Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x\_n$. Is there a limit formula for $supp(\mu)$, something like closure of limit points of $x\_n$ or similar, which allows kind of asymptotic recovering of $supp(\mu)$ from observed data.



Recovering measure support from the sequence of I.I.D random variables
By f.i. <https://encyclopediaofmath.org/wiki/Support_of_a_measure> $supp(\mu)$ is the smallest closed set $C \subset \mathbb{R}^n$ such that $\mu(C^c) = 0$, $C^c$ the complement of $C$. Let $y := (y\_n)\_{n \in \mathbb{N}}$ be any sample of $(x\_n)$ (independent realizations of $\mu$) and $A := \{y\_n \colon n \in \mathbb{N}\}$, $A\_N := \{y\_n \colon n \leq N\}$ for $N \in \mathbb{N}$. Then by Borel-Cantelli with probability $1$ $A \subset C$, $A\_N \uparrow A$ and $A$ is dense in $C$, in particular $\bar A = C$. Thus each sequence of iid random variables recovers the support (a.s.).


