1
$\begingroup$

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.

  • 0
    @Nate: I'm happy to use whatever terminology is used by a suitable reference...!2011-09-15

2 Answers 2

3

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.

1

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.

  • 0
    Good - this is very similar to the proof I have. My desire, though, is to simply cite this fact from a suitable source, rather tha$n$ $r$edo known work. Even a source that provides some (more?) standard terminology would be helpful.2011-09-15