We know that for independent random variables $X,Y$ we have: $\varphi_{X+Y}(t)=\varphi_X(t)\cdot \varphi_Y(t)$.

Now I'm searching for an example that shows that the reversal of the statement is not true, so I want $X$ and $Y$ which fulfill $\varphi_{X+Y}(t)=\varphi_X(t)\cdot \varphi_Y(t)$ but are dependent.

I can't think of an example that shows this.


1 Answer 1


Consider the characteristic function of $nX$, $n \in \mathbb N$. We have

$$ \varphi_{nX}(t) = \int_\mathbb R e^{itx}f_{nX}(x)dx. $$ $f_{nX}(x) = f_X(\frac{x}{n})$ and so $$ \varphi_{nX}(t) = \frac{1}{n}\int_\mathbb R e^{intx}f_X(x)dx = \frac{1}{n}\varphi_X(nt) $$

To find a counter example, it is sufficient to find a distribution for $X$ such that $$\varphi_X(t)^n \neq \frac{1}{n}\varphi_X(nt).$$ Take, for example, the normal distribution.


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