TY - JOUR

T1 - Rates of convergence for extremes of geometric random variables and marked point processes

AU - Cipriani, Alessandra

AU - Feidt, Anne

PY - 2016/3/1

Y1 - 2016/3/1

N2 - We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.

AB - We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.

KW - Marked point process of extremes

KW - Marshall-Olkin geometric distribution

KW - Maxima of geometric random variables

KW - Poisson approximation

KW - Stein-Chen method

UR - http://www.scopus.com/inward/record.url?scp=84956638471&partnerID=8YFLogxK

U2 - 10.1007/s10687-015-0229-x

DO - 10.1007/s10687-015-0229-x

M3 - Article

AN - SCOPUS:84956638471

VL - 19

SP - 105

EP - 138

JO - Extremes: statistical theory and applications in science, engineering and economics

JF - Extremes: statistical theory and applications in science, engineering and economics

SN - 1386-1999

IS - 1

ER -