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Let $(m_i, i \in \mathbb{N})$ be positive weights with $\sum_{i \in \mathbb{N}} m_i^2 < 0.1$. Consider a subcritical branching process in discrete time and continuous space, started from some initial mass $x>0$, and with branching mechanism as follows: given mass $m$ in generation $n$, generation $n+1$ has total mass $$ \sum_{i \in \mathbb{N}} m_i N_i, $$ where the $N_i$ are independent and $N_i$ has distribution $\mathrm{Poisson}(m\cdot m_i)$. (One can think of a total number $N_i$ of copies of mass $m_i$ in generation $n+1$.)

Let $Z_n$ be the total mass at level $n$. Does $\sum_{n=0}^{\infty} Z_n^{1/2}$ then have exponential tails?

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I have now answered this question in the affirmative, essentially by looking at the sequence of random times $(N_k)_{k \geq 0}$, where $N_0$ is the last time $n$ that $Z_n$ has mass more than $Z_0$, and for $k \geq 0$, $N_{k+1}$ is the last time $n > N_k$ that $Z_n$ has mass more than $Z_0/2^k$. By considering such a dyadic expansion the problem reduces, with a little work, to that of finding exponential tails for the total mass $\sum_{n=0}^{\infty} Z_n$, which is not hard. I am not posting this as an answer because I am still hoping the result is a consequence of some more general fact. –  Louigi Addario-Berry Aug 26 '10 at 20:23

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