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August, 1974 On the Central Limit Theorem for Sample Continuous Processes
Evarist Gine M.
Ann. Probab. 2(4): 629-641 (August, 1974). DOI: 10.1214/aop/1176996609


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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Evarist Gine M.. "On the Central Limit Theorem for Sample Continuous Processes." Ann. Probab. 2 (4) 629 - 641, August, 1974.


Published: August, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0288.60017
MathSciNet: MR370695
Digital Object Identifier: 10.1214/aop/1176996609

Primary: 60F05
Secondary: 60G99

Keywords: $\epsilon$-entropy (metric) , Central limit theorem for sample continuous processes , random Fourier series , random Taylor series

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 4 • August, 1974
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