Characteristic function problem

Question

first time poster so be nice! Here's the problem:

Let $\phi(t)$ be a characteristic function, then $e^{\lambda(\phi(t)-1)}$ is a characteristic function.

Pretty stuck, any help appreciated!

Answer 1

Suppose that $\phi(t)$ is CF of $Y$. Suppose $X=Y_1+...+Y_P$, sum of $P$ i.i.d. $Y_i$'s, where $P$ follows Poisson distribution of parameter $\lambda$. Then we have: $$ \mathbb E(e^{itX})=\sum_{k=0}^\infty \mathbb E(e^{itX};P=k)\frac{e^{-\lambda}\lambda^k}{k!}=\sum_{k=0}^\infty \phi(t)^k \frac{e^{-\lambda}\lambda^k}{k!}=e^{\lambda(\phi(t)-1)}. $$