Exchangeable random variables: Difference between revisions

Revision as of 22:27, 7 December 2023

Revision as of 13:33, 2 January 2024

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: <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|CesaroCesàro limit]] of the indicator functions. In cases where the CesaroCesàro 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>