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On the Central Limit Theorem for Sample Continuous Processes

Evarist Gine M.
The Annals of Probability
Vol. 2, No. 4 (Aug., 1974), pp. 629-641
Stable URL: http://www.jstor.org/stable/2959413
Page Count: 13
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On the Central Limit Theorem for Sample Continuous Processes
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Abstract

Let $\{X_k\}^\infty_{k = 1}$ be a sequence of independent centered random variables with values in $C(S)$ (i.e., sample continuous processes in $S$), $(S,d)$ being a compact metric space. This sequence is said to satisfy the central limit theorem if there exists a sample continuous Gaussian process on $S, Z$, such that $\mathscr{L}(\sum^n_{k = 1}X_k/n^{\frac{1}{2}})\rightarrow_{w^\ast} \mathscr{L}(Z)$ in $C'(C(S))$. In this paper some sufficient conditions are given for the central limit theorem to hold for $\{X_k\}^\infty_{k = 1}$; these conditions are on the modulus of continuity of the processes $X_k$ and they are expressed in terms of the metric entropy of distances associated to $\{X_k\}$. Then, in order to give some insight on these theorems, several results on the central limit theorem for particular processes (random Fourier and Taylor series, as well as more general processes on [0, 1]) are deduced.

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