Uniqueness of distribution of sum of i.i.d. random variables

Question

Suppose $\{X_i\}$ and $\{Y_i\}$ are two sequences of random variables, each being i.i.d.. If for some $p\ge 2$, we know $\sum_{i=1}^p X_i\overset{d}{=}\sum_{i=1}^p Y_i$, can one infer $X_i\overset{d}{=} Y_i$? Equivalently, this is asking if the characteristic functions satisfy $\phi_X^p=\phi_Y^p$, can one infer $\phi_X=\phi_Y$?

Answer 1

We can have two different characteristic functions whose squares are identical! Here is the example form Feller's book. Let $\phi_1(t)=1-|t|$ for $|t| \leq 1$ and extend the definition to $\mathbb R$ by making it periodic with period $2$. Let $\phi_2(t)=2[\phi_1(\frac t 2)-\frac 1 2]$. $\phi_1$ can be shown to be the characteristic function of a random variable which takes the value $0$ with probability $\frac 1 2$ and the values $\pm (2k+1)$ with probabilities $\frac 2 {[(2k+1)\pi]^{2}}$ and $\phi_2$ is obtained from $\phi_1$ by a simple transformation.