## 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 $$\{X_ k\}$$.

### MSC:

 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
Full Text: