Strong convergence of multivariate maxima

It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar's theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.


Published in:
Journal Of Applied Probability, 57, 1, 314-331
Year:
Mar 01 2020
Publisher:
Cambridge, CAMBRIDGE UNIV PRESS
ISSN:
0021-9002
1475-6072
Keywords:
Laboratories:




 Record created 2020-05-16, last modified 2020-05-18


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