The limit distribution of the maximum increment of a random walk with dependent regularly varying jump sizes
The authors investigate the maximum increment of a random walk \((S_{n})\) starting with \(S_{0}=0\) with regularly varying jump sizes \((X_{t})\). The maximum increment of the random walk \((S_{n})\) is defined by the quantities \[ \tilde{M}_{n}^{(\gamma )}=\max_{1\leq l\leq n}\frac{1}{f(l)}\max_{0\leq k\leq n-l}\left| S_{k+l}-S_{k}\right| ,\qquad n\geq 1, \] and their centered analogs \[ \tilde{T}_{n}^{(\gamma )}=\max_{1\leq l\leq n}\frac{1}{f(l(1-l/n))} \max_{0\leq k\leq n-l}\left| S_{k+l}-S_{k}-l\frac{S_{n}}{n}\right| ,\qquad n\geq 1, \] where \(f\) is a non-decreasing function on \([0,\infty )\) satisfying \(f(1)=1\) and \(f(l)\geq l^{\gamma }\) for \(l\geq 1\) such that the relation \[ \lim_{n\rightarrow \infty }\inf_{1\leq l\leq d_{n}}\frac{f(l(1-l/n))}{f(l)}=1 \] holds for any increasing sequence \((d_{n})\) of positive numbers satisfying \( d_{n}^{2}/n\rightarrow 0\). The first author and \textit{A. Račkauskas} [Bernoulli 16, No. 4, 1016--1038 (2010; Zbl 1215.60018)] have previously studied the case of a random walk \((S_{n})\) with i.i.d. regularly varying Banach-valued jump sizes \((X_{t})\). Consider real-valued random variables \((X_{t})\), which are assumed to be regularly varying, that is, the generic element \(X\) of this sequence satisfies the tail balance condition \[ \operatorname{P}\left( X>x\right) =\tilde{p}x^{-\alpha }L(x)\qquad \text{and}\qquad \operatorname{P}\left( X\leq -x\right) =\tilde{q}x^{-\alpha }L(x),\qquad x\rightarrow \infty , \] where \(L\) is a slowly varying function, \(\alpha >0\) is the index of regular variation, and \(\tilde{p},\tilde{q}\geq 0\), \(\tilde{p}+\tilde{q}=1\). Assuming \(\operatorname{E}X=0\) if \(\operatorname{E}\left| X\right| <\infty \), their result states that, for \(\gamma >\max (0,\;0.5-\alpha ^{-1})\) and for the normalizing sequence \((a_{n})\) chosen according to \(n\operatorname{P}\left( \left| X\right| >a_{n}\right) \rightarrow 1\), \[ \lim_{x\rightarrow \infty }\operatorname{P}\left( a_{n}^{-1}\tilde{M}_{n}^{(\gamma )}\leq x\right) =\lim_{x\rightarrow \infty }\operatorname{P}\left( a_{n}^{-1}\tilde{T} _{n}^{(\gamma )}\leq x\right) =\Phi _{\alpha }(x),\qquad x>0, \] where \(\Phi _{\alpha }(x)=\exp \left\{ -x^{-\alpha }\right\} \) for \(x>0\) denotes the Fréchet distribution function. The authors successively extend the study to the case of a moving average of finite order \(q\), i.e., \[ X_{t}=\sum_{i=1}^{q}\psi _{i}Z_{t-i},\qquad t\in \mathbb{Z}, \] for an iid regularly varying noise\ sequence \((Z_{t})\) and then to the case of a moving average of infinite order \(q=\infty \). Furthermore, they extend the study to the case in which jump sizes constitute a general strictly stationary regularly varying sequence \((X_{t})\) with index \(\alpha >0\) subject to some mixing and anti-clustering conditions, where the regular\ variation means that for every \(d\geq 1\), the vector \((X_{1},\ldots ,X_{d})\) follows a multivariate regular variation structure with index \(\alpha \). These new results show that the distributions of both \(a_{n}^{-1}\tilde{M} _{n}^{(\gamma )}\) and \(a_{n}^{-1}\tilde{T}_{n}^{(\gamma )}\) converge to \( \Phi _{\alpha }^{\xi }\) for some constant \(\xi >0\).
scientific article; zbMATH DE number 6176810
scientific article