On the ergodic theorems (III) (The random ergodic theorem)

The ergodic theorems

Ergodic Theorems

Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial way on a maximal inequality. This is no accident: it can in fact be shown that a maximal inequality is a necessary condition for an almost everywhere convergence theorem. We will refrain from carrying...

Random Ergodic Theorems with Universally Representative

When elements of a measure-preserving action of R d or Z d are selected in a random way, according to a stationary stochastic process, a.e. convergence of the averages of an L p function along the resulting orbits may almost surely hold, in every system; in such a case we call the sampling scheme universally representative. We show that i.i.d. integer-valued sampling schemes are universally rep...

Ergodic Theorems over Sparse Random Subsequences

We prove an L subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L-good sequences nearly as sparse as the set of squares. We extend this theorem to a more general setting of measure-preserving group actions. In addition, we use the same technique to prove an L almost everywhere convergence result for a modulate...

Ergodic Theorems for Random Group Averages

This is an earlier, but more general, version of ”An L Ergodic Theorem for Sparse Random Subsequences”. We prove an L ergodic theorem for averages defined by independent random selector variables, in a setting of general measure-preserving group actions. A far more readable version of this paper is in the works.