Identification of periodic and cyclic fractional stable motions

Pipiras, Vladas; Taqqu, Murad S.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 612-637.

Nous considérons une sous-classe de l'ensemble des processus autosimilaires stables non gaussiens à accroissements stationnaires. C'est la sous-classe des processus à moyenne mobile mixte. Appliquant une méthodologie introduite par Jan Rosiński, nous avons établi précédemment une correspondance entre les représentations minimales de ces processus et des flots non singuliers. Les processus associés aux flots dissipatifs et ceux associés au flot «identité» (qui est un flot conservatif) ont déjà été caractérisés. Nous étudions ici les processus associés aux flots périodiques et cycliques. Un flot est «périodique» s'il ramène tout point de l'espace à sa position de départ en un temps fini, positif ou nul. Ce flot est «cyclique» si ce temps de retour est strictement positif. Les flots périodiques et cycliques sont des flots conservatifs. Un processus autosimilaire stable à moyenne mobile mixte est appelé «périodique» (ou «cyclique») si sa representation minimale est associée à un flot périodique (ou cyclique). Il n'est toutefois pas toujours facile de déterminer la représentation minimale d'un processus, et, de plus, les processus autosimilaires sont souvent caractérisés par une représentation non minimale. C'est pouquoi nous offrons une méthode directe pour déterminer si ces processus sont périodiques (ou cycliques) sans devoir passer par l'intermédiaire des flots. Cette méthode fonctionne même si la representation des processus est non minimale. Nous obtenons finalement une décomposition des processus autosimilaires stables à moyenne mobile mixte qui inclue les processus périodiques et cycliques. Cette décomposition est plus fine que celles connues auparavant.

We consider an important subclass of self-similar, non-gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar stable mixed moving averages related to dissipative flows have already been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is on self-similar stable mixed moving averages related to periodic and cyclic flows. Periodic flows are conservative flows such that each point in the space comes back to its initial position in finite time, either positive or null. The flow is cyclic if the return time is positive. Self-similar mixed moving averages are called periodic, resp. cyclic, fractional stable motions if their minimal representations are generated by periodic, resp. cyclic, flows. In practice, however, minimal representations are not particularly easy to determine and, moreover, self-similar stable mixed moving averages are often defined by nonminimal representations. We therefore provide a way which is not based on flows, to detect whether these processes are periodic or cyclic even if their representations are nonminimal. These identification results lead naturally to a decomposition of self-similar stable mixed moving averages which includes the new classes of periodic and cyclic fractional stable motions, and hence is more refined than the one previously established.

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