Definition of sequence sampled from a measure

Question

Question: Exactly what does it mean for a sequence of points to be sampled from a given probability measure?

I have in mind statements such as «let the sequence $(x_k)$ be sampled with density $f$», where $f \ge 0$ is a Borel function on $\mathbb{R}^n$ such that $\int f = 1$, or for such $f$ defined on appropriate submanifolds of $\mathbb{R}^n$.

I am looking for the proper, abstract framework in which to formulate this concept, along with a formal definition in said context. Assume full knowledge of topology, measure theory, functional analysis, etc.

My intuition is something along the lines that this should hold iff for every open set $U \subseteq \mathbb{R}^n$,

$$\lim_m \, (\text{density of points } x_k \text{ for } k \le m \text{ in } U ) = \int_U f,$$

but it is not obvious to me how this notion of density should be defined or if this is sound.

Certainly the Euclidean structure has nothing to do with the phenomenon, and the concept should be expressible in suitable topological spaces (with a Borel measure) or even in general probability spaces.

Answer 1

What you are asking about is called equidistribution with respect to a given probability measure $\mu$ on a topological space $X$. Namely, a sequence $(x_n)$ is $\mu$-equidistributed if the empirical distributions $(\delta_{x_1} + \dots + \delta_{x_n})$ weakly converge to $\mu$. Whether $X=\mathbb R^n$, or whether $\mu$ has a density (either with respect to the Lebesgue measure, or with respect to any given reference measure in the general case) is completely irrelevant.

If you are interested in the "higher order" equidistribution as well (this question arises in the theory of pseudorandom number generators), then one should also require, for any $d>1$, the convergence of order $d$ empirical distributions (i.e., those of the "vectors" $(x_n,x_{n+1},\dots,x_{d+d-1})$) to the product of $\mu$ by itself $d$ times.

In the case when $X$ is finite, and $\mu$ is the uniform distribution on $X$, these are precisely the definitions of, respectively, the "simply normal" and the "normal" real numbers (identified with their base $|X|$ expansions).

However, as pointed out by Dieter Kadelka, one should realize that usually, when talking about a point or a sequence (which is a point in the space of sequences) sampled from a certain distribution $\mu$, one just means a random point whose distribution is the measure $\mu$.