Functional limit theorems and empirical entropy. I. (English. Russian original) Zbl 0627.60043

Kolchinskij, V. I.

Theory Probab. Math. Stat. 33, 35-45 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 31-42 (1985).

Let (A,\({\mathcal A},\mu)\) be a probability space, S an arbitrary set and F a finite subset of the real line. Given a random function \(\xi\) : \(A\times S\to F\), a pseudometric \(\tau\) in S is introduced by setting \(\tau (s,t)=\mu^{1/2}(\xi (s)\neq \xi (t))\), s,t\(\in S\). For a sequence \(\{\xi_ n\}_{n\geq 1}\) of independent copies of \(\xi\) set \[ S_ n(t)=\sum^{n}_{k=1}[\xi_ k(t)-E(\xi_ k(t))],\quad t\in S,\quad n\geq 1. \] If (S,\(\tau)\) is totally bounded and \(\{\xi_ n\}_{n\geq 1}\) satisfies certain measurability conditions, then the \(S_ n\) are random elements in the space \((D_ F(S),{\mathcal B}_ b)\), where \(D_ F(S)\) is the linear span of the space \(C_ 0(S)\) of all \(\tau\)-continuous functions on S and the set of functions \(x: S\to F\), and where \({\mathcal B}_ b\) is the \(\sigma\)-field generated by the balls with respect to the supremum norm in \(D_ F(S)\). Limit theorems for the sequence \(\{S_ n\}_{n\geq 1}\) are studied.
In particular, necessary and sufficient conditions for the validity of the strong law of large numbers and sufficient conditions for the central limit theorem to hold are obtained. Here by central limit theorem is meant that \(\{n^{1/2}S_ n\}_{n\geq 1}\) converges in law in \((D_ F(S),{\mathcal B}_ b)\) to a centered Gaussian random element taking values in \(C_ 0(S)\) and having correlation function \[ R(s,t)=E(\xi (s)\xi (t))-E(\xi (s))E(\xi (t)),\quad s,t\in R. \] The conditions are formulated in terms of empirical entropy characteristics of the sequence \(\{\xi_ n\}_{n\geq 1}\).

Reviewer: E.Häusler

MSC:

60F17	Functional limit theorems; invariance principles
28D20	Entropy and other invariants
60F05	Central limit and other weak theorems
60G30	Continuity and singularity of induced measures
60G50	Sums of independent random variables; random walks

Keywords:

functional limit theorems; strong law of large numbers; central limit theorem; empirical entropy characteristics