1,720,973 research outputs found
Asymptotic number of clusters for species sampling sequences with non-diffuse base measure
No full textWe investigate the asymptotic clustering structure of species sampling sequences (ξn)n, for which the base measure has atomic components. We prove a stochastic representation for (ξn)n in terms of a latent exchangeable random partition. Then, we study the asymptotic behaviour of the total number of blocks and of the number of blocks with fixed cardinality in the partition generated by (ξn)n
Asymptotic behavior of finite partially exchangeable random arrays: a central limit theorem
No full textWe study a central limit theorem for a sequence of finite arrays of partially exchangeable random variables
A central limit problem for partially exchangeable random variables. 1996.
No full textThe paper deals with the central limit problem for arrays of partially exchangeable random variables. It is shown that, under suitable negligibility conditions, the class of limiting laws coincides with that of exchangeable laws which are presentable as mixtures of infinitely divisible distributions. Moreover necessary and sufficient conditions for convergence to any specified element are provided. An analogous representation is given for mixtures of stable laws
Mixture of Species Sampling Models
Full text linkWe introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric priors recently introduced to provide sparsity. Furthermore, we show how mSSS arise while considering hierarchical species sampling random probabilities (e.g., the hierarchical Dirichlet process). Extending previous results, we prove that mSSS are obtained by assigning the values of an exchangeable sequence to the classes of a latent exchangeable random partition. Using this representation, we give an explicit expression of the Exchangeable Partition Probability Function of the partition generated by an mSSS. Some special cases are discussed in detail—in particular, species sampling sequences with general base measures and a mixture of species sampling sequences with Gibbs-type latent partition. Finally, we give explicit expressions of the predictive distributions of an mSSS
Exchangeability, predictive distributions and parametric models
No full textConditions are stated in order that parametric models turn out to be limiting forms of predictive distributions and parameters are limt of suitable predictive sufficient statistics
Central limit theorem in uniform metrics for generalized Kac equations
No full textThe aim of this paper is to give explicit rates for the speed of convergence to equilibrium of the solution of the generalized Kac equation in two strong metrics: the total variation distance (TV) and the uniform metric between characteristic functions (χ0). A fundamental role in our study is played by the probabilistic representation of the solution of the generalized Kac equation as marginal law of a stochastic process which is a weighted random sum of i.i.d. random variables, where the weights are positive and dependent. Exponential bounds for the total variation distance between the solution and the gaussian stationary state of the Kac equation have been proved by Dolera, Gabetta and Regazzini (2009). In our more general setting the equilibrium states are scale mixtures of stable distributions and hence not necessarily gaussian. Therefore we develop new tools based on ideal metrics that are used in the literature for quantitative central limit theorems for i.i.d. random variables in the domain of attraction of a stable distribution. We obtain first exponential bounds in the so-called ”r-smoothed total variation” and in the weighted χr-metric for a suitable r, then we deduce rates of convergence with respect to the “corresponding” uniform metrics TV and χ0
Self-similar solutions in one-dimensional kinetic models: a probabilistic view
No full textThis paper deals with a class of Boltzmann equations on the real line, extensions of the well-known Kac caricature. A distinguishing feature of the corresponding equations is that therein, the collision gain operators are defined by N-linear smoothing transformations. These kind of problems have been studied, from an essentially analytic viewpoint, in a recent paper by Bobylev, Cercignani and Gamba [Comm. Math. Phys. 291 (2009) 599–644]. Instead, the present work rests exclusively on probabilistic methods, based on techniques pertaining to the classical central limit problem and to the so-called fixed-point equations for probability distributions. An advantage of resorting to methods from the probability theory is that the same results—relative to self-similar solutions—as those obtained by Bobylev, Cercignani and Gamba, are here deduced under weaker conditions. In particular, it is shown how convergence to a self-similar solution depends on the belonging of the initial datum to the domain of attraction of a specific stable distribution. Moreover, some results on the speed of convergence are given in terms of Kantorovich–Wasserstein and Zolotarev distances between probability measures
- …
