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Setup:

A set of random variables $X_1, \ldots X_n$ is independent if for all Borel-sets $B_1,\ldots,B_n$ it is the case that $$P(X_1 \in B_1,\ldots,X_n \in B_n) = \prod_{i=1}^n P(X_i \in B_i)$$

An alternative definition is that the generated sigma-algebras $\sigma(X_1),\ldots,\sigma(X_n)$ should be independent.

Question:

Let $X, Y, Z$ be independent random variables. I am pretty sure that this implies that $XY$ and $Z$ are also independent, but I have not been able to prove this.

Is there an elementary way to prove this or is there a trick or a general theorem needed?

Thank you.

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Hint:

  1. Recall (or check) that $(X,Y)$ and $Z$ are independent.
  2. Set $g(x,y) := x \cdot y$, then $X \cdot Y = g(X,Y)$. Thus, $$\mathbb{P}(X \cdot Y \in B_1, Z \in B_2) = \mathbb{P}\big((X,Y) \in g^{-1}(B_1), Z \in B_2 \big)$$ for any two Borel sets $B_1$, $B_2$. (Note that $g^{-1}(B_1)$ is the inverse image of $B_1$ under $g$. No need for an inverse function!) Now use step 1 to write the right-hand side as a product of suitable probabilities.
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  • $\begingroup$ Thank you! Does this work? For all $a, b, c \in \mathbb R$, $$P((X,Y) \in (-\infty, a]\times (-\infty, b], Z \in (-\infty, c]) = P(X \in (-\infty, a], Y \in (-\infty, b], Z \in (-\infty, c]) = P(X \in (-\infty, a]) P(Y \in (-\infty, b]) P(Z \in (-\infty, c]) = P((X,Y) \in (-\infty, a]\times (-\infty, b]) P(Z \in (-\infty, c]).$$ Since the rectangles used above are $\pi$-stable generators of $\mathscr B^2$ and $\mathscr B^1$, respectively, and $(X,Y)$ and $Z$ are independent on those rectangles, they are independent. Now, since multiplication is measurable, the claim follows. $\endgroup$ – Epiousios Aug 31 '17 at 7:53
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    $\begingroup$ @1524 Yes, that's correct. $\endgroup$ – saz Aug 31 '17 at 8:57

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