Let $\phi(t)$ be a characteristic equation of a discrete r.v. X. Show that the function $e^{\mu (\phi(t) -1)}$ is a characteristic function. (Hint: Consider the sum of iid R.v.)
Attempt: I was told that the function could be shown to be a the sum of characteristic function of iid Poisson rv. I know the Poisson characteristic for the sum of 2 rv to be $e^{(\mu + \lambda)(e^{it}-1)}$. I am having trouble splitting this function $e^{\mu (\phi(t) -1)}$ to $e^{(\mu + \lambda)(e^{it}-1)}$. I was trying $e^{\mu (\phi(t) -1)} = \sum \frac{(\mu (\phi (t)-1))^x}{x!}$. But I got stuck as that's an infinite sum.
Best Answer
Let $\{X_i\}$ be i.i.d with characteristic function $\phi$. Let $N$ be a Poisson random variable with parameter $\mu$ independent of $X_i$'s. Then $Ee^{it(X_1+X_2+...+X_N)}=\sum_{n=0}^{\infty} Ee^{it(X_1+X_2+...+X_n)}e^{-\mu} \frac {\mu ^{n}} {n!}=\sum_{n=0}^{\infty} \phi (t)^{n}e^{-\mu} \frac {\mu ^{n}} {n!}=e^{\mu (\phi(t)-1)}$.