PhD Defense Jorge Yslas Altamirano
Point process convergence of random walks and the estimation of multivariate heavy–tailed distributions
This thesis is concerned with point process convergence for sequences of random walks and the estimation of inhomogeneous phase–type distributions.
First, we study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fréchet distributions.
We make heavily use of precise large deviation results for sums of i.i.d. random variables. In particular, we show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2 + δ)th moment, and as a consequence, we derivethe joint convergence of the off–diagonal entries in sample covariance and correlation matrices of a high–dimensional sample whose dimension increases with the sample size.
Then, we provide a fitting procedure for the class of inhomogeneous phase–type distributions introduced in . We propose a multivariate extension of inhomogeneous phase–type distributions as functionals of elements of Kulkarni’s multivariate phasetype class , and study parameter estimation for the resulting new and flexible class of multivariate distributions.
Supervisor: Prof. Thomas Mikosch, University of Copenhagen
Co-supervisor: Prof. Mogens Bladt, University of Copenhagen
Prof. Jostein Paulsen (chairman), University of Copenhagen
Prof. Olivier Wintenberger, Université Pierre & Marie Curie Paris (Sorbonne)
Prof. Søren Asmussen, Aarhus University