To cite this version:Elisaveta Pancheva, Ivan K. Mitov, Kosto V. Mitov. Limit theorems for extremal processes generated by a point process with correlated time and space components. Statistics and Probability Letters, Elsevier, 2009, 79 (3) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPTLimit theorems for extremal processes generated by a point process with correlated time and space components
AbstractThe point process N = {(T k , X k ), k = 0, 1, 2, 3 . . .} defines the sequence of maxima M(t) = {k:T k ≤t} X k . Using time and space scaling it is possible to define different sequences of random time changed extremal processes. The convergence of such sequences to non degenerate extremal processes is proved in case where the time and space components of the point process are correlated.
Given an extremal process X : [0, ∞) → [0, ∞) d with lower curve C and associated point process N = {(t k , X k ) : k ≥ 0}, t k distinct and X k independent, given a sequence ζn = (τn, ξn), n ≥ 1 of time-space changes (max-automorphisms of [0, ∞) d+1 ), we study the limit behaviour of the sequence of extremal processesunder a regularity condition on the norming sequence ζn and asymptotic negligibility of the max-increments of Yn. The limit class consists of self-similar (w.r.t. a group ηα = (σα, Lα), α > 0, of time-space changes) extremal processes. Under self-similarity here we understand the property LαThe univariate marginals of Y are max-selfdecomposable. If additionally the initial extremal process X is supposed to have homogeneous max-increments then the limit process is max-stable with homogeneous max-increments.
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