I have been reading Grimmett and Stirzaker for independent learning and I do not understand why one of the conclusions they draw is important. For the martingale $W_n = Z_n / \mathbb{E}(Z_n)$, they show that there exists a random variable $W$ such that $W_n \to W$ in distribution. They then use the continuity theorem (theorem 5.9.5) to show that there exists a characteristic function $\phi_W$ such that $\phi_{W_n} \to \phi_{W}$. They conclude with the following equation:
$$ \phi_W(\mu t) = G(\phi_{W}(t)) $$
Where $G(\phi_W(t))$ is the generating function of $\phi_W(t)$. I understand how they arrive at this equation, but are there any proofs/concepts that make use of it? I can't find a place in the book where they reference it again.