3

This is apparently asked many times but there is still something I do not understand. Taken from Mathematical description of a random sample:

Mathematical description of a random sample: which one is it and why?

  1. $X_1(\omega), X_2(\omega), ..., X_n(\omega)$, where $X_1, ..., X_n$ are different but i.i.d. random variables.
  2. $X(\omega_1), X(\omega_2), ..., X(\omega_n)$, where $X$ is a (single) random variable.

I would like to add:

  1. $X_1(\omega_1), X_2(\omega_2), ..., X_n(\omega_n)$, where $X_1, ..., X_n$ are different but i.i.d. random variables.

Why not use 3.?

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After a bit of research told me that nothing of the above is true; see for example the book of Dudley "Real analysis and Probability" (section 11.4). The empirical measure on a measure space $(\Omega,\mathcal F,\mu)$ is defined via a sequence of iid random variables $X_i$ defined on $\Omega^{\mathbb N}$, by the mapping $$A\mapsto \frac{1}{n}\sum_{i=1}^n\delta_{X_i(\omega)}(A)\text{ for }\omega\in \Omega^{\mathbb N}.$$ The approximation property says that for almost all $\omega\in \Omega^{\mathbb N}$, the empirical measure converges weakly (or weakly-$*$ depending on your moods ;-) ) to $\mu$.

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