Parthanil Roy
Indian Statistical Institute, Kolkata
September 1, 2016
Extreme value theory for stable random fields indexed by finitely generated free groups: In this work, we investigate the extremal behaviour of left-stationary emph{symmetric $alpha$-stable} (S$alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls (in the Cayley graph) of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying quasi-invariant action of the free group but is different from what happens in the case of S$alpha$S random fields indexed by $mathbb{Z}^d$. The presence of this new dichotomy is confirmed by the study of stable random fields generated by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a new class on point processes that we have termed as emph{random
Indian Statistical Institute, Kolkata
September 1, 2016
Extreme value theory for stable random fields indexed by finitely generated free groups: In this work, we investigate the extremal behaviour of left-stationary emph{symmetric $alpha$-stable} (S$alpha$S) random fields indexed by finitely generated free groups. We begin by studying the rate of growth of a sequence of partial maxima obtained by varying the indexing parameter of the field over balls (in the Cayley graph) of increasing size. This leads to a phase-transition that depends on the ergodic properties of the underlying quasi-invariant action of the free group but is different from what happens in the case of S$alpha$S random fields indexed by $mathbb{Z}^d$. The presence of this new dichotomy is confirmed by the study of stable random fields generated by the canonical action of the free group on its Furstenberg-Poisson boundary with the measure being Patterson-Sullivan. When the action of the free group is dissipative, we also establish that the scaled extremal point process sequence converges weakly to a new class on point processes that we have termed as emph{random