69 research outputs found
On a class of self-similar processes with stationary increments in higher order Wiener chaoses
We study a class of self-similar processes with stationary increments
belonging to higher order Wiener chaoses which are similar to Hermite
processes. We obtain an almost sure wavelet-like expansion of these processes.
This allows us to compute the pointwise and local H\"older regularity of sample
paths and to analyse their behaviour at infinity. We also provide some results
on the Hausdorff dimension of the range and graphs of multidimensional
anisotropic self-similar processes with stationary increments defined by
multiple Wiener integrals.Comment: 22 page
On Stein's Method for Infinitely Divisible Laws With Finite First Moment
We present, in a unified way, a Stein methodology for infinitely divisible
laws (without Gaussian component) having finite first moment. Based on a
correlation representation, we obtain a characterizing non-local Stein operator
which boils down to classical Stein operators in specific examples. Thanks to
this characterizing operator, we introduce various extensions of size bias and
zero bias distributions and prove that these notions are closely linked to
infinite divisibility. Combined with standard Fourier techniques, these
extensions also allow obtaining explicit rates of convergence for compound
Poisson approximation in particular towards the symmetric -stable
distribution. Finally, in the setting of non-degenerate self-decomposable laws,
by semigroup techniques, we solve the Stein equation induced by the
characterizing non-local Stein operator and obtain quantitative bounds in weak
limit theorems for sums of independent random variables going back to the work
of Khintchine and L\'evy.Comment: 58 pages. Minor changes and new results in Sections 5 and
A stroll along the gamma
We provide the first in-depth study of the "smart path" interpolation between
an arbitrary probability measure and the gamma-
distribution. We propose new explicit representation formulae for the ensuing
process as well as a new notion of relative Fisher information with a gamma
target distribution. We use these results to prove a differential and an
integrated De Bruijn identity which hold under minimal conditions, hereby
extending the classical formulae which follow from Bakry, Emery and Ledoux's
-calculus. Exploiting a specific representation of the "smart path", we
obtain a new proof of the logarithmic Sobolev inequality for the gamma law with
as well as a new type of HSI inequality linking relative
entropy, Stein discrepancy and standardized Fisher information for the gamma
law with .Comment: Typos correcte
From almost sure local regularity to almost sure Hausdorff dimension for Gaussian fields
Fine regularity of stochastic processes is usually measured in a local way by
local H\"older exponents and in a global way by fractal dimensions. Following a
previous work of Adler, we connect these two concepts for multiparameter
Gaussian random fields. More precisely, we prove that almost surely the
Hausdorff dimensions of the range and the graph in any ball are
bounded from above using the local H\"older exponent at . We define the
deterministic local sub-exponent of Gaussian processes, which allows to obtain
an almost sure lower bound for these dimensions. Moreover, the Hausdorff
dimensions of the sample path on an open interval are controlled almost surely
by the minimum of the local exponents.
Then, we apply these generic results to the cases of the multiparameter
fractional Brownian motion, the multifractional Brownian motion whose
regularity function is irregular and the generalized Weierstrass function,
whose Hausdorff dimensions were unknown so far.Comment: 28 page
Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods
In these notes, we obtain new stability estimates for centered
self-decomposable probability measures on with finite second
moment and for non-degenerate symmetric -stable probability measures on
with . These new results are refinements of
the corresponding ones available in the literature. The proofs are based on
Stein's method for self-decomposable laws, recently developed in a series of
papers, and on closed forms techniques together with a new ingredient: weighted
Poincar\'e-type inequalities. As applications, rates of convergence in
Wasserstein-type distances are computed for several instances of the
generalized central limit theorems (CLTs). In particular, a
-rate is obtained in -Wasserstein distance when the target
law is a non-degenerate symmetric -stable one with .
Finally, the non-degenerate symmetric Cauchy case is studied at length from a
spectral point of view. At last, in this Cauchy situation, a -rate of
convergence is obtained when the initial law is a certain instance of layered
stable distributions.Comment: 102 page
Stein's method on the second Wiener chaos : 2-Wasserstein distance
In the first part of the paper we use a new Fourier technique to obtain a
Stein characterizations for random variables in the second Wiener chaos. We
provide the connection between this result and similar conclusions that can be
derived using Malliavin calculus. We also introduce a new form of discrepancy
which we use, in the second part of the paper, to provide bounds on the
2-Wasserstein distance between linear combinations of independent centered
random variables. Our method of proof is entirely original. In particular it
does not rely on estimation of bounds on solutions of the so-called Stein
equations at the heart of Stein's method. We provide several applications, and
discuss comparison with recent similar results on the same topic
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