A Note on Uniform Laws of Averages for Dependent Processes

If for a ‘permissible’ family of functions F and an i.i.d. process {Xi} ∞ i=0 lim n→∞ sup f∈F ∣ ∣ ∣ ∣ ∣ 1 n n−1 ∑ i=0 f(Xi)− Ef(X0) ∣ ∣ ∣ ∣ ∣ = 0 with probability one, then the same holds for every absolutely regular (weakly Bernoulli) process having the same marginal distribution. In particular, for any class of sets C having finite V-C dimension and any absolutely regular process {Xi} ∞ i=0 lim n→∞ sup C∈C ∣ ∣ ∣ ∣ ∣ 1 n n−1 ∑ i=0 IC(Xi)− IP{X0 ∈ C} ∣ ∣ ∣ ∣ ∣ = 0 with probability one. Appears in Statistics and Probability Letters, 17 169-172, 1993. Andrew Nobel is with the Beckman Institute, University of Illinois at Urbana-Champaign, 405 N. Mathews, Urbana, IL. 61801 Email: [email protected] Amir Dembo is with the Statistics and Mathematics Departments, Stanford University, Stanford, CA 94305. This work was partially supported by the National Science Foundation under Grant NCR-89-14538. 1