The fundamental theorem of mutual insurance

The essence of mutual insurance is the notion that re-distributing risk in a pool of risks is more beneficial than taking the risk alone. Interpreting ‘more beneficial’ as an increase in utility and considering sequences of exchangeable risks, we are able to formalize this notion from the policyholder’s perspective and demonstrate its validity for various alternative preference functionals (e.g., expected utility, Choquet expected utility, and distortion risk measures). To obtain this result, we exploit that for a sequence of exchangeable risks the corresponding sequence of arithmetical averages is a reversed martingale.We conclude that pooling risks is fundamental for understanding the mechanisms of insurance because it favourably affects the utility of policyholders, and we refer to this phenomenon as the ‘utility-improving effect of risk pooling’. Moreover, we demonstrate that the utility of the policyholder is (strictly) increasing with the size of the risk pool.     

Nonparametric predictive utility inference

We consider the natural combination of two strands of recent statistical research, i.e., that of decision making with uncertain utility and that of Nonparametric Predictive Inference (NPI). In doing so we present the idea of Nonparametric Predictive Utility Inference (NPUI), which is suggested as a possible strategy for the problem of utility induction in cases of extremely vague prior information. An example of the use of NPUI within a motivating sequential decision problem is also considered for two extreme selection criteria, i.e., a rule that is based on an attitude of extreme pessimism and a rule that is based on an attitude of extreme optimism.     

Symmetric Distributions of Random Measures in Higher Dimensions

An infinite sequence of random variables X=(X ₁, X ₂,...) is said to be spreadable if all subsequences of X have the same distribution. Ryll-Nardzewski showed that X is spreadable iff it is exchangeable. This result has been generalized to various discrete parameter and higher dimensional settings. In this paper we show that a random measure on the tetrahedral space is spreadable, iff it can be extended to an exchangeable random measure on . The result is a continuous parameter version of a theorem by Kallenberg.     

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 weaker than exchangeability, but equivalent to it in the presence of stationarity. We extend such 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, which is shown to be 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 at the foundations of Bayesian statistics. Natural examples of partially c.i.d. constructions are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.     

Exchangeable measures for subshifts

Let Ω be a Borel subset of where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when . We apply the ergodic theory of equivalence relations to study the case , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0,1] (a non-Markovian constraint).     

A noncommutative extended de Finetti theorem

The extended de Finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem.In contrast to the classical result of Ryll-Nardzewski, exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and Kümmerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.     

When is the mean self-consistent?

We study the conditions under which the sample mean is self-consistent, and therefore an optimal predictor, for an arbitrary observation in the sample.     

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ${\tilde{\xi }}_1,{\tilde{\xi }}_2,\dots$, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution ${\tilde{\mathfrak{e}}}_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{{\tilde{\xi }}_i}$ of the n past observations, while the latter is compared to the m-fold product ${\tilde{\mathfrak{e}}}_n^m$. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts ${\delta }_{{\tilde{\mathfrak{e}}}_n}$ and ${\tilde{\mathfrak{e}}}_n^m$, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ${\tilde{\xi }}_n$'s, and not under hypothetical product laws ${\mathfrak{p}}_0^{\infty }$, as ${\mathfrak{p}}_0$ varies among the admissible determinations of a random probability measure.     

A De Finetti-type theorem for nonexchangeable finite-valued random variables

We introduce the concept of a random assignment process (roughly speaking, such a process assigns, according to some weight function, labels to the points of an arbitrary set), and we impose conditions on the weight function ensuring that a De Finetti-type theorem is satisfied. In particular, this provides a wide class of finite-valued random variables which, despite they are neither exchangeable nor identically distributed, verify a De Finetti-type theorem (i.e., they are conditionally independent).     

Laws of Large Numbers for Exchangeable Random Sets in Kuratowski-Mosco Sense

Abstract

Most of the results for laws of large numbers based on Banach space valued random sets assume that the sets are independent and identically distributed (IID) and compact, in which Rådström embedding or the refined method for collection of compact and convex subsets of a Banach space plays an important role. In this paper, exchangeability among random sets as a dependency, instead of IID, is assumed in obtaining strong laws of large numbers, since some kind of dependency of random variables may be often required for many statistical analyses. Also, the Hausdorff convergence usually used is replaced by another topology, Kuratowski-Mosco convergence. Thus, we prove strong laws of large numbers for exchangeable random sets in Kuratowski-Mosco convergence, without assuming the sets are compact, which is weaker than Hausdorff sense.