Let (X_n) be a stationary sequence. We prove the following

(i) If the variables (X_n) are iid and E (∣X₁∣)<∞ then

lim_p→1⁺ ((p-1) ∑^∞_n=1 (∣X_n(x)∣^p/n^p))^1/p = E (∣X₁∣), a.e.

(ii) If X_n(x) = f(Tⁿx) where (X,F,μ,T) is an ergodic dynamical system, then

lim_p→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, sup_{1<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)= ∥((X_n(x)/n))∥_1,∞.