A Central Limit Theorem Under Metric Entropy with L₂ Bracketing

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 ᵢ: i ≥ 1\ be a sequence of independent copies of V. The limiting behavior of the process Sₙ(s) = n-1/2∑ⁿᵢ₌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).