arXiv subjects

Sandra Fortini

Publications and source records attributed to Sandra Fortini.

21 records · Page 2Linked to original sources

arXiv2017

On a notion of partially conditionally identically distributed sequences

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence that is weaker than exchangeability, but is equivalent to exchangeability for stationary sequences. In this article we extend this notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, that is equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies in the foundations of Bayesian statistics. Natural examples are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.

math.PR↗

arXiv2015

Predictive Characterization of Mixtures of Markov Chains

Predictive constructions are a powerful way of characterizing the probability law of stochastic processes with certain forms of invariance, such as exchangeability or Markov exchangeability. When de Finetti-like representation theorems are available, the predictive characterization implicitly defines the prior distribution, starting from assumptions on the observables; moreover, it often helps designing efficient computational strategies. In this paper we give necessary and sufficient conditions on the sequence of predictive distributions such that they characterize a Markov exchangeable probability law for a discrete valued process X. Under recurrence, Markov exchangeable processes are mixtures of Markov chains. Thus, our results help checking when a predictive scheme characterizes a prior for Bayesian inference on the unknown transition matrix of a Markov chain. Our predictive conditions are in some sense minimal sufficient conditions for Markov exchangeability; we also provide predictive conditions for recurrence. We illustrate their application in relevant examples from the literature and in novel constructions.

stat.ME↗

arXiv2012

Central Limit Theorem with Exchangeable Summands and Mixtures of Stable Laws as Limits

The problem of convergence in law of normed sums of exchangeable random variables is examined. First, the problem is studied w.r.t. arrays of exchangeable random variables, and the special role played by mixtures of products of stable laws - as limits in law of normed sums in different rows of the array - is emphasized. Necessary and sufficient conditions for convergence to a specific form in the above class of measures are then given. Moreover, sufficient conditions for convergence of sums in a single row are proved. Finally, a potentially useful variant of the formulation of the results just summarized is briefly sketched, a more complete study of it being deferred to a future work.

math.PR↗