An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions

Abstract

Let (X, α, P) be a probability space, x(1),x(2), ... coordinates for the countable product of copies of X, and ℱ ⊂ ℒ² (X, α, P). Let v_n be the normalized empirical measure n^−1/2(δ_x(1)+...+δ_x(n)-nP). Let ℓ^∞(ℱ) be the set of all bounded real functions G on ℱ with norm ‖G‖_ℱ := sup_fεℱ ¦G(f)¦. Let μ(f) := εfdμ for any measure μ. We are interested in central limit theorems where v_n converges in law for ‖·‖_ℱ. The limit is a Gaussian process G_p with mean 0 and covariance

$$EG_P (f)G_P (g) = P(fg) - P(f)P(g): = (f,g)_{P,O} ,$$

f,g ε ℱ. Say ℱ is G_PBUC if G_P can be chosen to have each sample function bounded and uniformly continuous for (.,.)_P,0. Say the central limit theorem holds for ℱ if ℱ is G_PBUC and we have convergence of upper integrals lim_n→∞ ∝*H(v_n)dpⁿ = EH(G_P) for every bounded continuous real function H on ℓ^∞(ℱ). (Thus, as noted by J. Hoffmann-Jørgensen, convergence "in law" does not require definition of laws.) In Sec. 4 an extended Wichura’s theorem is proved: given such convergence in law, there exist almost surely convergent realizations. Sec. 5 shows that the new definition of central limit theorem holding is equivalent to the previous definition of "functional Donsker class." Secs. 6–7 treat the case where ℱ={M1_h≥M: 0<M<∞} for a random variable h. This case reduces to the study of weighted empirical distribution functions. Conditions for the central limit theorem are collected and made precise.

Partially supported by National Science Foundation Grant MCS-8202122