Ergodic properties of stationary stable processes
A real random variable Y is called symmetric \(\alpha\)-stable (S\(\alpha\) S) for some \(\alpha\in (0,2]\), if E exp(irY)\(=\exp (-c_ Y| r|^{\alpha})\) for all r and some \(c_ Y\geq 0\). A process \(\{X_ t\}\) is called \(S\alpha\) S if all finite linear combinations \(\sum a_ jX_{t_ j}\) are \(S\alpha\) S. The authors derive conditions for stationary \(S\alpha\) S processes to be metrically transitive and mixing. Sub-Gaussian stationary processes and \(S\alpha\) S processes with a harmonic spectral representation are never metrically transitive. However, the latter satisfy a strong law of large numbers (SLLN) even though they are not generally stationary. Doubly stationary \(S\alpha\) S processes are defined in the paper and conditions for their metric transitivity, mixing and SLLN are given.
scientific article; zbMATH DE number 3990511
scientific article