Limit theorems for extremal processes generated by a point process with correlated time and space components

Abstract

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.

On randomly spaced observations and continuous-time random walks

“…In section 3, we present the limiting distribution of all extreme observations until a given time, discussing asymptotic tail independence and asymptotic full tail dependence between the observations and interarrival times in detail. Our main result extends previously mentioned results in this context, as well as some more recent results in Meerschaert and Stoev [17] and Pancheva et al [19] for instance. In the asymptotic full tail dependence case.…”

Extremal behavior of a coupled continuous time random walk

“…But for strongly coupled CTRW (e.g., when large jumps are correlated with long waiting times) the coupled CTRM presents a significant improvement over traditional methods. We note that some underlying theory for coupled CTRM was already present in the work of Silvestrov and Teugels [24] and Pancheva et al [25]. The CTRM framework parallels that of the well known CTRW.…”

Extremes of random variables observed in renewal times

“…For the renewal process with infinite mean, but with regularly varying steps, one can still determine the asymptotic distribution of the maximum, see [4]. In such a setting, the convergence of (M τ (t)) was shown at the level of stochastic processes, see [12,13]. In the rest of the paper we show how one can move beyond the maxima and extend those results to all upper order statistic in both finite and infinite mean case.…”

“…A partial results in this direction appears in [13] where only the convergence of the onedimensional distributions is proved. For t > 0, consider the random time changed extremal process…”

“…, X t,i , where X t,i is defined by (15) X (3) and (13). The following theorem describes the asymptotic behaviour of all upper order statistics in the sequence of observations (X n ) separated by regularly varying waiting times of infinite mean and independent of observations (X n ).…”