On Variation Functions for Subsequence Ergodic Averages

 Suppose denote the ergodic averages for the natural numbers . Let denote the corresponding maximal function and let for . We show that for if there exists such that then there exists such that . Similar weak (1,1) inequalities follow for V _q when you know them for M too also with q > 1. We also show this fails completely if q= 1. We also show that for certain polynomial like and random sequences , if and is of exponential growth then for a certain positive constant C. (Received 11 February 1998; in revised form 10 December 1998)