= Exponential total progeny tail for bounded subcritical offspring
If a <Galton-Watson process> starts from one ancestor and has <bounded> nonnegative <integer> offspring $Z$ with $\mathbb E Z<1$, its <total progeny> $T$ obeys $\mathbb P(T>n)\leq e^{-cn}$ for some $c>0$ and all <integers> $n\geq1$. Explore one individual at a time. The unexplored <queue> has size $1+\sum_{j=1}^t(Z_j-1)$ until its first hit of zero, whose <hitting time> is $T$. Hence $T>n$ implies $\sum_{j=1}^n(Z_j-1)\geq0$. <Boundedness> makes $M(\theta)=\mathbb E e^{\theta(Z-1)}$ finite near zero, with $M(0)=1$ and $M'(0)=\mathbb E Z-1<0$. Choose $\theta>0$ with $M(\theta)<1$. The exponential <Markov inequality> gives the desired bound with $c=-\log M(\theta)$. The same proof works whenever the offspring law has an <exponential moment> near zero, even without <boundedness>.
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