Exchangeable random variables: Difference between revisions

Revision as of 13:31, 30 July 2023

Revision as of 08:04, 6 September 2023

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* O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. ''International Statistical Review'' '''77(2)''', pp. 241–250.</ref>

'''The representation theorem:''' This statement is based on the presentation in O'Neill (2009) in references below. Given an infinite sequence of random variables <math>\mathbf{X}=(X_1,X_2,X_3,\ldots)</math> we define the limiting [[empirical distribution function]] <math>F_\mathbf{X}</math> by:

::::: <math>F_\mathbf{X}(x) = \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n I(X_i \le x).</math>

(This is the [[Cesaro summation|Cesaro limit]] of the indicator functions. In cases where the Cesaro limit does not exist this function can actually be defined as the [[Banach limit]] of the indicator functions, which is an extension of this limit. This latter limit always exists for sums of indicator functions, so that the empirical distribution is always well-defined.) This means that for any vector of random variables in the sequence we have joint distribution function given by:

::::: <math>\Pr (X_1 \le x_1,X_2 \le x_2,\ldots,X_n \le x_n) = \int \prod_{i=1}^n F_\mathbf{X}(x_i)\,dP(F_\mathbf{X}).</math>

If the distribution function <math>F_\mathbf{X}</math> is indexed by another parameter <math>\theta</math> then (with densities appropriately defined) we have:

::::: <math>p_{X_1,\ldots,X_n}(x_1,\ldots,x_n) = \int \prod_{i=1}^n p_{X_i}(x_i\mid\theta)\,dP(\theta).</math>

These equations show the joint distribution or density characterised as a mixture distribution based on the underlying limiting empirical distribution (or a parameter indexing this distribution).