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July, 1987 A Central Limit Theorem Under Metric Entropy with $L_2$ Bracketing
Mina Ossiander
Ann. Probab. 15(3): 897-919 (July, 1987). DOI: 10.1214/aop/1176992072


Let $(\mathbf{S}, \rho)$ be a metric space, $(\mathbf{V}, \mathscr{V}, \mu)$ be a probability space, and $f: \mathbf{S} \times \mathbf{V} \rightarrow \mathbb{R}$ be a real-valued function on $\mathbf{S} \times \mathbf{V}$ which has mean zero and is Lipschitz in $L_2(\mu)$ with respect to $\rho$. Let $V$ be a random variable defined on $(\mathbf{V}, \mathscr{V}, \mu)$, and let $\{V_i: i \geq 1\}$ be a sequence of independent copies of $V$. The limiting behavior of the process $S_n(s) = n^{-1/2}\sum^n_{i=1} f(s, V_i)$ is studied under an integrability condition on the metric entropy with bracketing in $L_2(\mu)$. This metric entropy condition is analogous to one which implies the continuity of the limiting Gaussian process. A tightness result is derived which, in conjunction with the results of Andersen and Dobric (1987), shows that a central limit theorem holds for the $S_n$-process. This result generalizes those of Dudley (1978), Dudley (1981) and Jain and Marcus (1975).


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Mina Ossiander. "A Central Limit Theorem Under Metric Entropy with $L_2$ Bracketing." Ann. Probab. 15 (3) 897 - 919, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0665.60036
MathSciNet: MR893905
Digital Object Identifier: 10.1214/aop/1176992072

Primary: 60F17
Secondary: 60F05

Keywords: Empirical processes , functional central limit theorems , Invariance principles , Law of the iterated logarithm , weak convergence

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 3 • July, 1987
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