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A strong invariance theorem for the tail empirical process. (English) Zbl 0664.60038

Let \(U_ 1,U_ 2,..\). be a sequence of independent uniform (0,1) random variables. Let \(\alpha_ n(s)\), \(0\leq s\leq 1\), be the empirical process based on the first n of these variables. Further, a(n) is a sequence of positive constants such that \(0<a(n)<1\), a(n)\(\to 0\), and \(k(n):=n a(n)\to \infty\) as \(n\to \infty\). The tail empirical process is \(w_ n(s)=a(n)^{-1/2}\alpha_ n(a(n)s),\) \(0\leq s\leq 1\). This paper essentially characterises those sequences a(n) for which the tail empirical process can be closely approximated by a sequence of Wiener processes.
Specifically, let \(W_ m\) denote independent standard Wiener processes, and define \(l(n):=2 \log \log (n\vee 3)\). If l(n)/k(n)\(\to 0\), as \(n\to \infty\), Theorem 1 of this paper states that a single probability space can be constructed which supports all the required random variables, and such that with probability one: \[ \lim_{n\to \infty}\sup_{0\leq s\leq 1}l(n)^{-1/2}| w_ n(s)-k(n)^{-1/2}\sum^{n}_{m=1}W_ m(a(n)s)| =0. \] A functional law of the iterated logarithm for tail empirical processes is a corollary of this approximation result. If \(k(n)=c_ n \log \log n\), where \(c_ n\to c>0\), Corollary 1 shows that the approximation theorem cannot hold. Other corollaries and remarks provide analogues to the above results for tail quantile processes, and for weighted versions of the tail empirical process.
The key approximation result of the proofs is a refinement [cf. the author and W. R. van Zwet, Ann. Probab. 15, 871-884 (1987; Zbl 0638.60040)] of the well-known Komlós, Major and Tusnády approximation theorem. This is combined with inequalities and techniques drawn from a number of sources.

MSC:

60F17 Functional limit theorems; invariance principles
60F99 Limit theorems in probability theory

Citations:

Zbl 0638.60040
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