Large deviations for the branching random walk with heavy-tailed associated random walk - a principle of one big jump
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 $Z_n = \sum_{|u|=n} e^{-V_u} δ_{V_u}$ 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 $Z_n((n\mathbb{E}[X] + t_n , \infty)) = W n\mathbb{P}(X > t_n )(1 + o(1))$ 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.