New York Journal of Mathematics
Volume 3A (1997-1998) 15-30


I. Assani

Convergence of the p-Series for Stationary Sequences

Published: November 26, 1997
Keywords: p-Series, maximal function, iid random variables and stationary sequences
Subject: 28D05, 60F15, 60G50

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)∣p/np))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)/n)p)1/p = ∫f dμ a.e.
for f≧0, f∈ L log L. Furthermore the maximal function,
sup1<p< ∞ (p-1)1/p (∑n=1 (f(Tnx)/n)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)/n))∥1,∞.

Author information

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599