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Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when 1 is a simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤1/2; otherwise, it is called large).
Strassen's invariance principle for additive functionals of Markov chains with spectral gap in the Wasserstein metric is proved.
Invariance principle in is studied using signed random measures. This approach to the problem uses an explicit isometry between and a reproducing kernel Hilbert space giving a very convenient setting for the study of compactness and convergence of the sequence of Donsker functions. As an application, we prove a version of the invariance principle in the case of -mixing random variables. Our result is not available in the -setting.
We show a result of approximation in law of the d-parameter fractional Brownian sheet in the space of the continuous functions on [0,T]d. The construction of these approximations is based on the functional invariance principle.
If a stochastic process can be approximated with a Wiener process with positive drift, then its maximum also can be approximated with a Wiener process with positive drift.
We determine the asymptotic behavior of the realized power variations, and more generally of sums of a given function f evaluated at the increments of a Lévy process between
the successive times iΔn for i = 0,1,...,n. One can elucidate completely the first order behavior, that is the convergence in probability of such sums, possibly after normalization and/or
centering: it turns out that there is a rather wide variety of possible behaviors, depending on the structure of jumps and on the chosen...
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