Alexander Gnedin, Grigori Olshanski
A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois field F_q that are invariant under the natural action of the infinite group of invertible matrices with coefficients from F_q.
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Finite version of the $q$-analogue of de Finetti's theorem
本文提出了 de Finetti 定理 $q$-类比的渐近版本，利用 $q$-可交换概率测度空间的凸结构，证明了其最优收敛速率阶为 $q^n$。
We show an organized form of quantum de Finetti theorem for Boolean independence. We define a Boolean analogue of easy quantum groups for the categories of interval partitions, which is a family of sequences of quantum semigroups. We construct the Haar states on those quantum semigroups. The proof of our de Finetti theorem is based on the analysis of the Haar states. [Modified]Definition of the Boolean quantum semigroups on categories of interval partitions [Delete]Classification of categories of interval partitions [Add]Proof of the positiveness of the Haar functionals (in particular they are…
In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {\it axiom} for the probability of the disjunction of two incompatible events becomes a {\it consequence} of de Finetti's logic-operational consistency notion. Working in the context of boolean algebras, we prove de Finetti's theorem. The mathematical background required is little more than that which is taught in high school. As a preliminary step we prove what de Finetti called ``the Fundamental Theorem of Probability'', his main contribution both to Boole's probabilistic inferen…
A sequence of random variables is called exchangeable if the joint distribution of the sequence is unchanged by any permutation of the indices. De Finetti's theorem characterizes all $\{0,1\}$-valued exchangeable sequences as a "mixture" of sequences of independent random variables. We present an new, elementary proof of de Finetti's Theorem. The purpose of this paper is to make this theorem accessible to a broader community through an essentially self-contained proof.
We show that boundary theory for transient Markov chains, as initiated by Doob, can be used to prove de Finetti's classical representation result for exchangeable random sequences. We also include the relevant parts of the theory, with full proofs.
We provide a characterization of continuous semimartingales whose law is invariant with respect to predictable random rotations. In particular we prove that all such semimartingales are obtained by integrating a predictable process with respect to an independent $n$ dimensional Brownian motion.
We consider a notion of uniform thinning for a finite sequence of random variables $(X_1,...,X_n)$ obtained by removing one random variable, uniformly at random. If a triangular array of random variables $(X_{n,k} : n \in \mathbb{N}_+, 1 \le k \le n)$ satisfies that the law of $(X_{n,1},...,X_{n,n})$ is obtained by uniformly thinning $(X_{n+1,1},...,X_{n+1,n+1})$, then we call the array thinning-invariant. We give a representation for the Choquet simplex of all thinning-invariant triangular arrays of random variables, when all random variables take values in a compact metric space (with Borel …
A classic and fundamental result about the decomposition of random sequences into a mixture of simpler ones is de Finetti's Theorem. In its original form it applies to infinite 0-1 valued exchangeable sequences. Later it was extended and generalized in numerous directions. After reviewing this line of development, we present our new decomposition theorem, covering cases that have not been previously considered. We also introduce a novel way of applying these types of results in the analysis of random networks. For self-containment, we provide the introductory exposition in more details than us…
A nonstandard proof of de Finetti's theorem
本文为可交换的Bernoulli随机变量序列提供了de Finetti定理的非标准分析证明，通过组合论证证明该序列的概率分布由超有限样本均值诱导，从而可唯一表示为iid序列的混合。
de Finetti's theorem and the existence of regular conditional distributions and strong laws on exchangeable algebras
本文推广了de Finetti定理，证明随机元素可交换序列条件i.i.d.等价于每个元素在可交换σ-代数上有正则条件分布，并由此推出Radon分布序列的条件i.i.d.性质。
A finite form of de Finetti's representation theorem is established using elementary information-theoretic tools: The distribution of the first $k$ random variables in an exchangeable binary vector of length $n\geq k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.
The extended de Finetti theorem characterizes exchangeable infinite random sequences 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-Nadzewski, exchangeability turns out to be stronger than spreadability for infinite noncommutative random sequences. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of c…
We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finetti's Theorem for standard Borel spaces is an instance of this result.
