Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump

Abstract

We prove a version of Nagaev's theorem for the branching random walk with heavy-tailed associated random walk. For a branching random walk on $\mathbb{R}$ we consider the random measure $Z_n={\sum }_{|u|=n}{e}^{-V_u}{\delta }_{V_u}$ where $V_u$, $|u|=n$ denote the positions of the particles in the $n$-th generation. Under the assumption that $\mathbb{E}[Z_1(\cdot )]$ is a probability distribution with regularly varying tail, we prove that $Z_n((n\mathbb{E}[X]+t_n,\infty ))=Wn\mathbb{P}(X>t_n)(1+o(1))$ in $L^1$ as $n\to \infty$ where $W$ is a non-zero random variable, $t_n\uparrow \infty$ grows suitably fast, and $X$ has law $\mathbb{E}[Z_1(\cdot )]$. The result is explained probabilistically by a principle of one big jump for the branching random walk.