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A Central Limit Theorem Under Metric Entropy with L2 Bracketing

Mina Ossiander
The Annals of Probability
Vol. 15, No. 3 (Jul., 1987), pp. 897-919
Stable URL: http://www.jstor.org/stable/2244030
Page Count: 23
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A Central Limit Theorem Under Metric Entropy with L2 Bracketing
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Abstract

Let (S, ρ) be a metric space, (V, V, μ) be a probability space, and f: S × V → R be a real-valued function on S × V which has mean zero and is Lipschitz in L2(μ) with respect to ρ. Let V be a random variable defined on (V, V, μ), and let {Vi: i ≥ 1} be a sequence of independent copies of V. The limiting behavior of the process Sn(s) = n-1/2∑n i=1 f(s, Vi) is studied under an integrability condition on the metric entropy with bracketing in L2(μ). 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 Sn-process. This result generalizes those of Dudley (1978), Dudley (1981) and Jain and Marcus (1975).

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