4,141 research outputs found
Analysis of the Rosenblatt process
We analyze {\em the Rosenblatt process} which is a selfsimilar process with
stationary increments and which appears as limit in the so-called {\em Non
Central Limit Theorem} (Dobrushin and Major (1979), Taqqu (1979)). This process
is non-Gaussian and it lives in the second Wiener chaos. We give its
representation as a Wiener-It\^o multiple integral with respect to the Brownian
motion on a finite interval and we develop a stochastic calculus with respect
to it by using both pathwise type calculus and Malliavin calculus
Multivariate limit theorems in the context of long-range dependence
We study the limit law of a vector made up of normalized sums of functions of
long-range dependent stationary Gaussian series. Depending on the memory
parameter of the Gaussian series and on the Hermite ranks of the functions, the
resulting limit law may be (a) a multivariate Gaussian process involving
dependent Brownian motion marginals, or (b) a multivariate process involving
dependent Hermite processes as marginals, or (c) a combination. We treat cases
(a), (b) in general and case (c) when the Hermite components involve ranks 1
and 2. We include a conjecture about case (c) when the Hermite ranks are
arbitrary
Four moments theorems on Markov chains
We obtain quantitative Four Moments Theorems establishing convergence
of the laws of elements of a Markov chaos to a Pearson distribution,
where the only assumptionwemake on the Pearson distribution is that it admits
four moments. While in general one cannot use moments to establish convergence
to a heavy-tailed distributions, we provide a context in which only the
first four moments suffices. These results are obtained by proving a general
carré du champ bound on the distance between laws of random variables in the
domain of a Markov diffusion generator and invariant measures of diffusions.
For elements of a Markov chaos, this bound can be reduced to just the first four
moments.First author draf
Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes
\noindent The paper establishes weak convergence in of normalized
stochastic processes, generated by Toeplitz type quadratic functionals of a
continuous time Gaussian stationary process, exhibiting long-range dependence.
Both central and non-central functional limit theorems are obtained
Four moments theorems on Markov chaos
We obtain quantitative Four Moments Theorems establishing convergence of the
laws of elements of a Markov chaos to a Pearson distribution, where the only
assumption we make on the Pearson distribution is that it admits four moments.
While in general one cannot use moments to establish convergence to a
heavy-tailed distributions, we provide a context in which only the first four
moments suffices. These results are obtained by proving a general carr\'e du
champ bound on the distance between laws of random variables in the domain of a
Markov diffusion generator and invariant measures of diffusions. For elements
of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page
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