On the Spectral SLLN and Pointwise Ergodic Theorem in $L^\alpha$

Abstract

We obtain criteria for the SLLN to hold for processes which are Fourier transforms of random measures. With this spectral approach, we also give criteria for the pointwise ergodic theorem to hold, for some classes of operators between $L^\alpha$-spaces, $1 \leq \alpha < + \infty$. These results apply in particular to contractions on $L^2$. Some random fields extensions are also studied.