Hello Math Stack Exchange community,
I am currently studying the concept of characteristic functions in probability theory and came across an interesting problem. I understand that for independent random variables, the characteristic function of their sum is the product of their individual characteristic functions, i.e., $\phi_{X+Y}(t) = \phi_X(t) \phi_Y(t)$ for all $t$. However, I am curious about whether this property can hold for dependent random variables as well.
Problem: Find two dependent random variables $X$ and $Y$ such that $\phi_{X+Y}(t) = \phi_X(t) \phi_Y(t)$ for all $t$.
Here is what I know that might help:
The characteristic function of a random variable $X$ is defined as $\phi_X(t) = E[e^{itX}]$, where $i$ is the imaginary unit, and $E[\cdot]$ represents the expected value.
The characteristic function of a sum of two random variables is given by $\phi_{X+Y}(t) = E[e^{it(X+Y)}]$.
With this knowledge, I attempted to answer the problem:
Let $X$ and $Y$ be two random variables with known characteristic functions $\phi_X(t)$ and $\phi_Y(t)$. We want to find a pair of dependent random variables $X$ and $Y$ that satisfy the given condition.
At this point, I got stuck. I tried to relate the expected values of $X$ and $Y$ and their characteristic functions, but couldn't make any progress. I am not sure how to approach this problem further or whether there are specific properties of dependent random variables that can help me. Thank you!
Best Answer
Answer for the question in the title: Let $X$ have Cauchy distribution and $Y=X$. Then $X$ and $Y$ are dependent, $Ee^{it(X+Y)}=Ee^{2it X}=e^{-|2t|}$ and $Ee^{itX}Ee^{itY}=e^{-|t|}e^{-|t|}=e^{-2|t|}$.