pr.probability - Exponential tails for a functional of a subcritical branching process.

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 $mathrm{Poisson}(mcdot m_i)$. (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?