I need to prove that if $\phi(t)$ if a characteristic function then so is
$e^{\lambda(\phi(t) -1)}$ for $\lambda$ > 0
My problem is that I'm stuck at proving uniform continuity. Is it sufficient to say that it follows since $\phi$ and $e^x$ are uniformly continuous? That isn't really a proof though.
Let $X_1,X_2,\dots$ be a sequence of i.i.d. random variables with characteristic function $\phi$. Let $N$ be a Poisson random variable with parameter $\lambda$, independent with the $X_i$. Then the random variable $$ Z = \sum_{i=1}^N X_i \overset{\text{def}}{=} \sum_{i=1}^\infty X_i\,1_{N \geq i} $$ has characteristic function $e^{\lambda(\phi(t)-1)}$. See this article for reference, or this answer on m.s.e. for a very similar computation (concerning the moment generating function).
Actually the function $x \mapsto e^x$ is not uniformly continuous on the real line (can you prove why not?). So you will have to use properties of a characteristic function for your proof.
