1. A.B. Nobel. "Evaluating the performance of a simple inductive procedure in the presence of overfitting error", In Proceedings of the Fourth Annual Conference on Computational Learning Theory, Santa Cruz, CA, pp.267-274, 1991.

Summary: This paper studies the approximate consistency of the empirical risk minimization procedure when it is applied to a class of functions for which the Glivenko-Cantelli property fails to hold. A result of Steele shows that the relevant asymptotic behavior of a given class of functions can be described by a non-negative constant h, which bounds the performance of empirical minimization. It is shown how one can obtain bounds on the value of h from any absolutely regular (weakly Bernoulli) sequence of observations.

2. A.B. Nobel and A. Dembo, "A note on uniform laws of averages for dependent processes", Statistics and Probability Letters, vol.17, pp.169-172, 1993.

Summary: If for some i.i.d. sequence of random variables the sample averages of functions in a class F converge uniformly to their corresponding expected values, then the same uniform behavior holds for every absolutely regular (weakly Bernoulli) sequence of random variables with the same one dimensional distribution. In particular, for every family of sets C with finite VC-dimension and every absolutely regular process, the relative frequencies of sets C in C converge uniformly to their probabilities.

3. A.B. Nobel, "A counterexample concerning uniform ergodic theorems for a class of functions", Statistics and Probability Letters, vol.24, pp.165-168, 1995.

Summary: For i.i.d. processes, Vapnik and Chervonenkis, and later Talagrand, gave necessary and sufficient conditions under which the sample averages of functions in a class F converge uniformly to their corresponding expected values. It is shown by a simple construction that their conditions fail in a fundamental way to guarantee such convergence for strongly dependent processes.