Convergence of the p-Series for Stationary Sequences

摘要

Let (Xn) be a stationary sequence. We prove the following(i) If the variables (Xn) are iid and E (∣X1∣)∞ then limp→1+ ((p-1) ∑∞n=1 (∣Xn(x)∣pp))1/p = E (∣X1∣), a.e. (ii) If Xn(x) = f(Tnx) where (X,F,μ,T) is an ergodic dynamical system, then limp→1+ ((p-1) ∑∞n=1 (f(Tnx))p)1/p = ∫f dμ a.e. for f≧0, f∈ L log L. Furthermore the maximal function, sup1p ∞ (p-1)1/p (∑∞n=1 (f(Tnx))p)1/p is integrable for functions, f≧0, f∈ L log L. These limits are linked to the maximal function N*(x)= ∥((Xn(x)))∥1,∞.