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.
In this paper we introduce multitype branching processes with inhomogeneous Poisson immigration, and consider in detail the critical Markov case when the local intensity r(t) of the Poisson random measure is a regularly varying function. Various multitype limit distributions (conditional and unconditional) are obtained depending on the rate at which r(t) changes with time. The asymptotic behaviour of the first and second moments, and the probability of nonextinction are investigated.
We derive a formula for the product moment EX m 1 1 Á Á Á X m p p , m 1 ! 1, . . . , m n ! 1 in terms of the joint survival function when ðX 1 , . . . , X p Þ is a non-negative random vector. In the course of the derivation, we present an independent approach for deriving the formula for EX m 1 1 X m 2 2 (which is already known in the literature). These formulas will be useful tools for deriving explicit expressions for the product moments for the various bivariate and multivariate distributions in statistics specified by the joint tails of the cumulative distribution function. We illustrate their use by deriving product
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