Consider the Galton-Watson tree with offspring distribution $X$ given by $P(X=k) = (1-p)^kp$. Let $D$ be the depth of the process. My question is how to calculate for which values of $p$ we have $\mathbb{E}D<\infty$.
Explanation of terms: a Galton-Watson tree consists of generations. For example the first generation consists of $2$ people (probability $p(1-p)^2$), the second one consists of the children of these two and their amount is also geometric($p$) distributed, and so on, so they continue multiplying in a geometric way.
The depth is defined by the largest generation which has $>0$ children. A result given in my lecture notes is that $P(D\leq t) = G\circ ...\circ G(s)(0)$ where there are $t$ times a $G$, and $G(s)$ is the pgf. So here it looks like a continued fraction.
The lecture's first question is to calculate $P(D=t)$, so I did $P(D\leq t) - P(D<t)$. The next question is for which values $\mathbb{E}D<\infty$. I know that for $p>1/2$ the extinction probability is $1$, but unfortunaltely I calculated that for $p=1/2$, $\mathbb{E}D=\infty$, otherwise it was done.
Is there anyone who can help? Thanks in advance.