## Countably many random vectors and related problems.

Say I have a countably infinite number of iid random vectors $X_i:i\in\mathbb{N}$, each uniformly distributed on $[0,1]^k$ with say, $k=2$.

I need to evaluate stuff like:

$E_{X_0,X_1,\ldots}[\int_{[0,1]^k}\inf_{i\in\mathbb{N}}\|X_i-y\|^2dy]$,

which I "believe" is equal to $0$ (this is indeed a random vector quantizer). Regarding my belief, I can at least show that

$E_{X_0,X_1,\ldots,X_n}[\int_{[0,1]^k}\min_{0 \leq i \leq n}\|X_i-y\|^2dy] \rightarrow 0$ as $n\rightarrow\infty$.

Remark: Obviously writing the first expectation above does not make sense unless one defines how to do it; I do not know how to do it.

Is there any reference that might be relevant and covers similar problems?

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The first expectation makes perfect sense; the underlying probability space is the product of countably many copies of $[0,1]^k$. Unless I'm overlooking something all you need here is the dominated convergence theorem. – Mark Meckes Sep 25 at 0:13

1) Yes, you can define properly the first expectation, see e.g. http://planetmath.org/encyclopedia/TotallyFiniteMeasure.html

2) Then you have with your notations $$\mathbb E_{X_i : i\in\mathbb N}\int_{[0,1]^k}\inf_{i\in\mathbb N}\|X_i-y\|^2dy\leq \mathbb E_{X_i : i\in\mathbb N}\int_{[0,1]^k}\min_{1\leq i \leq N}\|X_i-y\|^2dy = \mathbb E_{X_1,\ldots,X_N}\int_{[0,1]^k}\min_{1\leq i \leq N}\|X_i-y\|^2dy$$ and use what you know.

Anyway, questions like this should be first asked on math stack exchange, they are not "research level questions".

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Wait - why so complicated ? Don't you have that the sequence $(X_n)$ is a.s. everywhere dense in the cube ? Then the inf in the expectation you want to compute is always equal to $0$ ...