An asymptotic independent representation in limit theorems for maxima of nonstationary random sequences. (English) Zbl 0781.60042

Let \(\{X_ k: k \in {\mathbb{N}}\}\) and \(\{\widetilde{X}_ k: k\in\mathbb{N}\}\) be two sequences of random variables, and \(M_ n,\widetilde{M}_ n\) be the \(n\)-th partial maxima for \(\{X_ k\}\) and \(\{\widetilde{X}_ k\}\), respectively. If \(\widetilde{X}_ 1,\widetilde{X}_ 2,\dots\) are independent r.v.’s and \[ \sup_{x \in {\mathbb{R}}^ 1} | P(M_ n\leq x)-P(\widetilde{M}_ n\leq x)| \to 0,\quad \text{ as }n\to\infty,\tag{*} \] we say that \(\{\widetilde{X}_ k: k\in \mathbb{N}\}\) is an asymptotic independent representation (a.i.r.) for maxima of \(\{X_ k: k\in\mathbb{N}\}\). If the sequence \(\{X_ k\}\) is stationary, it is naturally to look for an independent identically distributed sequence satisfying (*). This problem was investigated in [author, Stochastic Processes Appl. 37, No. 2, 281-297 (1991; Zbl 0743.60051)]. In the paper under review the problem of the existence of a.i.r. for maxima is solved for some classes of nonstationary sequences \(\{X_ k\}\).


60G70 Extreme value theory; extremal stochastic processes
60F99 Limit theorems in probability theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)


Zbl 0743.60051
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