# If two random variables $X_1$ and $X_2$ are dependent then must $X_1^2$ and $X_2^2$ be dependent?

If two random variables $$X_1$$ and $$X_2$$ are dependent then $$X_1^2$$ and $$X_2^2$$ be dependent.

I believe this statement to be false. Considering that $$X_1$$ and $$X_2$$ being dependent implies

$$\sigma(X_1)$$ is dependent of $$\sigma(X_2)$$ that is the sigma algebras generated by each rv are dependent, but since $$\sigma(X_1^2)\subset \sigma(X_1)$$ and $$\sigma(X_2^2)\subset \sigma(X_2)$$ the reduction could potentially lead to independent sigma algebras.

The counter example I came up with is

let:

$$X_1\sim \text{Unif}(0,1)$$ and $$X_2|X_1 = \begin{cases} 1 & X_1\in[0,\frac{1}{2})\\ -1 & X_1\in[\frac{1}{2},1]\\ \end{cases}$$

Note these two random variables are highly dependent but when I square both $$X_1\sim \frac{1}{2\sqrt{x_1}}$$ and $$X_1|X_1=1$$ thus the two squared random variables are independent. Is this counterexample sound?

• Does "dependent" mean "not independent" or "one determines the other"? Aug 16 '20 at 2:44
• @Henry Dependent as not independent Aug 16 '20 at 2:46

Your counter-example works, thought since your $$X_2^2$$ is constant it is not very revealing, as it is independent of everything
Another might be to have $$A$$ and $$B$$ independently standard normal (mean $$0$$, variance $$1$$) and
$$X_1=A$$ while $$X_2=\text{sign}(A)\, |B|$$.
Then $$X_1$$ and $$X_2$$ are positively correlated normal distributions while $$X_1^2$$ and $$X_2^2$$ are independent chi-squared distributions