Characterization of Almost Surely Continuous 1-Stable Random Fourier Series and Strongly Stationary Processes

Abstract

We complete the results of M. Marcus and G. Pisier by showing that a strongly stationary 1-stable process $(X_t)_{t \in G}$ defined on a locally compact group has a version with sample continuous paths if (and only if) the entropy integral $\int^\infty_0 \log^+ \log N(K, d_X, \varepsilon) d\varepsilon$ is finite, where $K$ is a given neighborhood of the unit and $d_X$ is the distance induced by the process.