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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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2  
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 '12 at 0:13

2 Answers 2

up vote 2 down vote accepted

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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Thank you for your reply. In the FAQ, it says "MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books." I thought the problem that I asked could be considered to be "graduate level." Perhaps there should be a criterion of e.g. "having published at least 3 Annals of Mathematics papers" to be a member of MO. For the future, I will make sure that I take my supertrivial insignificant questions elsewhere. –  roork Sep 25 '12 at 2:26
    
BTW, if a question is "research level" and can be answered by an arbitrary human H when it is asked, it means that it is not "research level" as its answer is already known by H. I also observe that except a few, almost all the questions in this forum is settled by an expert, and therefore, almost none of the questions asked here are "research level." –  roork Sep 25 '12 at 2:41
    
@roork: A research level question is not necessarily an open problem. –  Stanislav Sep 25 '12 at 8:58
    
@roork: I apology if you felt there is some contempt my answer, that is not my point. The thing is the answer follows from the definition of this infinite product probability space, and in particular its expectation, and looking for a commonly used definition is more appropriate for MSE. Usually people on MO downvote and directly close such questions, I just wanted to warn you. –  Adrien Hardy Sep 25 '12 at 10:17

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 $0$ ...

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I like it too ! –  Adrien Hardy Sep 25 '12 at 11:58
    
@Vincent: Thank you for the answer. Even though it is elegant, I wanted to know a reference where I could read about these things (this was just a toy problem to ask my question). I thus chose Adrien's answer. –  roork Sep 25 '12 at 20:45

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