It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely...

Vapnik and Cervonenkis (1971), and Talagrand (1987), have characterized the Glivenko-Cantelli Property for independent random variables. We show that these characterizations fail to hold for stationary ergodic processes.

If for a 'permissible' family of functions and an i.i.d. process {Xi}[infinity]i=0, with probability one, then the same holds for away absolutely regular (weakly Bernoulli) process having the same marginal distribution. In particular, for any class of sets having finite V-C dimension and any...