Finite Exchangeable Sequences
Persi Diaconis,David A. Freedman +1 more
TLDR
In this paper, it was shown that the variation distance between the distribution of exchangeable random variables and the closest mixture of independent, identically distributed random variables is at most 2 ck/n, where c is the cardinality of the set.Abstract:
Let $X_1, X_2,\cdots, X_k, X_{k+1},\cdots, X_n$ be exchangeable random variables taking values in the set $S$. The variation distance between the distribution of $X_1, X_2,\cdots, X_k$ and the closest mixture of independent, identically distributed random variables is shown to be at most $2 ck/n$, where $c$ is the cardinality of $S$. If $c$ is infinite, the bound $k(k - 1)/n$ is obtained. These results imply the most general known forms of de Finetti's theorem. Examples are given to show that the rates $k/n$ and $k(k - 1)/n$ cannot be improved. The main tool is a bound on the variation distance between sampling with and without replacement. For instance, suppose an urn contains $n$ balls, each marked with some element of the set $S$, whose cardinality $c$ is finite. Now $k$ draws are made at random from this urn, either with or without replacement. This generates two probability distributions on the set of $k$-tuples, and the variation distance between them is at most $2 ck/n$.read more
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References
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An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
Journal ArticleDOI
An Introduction to Probability Theory and Its Applications.
Book ChapterDOI
Probability Inequalities for sums of Bounded Random Variables
TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
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Sums of Independent Random Variables
TL;DR: In this paper, the authors define the notion of infinitely divisible distributions as the limits of the distributions of sums of independent random variables, and show that the distribution of a sum of independent non-identically distributed random variables converges to a given infinitely-divisible distribution.