On the extendibility of partially and Markov exchangeable binary sequences

On the extendibility of finitely exchangeable probability measures

A length-$n$ random sequence $X_1,\ldots,X_n$ in a space $S$ is finitely exchangeable if its distribution is invariant under all $n!$ permutations of coordinates. Given $N > n$, we study the extendibility problem: when is it the case that there is a length-$N$ exchangeable random sequence $Y_1,\ldots, Y_N$ so that $(Y_1,\ldots,Y_n)$ has the same distribution as $(X_1,\ldots,X_n)$? In this paper, we give a necessary and sufficient condition so that, for given $n$ and $N$, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments t…

The infinite extendibility problem for exchangeable real-valued random vectors

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on $\mathbb{R}^d$, which is: Can a given random vector $\vec{X} = (X_1,\ldots,X_d)$ be represented in distribution as the first $d$ members of an infinite exchangeable sequence of random variables? This is the case if and only if $\vec{X}$ has a stochastic representation that is "conditionally iid" according to the seminal de Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model $\vec{X}$ is not one of conditional independence. After an…

无限可交换序列的刚性

On rigidity of infinite exchangeable sequences

该研究探究无限可交换序列的概率分布由有限维边缘唯一确定的条件，通过概率核混合、指数核等情形推导最优坐标阈值，改进Freedman定理并确定ℓ_q-范数指数律的唯一性阈值。

Countable Partially Exchangeable Mixtures

Partially exchangeable sequences representable as mixtures of Markov chains are completely specified by de Finetti's mixing measure. The paper characterizes, in terms of a subclass of hidden Markov models, the partially exchangeable sequences with mixing measure concentrated on a countable set, both for discrete valued and for Polish valued sequences.

An ergodic theorem for partially exchangeable random partitions

We consider shifts $Π_{n,m}$ of a partially exchangeable random partition $Π_\infty$ of $\mathbb{N}$ obtained by restricting $Π_\infty$ to $\{n+1,n+2,\dots, n+m\}$ and then subtracting $n$ from each element to get a partition of $[m]:= \{1, \ldots, m \}$. We show that for each fixed $m$ the distribution of $Π_{n,m}$ converges to the distribution of the restriction to $[m]$ of the exchangeable random partition of $\mathbb{N}$ with the same ranked frequencies as $Π_\infty$. As a consequence, the partially exchangeable random partition $Π_\infty$ is exchangeable if and only if $Π_\infty$ is stati…

Finite Partially Exchangeable Laws are Signed Mixtures of Product Laws

Given a partition $\{I_1,\ldots,I_k\}$ of $\{1,\ldots,n\}$, let $(X_1,\ldots,X_n)$ be random vector with each $X_i$ taking values in an arbitrary measurable space $(S,\mathscr{S})$ such that their joint law is invariant under finite permutations of the indexes within each class $I_j$. Then, it is shown that this law has to be a signed mixture of independent laws and identically distributed within each class $I_j$. The representation is unique if and only if the set of these signed measures is weakly compact. We provide a necessary condition for the existence of a nonnegative directing measure.…