A non-de Finetti theorem for countable Euclidean spaces
本研究针对德·菲内蒂定理的群不变测度分类问题，在Jahel--Tsankov的推广成果基础上，给出了一个非德·菲内蒂的非阿基米德群实例，拓展了相关理论研究范畴。
We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their…
We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Köstl…
We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.
The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant stat…
We introduce a family of quantum semigroups and their natural coactions on noncommutative polynomials. We present three invariance conditions, associated with these coactions, for the joint distribution of sequences of selfadjoint noncommutative random variables. For one of the invariance conditions, we prove that the joint distribution of an infinite sequence of noncommutative random variables satisfies it is equivalent to the fact that the sequence of the random variables are identically distributed and boolean independent with respect to the conditional expectation onto its tail algebra. Th…
This paper provides a quantitative version of de Finetti law of large numbers. Given an infinite sequence $\{X_n\}_{n \geq 1}$ of exchangeable Bernoulli variables, it is well-known that $\frac{1}{n} \sum_{i = 1}^n X_i \stackrel{a.s.}{\longrightarrow} Y$, for a suitable random variable $Y$ taking values in $[0,1]$. Here, we consider the rate of convergence in law of $\frac{1}{n} \sum_{i = 1}^n X_i$ towards $Y$, with respect to the Kolmogorov distance. After showing that any rate of the type of $1/n^α$ can be obtained for any $α\in (0,1]$, we find a sufficient condition on the probability distri…
Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on $\mathbb{N}$, on quasi-local algebras are defined and proved to be $\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which include the CAR algebra, are then addressed as particular examples of quasi…
A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $μ$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $μ$. For a wide class of probability measures $μ$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $μ$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between th…
Finite de Finetti bounds in relative entropy
本文回顾并扩展了相对熵下的有限 de Finetti 定理，建立了两个与空间大小和维度无关的紧界，并揭示了与有放回和无放回抽样差异的联系。
De Finetti's theorem and related results for infinite weighted exchangeable sequences
本文研究了无限加权可交换序列的 De Finetti 型表示，提出嵌套条件族并导出了零一律和大数定律的加权扩展。
This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finetti's representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the Giry monad. This new limit is used to identify among exchangeable coalgebras the final one.
Full classification of de Finetti type theorems for *-random variables in classical and free probability
本文研究了*随机变量在经典和自由概率中de Finetti型定理的分类，证明了这些定理仅存在有限类型，并与Banica和Speicher引入的容易量子群相关。
We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mi…
A Third Information-Theoretic Approach to Finite de Finetti Theorems
本文利用基本的信息论工具建立了 de Finetti 表示定理的一种新有限形式，证明可交换向量中前 k 个随机变量的分布接近乘积分布的混合，以相对熵衡量并给出显式界限，该界限比早期证明更紧且适用于一般空间。
We construct several new spaces of quantum sequences and their quantum families of maps in sense of Sołtan. Then, we introduce noncommutative distributional symmetries associated with these quantum maps and study simple relations between them. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by inte…
Connecting classical finite exchangeability to quantum theory
本文通过量子理论中的boson对称密度矩阵形式，提出了一种有限交换序列的德芬尼定理表示，并探讨其在玻色子纠缠定义中的应用。
We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.
Generalizing the de Finetti--Hewitt--Savage theorem
本研究利用非标准分析工具，将de Finetti–Hewitt–Savage定理推广至任意豪斯多夫空间上的Radon分布可交换序列，还给出非标准分析入门介绍及Prokhorov定理的新推广。
In this article we give a proof of a q-analogue of the celebrated four functions theorem. This analogue was conjectured by Bjorner and includes as special cases both the four functions theorem and also Bjorner's q-analogue of the FKG inequality.
This paper has been withdrawn by the author; a revised version is part of the author's phd-thesis "Quasi-logarithmic structures" (Zurich, 2007).
In the present paper we prove that every k-dimensional Cartesian product of Kingman convolutions can be embedded into a k-dimensional symmetric convolution (k=1, 2, ...) and obtain an analogue of the Cramér-Lévy theorem for multi-dimensional Rayleigh distributions.
Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analog of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.
In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.
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