\xi _ +,\xi _ -\geq 0\). For \(\theta >\mu \) define \(R_ n=\sup \{j-i: 0\leq i<j\leq n, (j-i)^ {-1} (X_ {i+1}+\cdots +X_ j)>\theta \}\). The variable \(R_ n\) can be called ``the length of a long strange interval''. The main result of the paper is a derivation of the asymptotic distribution of the appropriately normalized \(R_ n\). Roughly speaking, \(R_ n\) grows as \(n^ {1/\alpha h}\) if \(h<1\) and as \(n^ {1/\alpha }\) if \(h>1\).
scientific article; zbMATH DE number 2119493