44 research outputs found
Spatially homogeneous Maxwellian molecules in a neighborhood of the equilibrium
This note deals with the long-time behavior of the solution to the spatially
homogeneous Boltzmann equation for Maxwellian molecules, when the initial datum
belongs to a suitable neighborhood of the Maxwellian equilibrium. In
particulary, it contains a quantification of the rate of exponential
convergence, obtained by simple arguments
The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation
In Dolera, Gabetta and Regazzini [Ann. Appl. Probab. 19 (2009) 186-201] it is
proved that the total variation distance between the solution of
Kac's equation and the Gaussian density has an upper bound which
goes to zero with an exponential rate equal to -1/4 as . In the
present paper, we determine a lower bound which decreases exponentially to zero
with this same rate, provided that a suitable symmetrized form of has
nonzero fourth cumulant . Moreover, we show that upper bounds like
are valid for some
vanishing at infinity when
for some in
and . Generalizations of this statement are presented,
together with some remarks about non-Gaussian initial conditions which yield
the insuperable barrier of -1 for the rate of convergence.Comment: Published in at http://dx.doi.org/10.1214/09-AAP623 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences
This paper deals with the problem of quantifying the approximation a
probability measure by means of an empirical (in a wide sense) random
probability measure, depending on the first n terms of a sequence of random
elements. In Section 2, one studies the range of oscillation near zero of the
Wasserstein distance
^{(p)}_{\pms} between \pfrak_0 and
\hat{\pfrak}_n, assuming that the \xitil_i's are i.i.d. with \pfrak_0 as
common law. Theorem 2.3 deals with the case in which \pfrak_0 is fixed as a
generic element of the space of all probability measures on (\rd,
\mathscr{B}(\rd)) and \hat{\pfrak}_n coincides with the empirical measure.
In Theorem 2.4 (Theorem 2.5, respectively) \pfrak_0 is a d-dimensional Gaussian
distribution (an element of a distinguished type of statistical exponential
family, respectively) and \hat{\pfrak}_n is another -dimensional Gaussian
distribution with estimated mean and covariance matrix (another element of the
same family with an estimated parameter, respectively). These new results
improve on allied recent works (see, e.g., [31]) since they also provide
uniform bounds with respect to , meaning that the finiteness of the p-moment
of the random variable \sup_{n \geq 1} b_n
^{(p)}_{\pms}(\pfrak_0,
\hat{\pfrak}_n) is proved for some suitable diverging sequence b_n of positive
numbers. In Section 3, under the hypothesis that the \xitil_i's are
exchangeable, one studies the range of the random oscillation near zero of the
Wasserstein distance between the conditional distribution--also called
posterior--of the directing measure of the sequence, given \xitil_1, \dots,
\xitil_n, and the point mass at \hat{\pfrak}_n. In a similar vein, a bound
for the approximation of predictive distributions is given. Finally, Theorems
from 3.3 to 3.5 reconsider Theorems from 2.3 to 2.5, respectively, according to
a Bayesian perspective
A Berry-Esseen theorem for Pitman's -diversity
This paper is concerned with the study of the random variable denoting
the number of distinct elements in a random sample of
exchangeable random variables driven by the two parameter Poisson-Dirichlet
distribution, . For , Theorem 3.8 in
\cite{Pit(06)} shows that
as . Here, is a
random variable distributed according to the so-called scaled Mittag-Leffler
distribution. Our main result states that \sup_{x \geq 0} \Big|
\ppsf\Big[\frac{K_n}{n^{\alpha}} \leq x \Big] - \ppsf[S_{\alpha,\theta} \leq x]
\Big| \leq \frac{C(\alpha, \theta)}{n^{\alpha}} holds with an explicit
constant . The key ingredients of the proof are a novel
probabilistic representation of as compound distribution and new, refined
versions of certain quantitative bounds for the Poisson approximation and the
compound Poisson distribution
A Bayesian nonparametric approach to count-min sketch under power-law data streams
The count-min sketch (CMS) is a randomized data structure that provides estimates
of tokens’ frequencies in a large data stream
using a compressed representation of the
data by random hashing. In this paper,
we rely on a recent Bayesian nonparametric (BNP) view on the CMS to develop a
novel learning-augmented CMS under powerlaw data streams. We assume that tokens
in the stream are drawn from an unknown
discrete distribution, which is endowed with a
normalized inverse Gaussian process (NIGP)
prior. Then, using distributional properties
of the NIGP, we compute the posterior distribution of a token’s frequency in the stream,
given the hashed data, and in turn corresponding BNP estimates. Applications to synthetic
and real data show that our approach achieves
a remarkable performance in the estimation of
low-frequency tokens. This is known to be a
desirable feature in the context of natural language processing, where it is indeed common
in the context of the power-law behaviour of
the data
Frequentistic approximations to Bayesian prevision of exchangeable random elements
Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements,
let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive
distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an
upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n
d_[X^m](p_m(\xi^(n)), \empiricn^m), where \empiricn is the empirical measure,
b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]]
and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m],
respectively, with the proviso that [S] denotes the space of all probability
measures on S. A characteristic feature of our work is that the aforesaid
bounds are established under the law of the \xi_n's, unlike the more common
literature on Bayesian consistency, where they are studied with respect to
product measures (p_0)^\infty, as p_0 varies among the admissible
determinations of a random probability measure
De Finetti's theorem: rate of convergence in Kolmogorov distance
This paper provides a quantitative version of de Finetti law of large
numbers. Given an infinite sequence of exchangeable
Bernoulli variables, it is well-known that , for a suitable random variable taking
values in . Here, we consider the rate of convergence in law of
towards , with respect to the Kolmogorov
distance. After showing that any rate of the type of can be
obtained for any , we find a sufficient condition on the
probability distribution of for the achievement of the optimal rate of
convergence, that is . Our main result improve on existing literature: in
particular, with respect to \cite{MPS}, we study a stronger metric while, with
respect to \cite{Mna}, we weaken the regularity hypothesis on the probability
distribution of