RANDOM ERGODIC-THEOREMS WITH UNIVERSALLY REPRESENTATIVE SEQUENCES

When elements of a measure-preserving action of R(d) or Z(d) are selec ted 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. i nteger-valued sampling schemes are universally representative (with p > 1) if and only if they have nonzero mean, and we discuss a variety o f other sampling schemes which have or lack this property.