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

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.HĂ¤usler

### 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