Let $(X_n)$ be a stationary sequence. We prove the following \flushpar (i) $\$ If the variables $(X_n)$ are iid and $\Bbb E (\vert X_1\vert)<\infty$ then $$\lim_{p\to1^+} \biggl((p-1) \biggl(\sum^\infty_{n=1} \frac{\vert X_n(x) \vert^p}{n^{p}}\biggr) \biggr) ^{1/p} = \Bbb E (\vert X_1 \vert), a.e.$$ \flushpar (ii) $\$ If $X_n(x) = f(T^n x)$ where $(X,\Cal F ,\mu ,T)$ is an ergodic dynamical system, then $$\lim_{p\to1^+} \biggl((p-1) \biggl(\sum^\infty_{n=1} \biggl(\frac{f (T^n x)}{n} \biggr)^p \biggr)\biggr) ^{1/p} = \int\,f d\mu \qquad \text{a.e.~for } f\geq 0,\ f\in L\,\log\,L.$$ Furthermore the maximal function,  \sup_{1