# Good reference for strictly positive random variables

I'm a number theorist finding myself needing to use some concepts from probability that are probably (no pun intended) quite basic to experts; I would rather cite a readily available source than reinvent the wheel myself.

Specifically, I would like to find a source (monograph/graduate textbook, perhaps) that gives a definition of a "strictly positive" random variable (one that assigns positive probability to every nonempty open set - not one that takes values in $\mathbb R_{>0}$). Ideally, I would like to find a source that already contains a proof of the following lemma: if $X$ and $Y$ are independent random variables taking values in the same space ($\mathbb R^n$, say), and if $X$ is strictly positive, then $X+Y$ is also strictly positive.

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You'll need an assumption that $X$ and $Y$ are independent -- if $X=-Y$, for example, you'll be out of luck. – Henning Makholm Sep 14 '11 at 23:34
I suspect that a much weaker assumption than independence would be plenty. – Michael Hardy Sep 14 '11 at 23:45
Probably, but harder to state. – Henning Makholm Sep 14 '11 at 23:48
Please don't call them "strictly positive". That name is very confusing. If these are continuous random variables, you could say they have "positive density". Or you could state it as the property that the distribution measures have full support. – Nate Eldredge Sep 15 '11 at 1:10
@Henning: They should indeed be independent, thanks - edited. – Greg Martin Sep 15 '11 at 1:32

The following is Proposition 2.1.3 (page 23) in Werner Linde's book Probability in Banach Spaces - Stable and Infinitely Divisible Distributions. It should give you what you need when $X$ and $Y$ are independent. Let $\mu$ be the distribution of $X$ and $\nu$ the distribution of $Y$, so that $\mu*\nu$ is the distribution of $X+Y$.

If $\mu,\nu$ are Radon measures on the Borel sets ${\cal B}(E)$ of a Banach space $E$, then $$\mbox{supp}(\mu*\nu)=\overline{\mbox{supp}(\mu)+\mbox{supp}(\nu)}.$$

For a proof, the reader is referred to Theorem 1.2.1 of Probability Measures on Locally Compact Groups by H. Heyer. I think that this would be a good, standard reference, though I don't own Heyer's book so I can't check it.

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No reference, but here goes, assuming that $X$ and $Y$ are independent and take values in a finite-dimensional real vector space:

It is enough to prove the property for an arbitrary open ball $B_r(x)$. The entire value space is covered by countably many open balls of radius $r/2$; by countable additivity at least one of these balls, call it $B_{r/2}(y)$, must contain a positive probability mass for $Y$.

Now, by assumption $P(X\in B_{r/2}(x-y))$ is also positive, and $P(X+Y\in B_r(x))$ must be at least the product of these probabilities.

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Good - this is very similar to the proof I have. My desire, though, is to simply cite this fact from a suitable source, rather than redo known work. Even a source that provides some (more?) standard terminology would be helpful. – Greg Martin Sep 15 '11 at 1:34