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Necessary and Sufficient Conditions for the Uniform Law of Large Numbers in the Stationary Case

Necessary and sufficient conditions for the uniform law of large numbers for stationary ergodic sequences of random variables are given. Three different types of conditions are investigated and established. Firstly it is shown that eventually total boundedness in mean is necessary and sufficient. This fact enables one to deduce the equivalence among almost sure convergence, convergence in mean, and convergence in probability in the uniform law under consideration. Secondly it is shown that eventual tightness is necessary and sufficient implying that the preceding convergence notions coincide with a weak convergence. Finally it is shown that the uniform convergence under consideration might be characterized in terms of the existence of certain totally bounded pseudo-metrics having a stochastic and mean control on increments of the process in question. Methods used in the proofs rely upon techniques established in the case of independent and identically distributed random variables interacting with those coming from ergodic theory. Given collaboration is significant and in this way some new facts are also obtained in the case of independent and identically distributed random variables itself.