A Central Limit Theorem Under Metric Entropy with L₂ Bracketing

Let (S, ) be a metric space, (V, V, ) 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 () with respect to. Let V be a random variable defined on (V, V, ), and let \Vᵢ: i 1\ be a sequence of independent copies of V. The limiting behavior of the process Sₙ (s) = n^-1/2ⁿ₈=₁ f (s, Vᵢ) is studied under an integrability condition on the metric entropy with bracketing in L₂ (). 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).