 

I. Assani
Convergence of the pSeries for Stationary Sequences


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


Abstract
Let (X_{n}) be a stationary sequence. We prove the
following
(i) If the variables (X_{n}) are iid
and E (∣X_{1}∣)<∞ then
lim_{p→1+} ((p1)
∑^{∞}_{n=1} (∣X_{n}(x)∣^{p}/n^{p}))^{1/p}
= E (∣X_{1}∣), a.e.
(ii) If X_{n}(x) = f(T^{n}x) where
(X,F,μ,T) is an ergodic dynamical system,
then
lim_{p→1+} ((p1)
∑^{∞}_{n=1} (f(T^{n}x)/n)^{p})^{1/p} = ∫f dμ
a.e.
for f≧0, f∈ L log L.
Furthermore the maximal function,
sup_{1<p< ∞} (p1)^{1/p}
(∑^{∞}_{n=1} (f(T^{n}x)/n)^{p})^{1/p}
is integrable for functions, f≧0, f∈ L log
L.
These limits are linked to the maximal function
N*(x)= ∥((X_{n}(x)/n))∥_{1,∞}.


Author information
Department
of Mathematics, University of North Carolina at
Chapel
Hill, Chapel Hill, NC 27599
assani@math.unc.edu

