Let ${(X_i)}_{i \in \mathbb{N}}$ be a sequence of random variables on the probability space $( \Omega, \mathcal{F}, P )$ and let $N : \Omega \rightarrow \mathbb{N}_0$ be an integer-valued random variable independent from ${(X_i)}_{i \in \mathbb{N}}$.
I am in interested in showing the following:
- $S := X_1 + \ldots + X_{N}$ is a random variable, i.e., $\mathcal{F}$-measurable.
- For $x \in \mathbb{R}$ we have $$P(S \leq x) = \sum_{k=0}^{\infty}P( N = k) P(X_1+\ldots+X_k \leq x).$$
An attempt at the second part:
\begin{align} P(S \leq x) = P \left( \sum_{i = 1}^{ N } X_i \leq x \right)= E \left( 1_{ \left\{ \sum_{i = 1}^{ N } X_i \leq x \right\} } \right) \overset{\text{Tower property}}{=} E \left( E \left( 1_{ \left\{ \sum_{i = 1}^{ N } X_i \leq x \right\} } \mid \sigma(N)\right) \right)=... \end{align}
How can one use the independence property to continue the chain of arguments? Is there maybe a better way to approach this? And, of course, how can one show that $S$ is itself a random variable in the first place?
For the first part observe that for every $x\in\mathbb R$ we have:$$\left\{ S\leq x\right\} =\bigcup_{k=0}^{\infty}\left(\left\{ N=k\right\} \cap\left\{ \sum_{i=1}^{k}X_{i}\leq x\right\} \right)$$ showing that $\{S\leq x\}$ is a measurable set.
This is enough to conclude that $S$ is a random variable.
Second part can be solved by:$$P\left(S\leq x\right)=\sum_{k=0}^{\infty}P\left(N=k\right)P\left(S\leq x\mid N=k\right)=\sum_{k=0}^{\infty}P\left(N=k\right)P\left(\sum_{i=1}^{k}X_{i}\leq x\mid N=k\right)=$$$$\sum_{k=0}^{\infty}P\left(N=k\right)P\left(\sum_{i=1}^{k}X_{i}\leq x\right)$$ where the last equality is based on independence.