I have trouble understanding the following identity: $\mathbb{E}[e^{i\theta\cdot T_t}]=\sum_{j\geq 0}{e^{-t}\frac{t^j}{j!}\mathbb{E}[e^{i\theta\cdot S_j}]}$. I see that somehow it uses the following lemma, which I tried to prove here.
Let $S_n$ be a discrete-time random walk with increment distribution p and $N_t$ be a Poisson process with parameter $1$.Then the process $T_t:=S_{N_t}$ has the distribution of a continuous-time random walk with increment distribution p.
Can someone give me more details between this step? I see it as a strange composition of random variables but I can't clarify this. Thanks