Left tail asymptotics for the density of supercritical branching processes in random environment

Let $(Z_n{)}_{n\ge 0}$ be a supercritical branching process in a random environment in which every individual produces at least one offspring, and let $W$ be the almost sure limit of the normalized $Z_n$. Write $p_1$ for the (random) probability that an individual has exactly one child and $m_0$ for the mean offspring number in a given environment. Assume that $p_1$ is positive almost surely and bounded away from $1$, that $m_0$ is bounded away from $1$ and from infinity together with a mild moment condition, and that $\log m_0$ is non-lattice. Let $a_0>0$ be the unique solution of the equation $\mathbb{E}[p_1{m}_0^{a_0}]=1$. Under these assumptions, we determine the precise left tail asymptotics of the annealed law of $W$: the distribution function $F(x)=\mathbb{P}(W\le x)$ satisfies $F(x)\sim D{x}^{a_0}$ as $x\downarrow 0$ with a constant $D\in (0,\mathrm{\infty })$ admitting an explicit representation, and the annealed density $w$ of $W$, which we show exists and can be chosen continuous on $(0,\mathrm{\infty })$, satisfies $w(x)\sim G{x}^{a_0-1}$ as $x\downarrow 0$ with $G=a_{0}D>0$. These results extend the classical power-law left tail behavior of the martingale limit of a Galton–Watson process in the literature to the random environment setting, where the exponent $a_0$ is determined by the averaged environmental law rather than by a fixed offspring distribution.