I'm basically asking an extension of the first part from this post a while back. To parse, it asked:
Given a branching process $Y = \{Y_{n} : n \geq 0\}$ and the corresponding family of random variables for the process $\{X_{k, n} : n, k \in \mathbb{N}\}$, suppose $Y_{0} = 0$ and for $n \geq 0$,
$$Y_{n+1} = X_{1, n+1} + ... + X_{Y_{n}, n+1}.$$
If $\mu = \mathbb{E}(X_{n, k})$ for any $n, k$, and $0 < \sigma^{2} = Var(X) < \infty$, then
$$E(Y_{n+1}^{2} | \mathcal{F}_{n}) = \mu^{2} Y_{n}^{2} + \sigma^{2} Y_{n}.$$ Show that this implies $G = \{G_{n} = Y_{n}/\mu^{n}\}$ is bounded in $\mathcal{L}^{2}$. I know how to show that $G_{n} = Y_{n}/\mu^{n}$ is a martingale wrt $\mathcal{\{F_{n}\}_{n}}$, but I can't figure this simple extension out. Thanks in advance!