An essay on the general theory of stochastic processes.
The Barnes G function and its relations with sums and products of generalized gamma convolution variables.
A new kind of augmentation of filtrations
Let (Ω, , (), ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to (the-algebra generated by ()) a coherent family of probability measures () indexed by , each of them being defined on . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer if...
The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles
The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables...
The distribution of eigenvalues of randomized permutation matrices
In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter ) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations....
A new kind of augmentation of filtrations
Let (Ω, , (), ) be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to (the -algebra generated by ()) a coherent family of probability measures () indexed by , each of them being defined on . It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, made right-continuous, but can have a negative answer...
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