Characteristic Function $\phi_{S_N}$ where $N$ is a random variable

Question

Let $X_1, X_2$, ... be i.i.d random variables all whose characteristic functions are $\phi_{X_i} = \phi(t), i = 1, 2, ...$

If $N$ is a random variable taking values in the positive integers, and if $N$ is independent of $X_1, X_2$, ..., determine $\phi_{S_N}$ where $S_N = X_1 + X_2 + ... + X_N.$

Attempt:

$$ \phi_{S_N}(t) = E(e^{itS_N}) = E(exp\{it(X_1 + X_2 + ... + X_N)\} $$

$$ = E(e^{itX_1}e^{itX_2}...e^{itX_N}) $$

$$ = E(e^{itX_1})...E(e^{itX_N})$$

$$ = \phi_{X_1}\phi_{X_2}...\phi_{X_N} $$

$$ = [\phi(t)]^N $$

Official Answer: $ E\Big([\phi(t)]^N\Big) $

Answer 1

The problem in your answer is that $\phi(t)^N$ is a random variable which may not be constant.

First observe that $e^{itS_N}=\sum_{n\geqslant 1}e^{itS_n}\mathbf 1\{N=n\}$. Then take the expectation on both sides. In order to compute $\mathbb E\left[e^{itS_n}\mathbf 1\{N=n\}\right]$, first use the independence of $N$ with $S_n$ to write it as $\mathbb E\left[e^{itS_n}\right]\mathbb P\{N=n\}$. Then use what you did for a fixed $n$.