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(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.) If the sequence <math>\bold{X}</math> is exchangeable then the elements of <math>\bold{X} | F_\bold{X}</math> are independent with distribution function <math>F_\bold{X}</math>. 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,...,X_n \le x_n) = \int \prod_{i=1}^n F_\bold{X}(x_i)\,dP(
If the distribution function <math>F_\bold{X}</math> is indexed by another parameter <math>\theta</math> then (with densities appropriately defined) we have:
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