On the maximal displacement of subcritical branching random walks with stretched exponential tail

Abstract

We study the maximal displacement of a one-dimensional subcritical branching random walk with offspring distribution $\{p_k\}$ and step size $X$ such that $m:={\sum }_{k=1}^{\infty }k{p}_k\in (0,1)$. Let $M_n$ denote the maximal position of all particles alive at time $n$ and let $M:={\sup }_{n\in \mathbb{N}}{M}_n$. First, we show that \[ \lim_{x \to +\infty} \frac{e^{λx^b}}{\ell(x) x^a } \, \mathbb{P}(M > x) = \frac{1 - p_0}{1 - m} \] whenever $\mathbb{P}(X>x)=\ell (x)x^a{e}^{-\lambda x^b}$ for some slowly varying function $\ell$, $b\in [0,1)$, and under further assumptions on $a$. Next, we prove that \[ \lim_{x \to +\infty} \frac{e^{λx^b+γx}}{\ell(x) x^a } \, \mathbb{P}(M > x) \quad \text{exists and belongs to } (0, \infty) \] provided that ${\sum }_{k=1}^{\infty }k(\log k)p_k<\infty$ and for some $x_{\ast }>0$, $\mathbb{P}(X>x)={\int }_x^{\infty }\ell (y)y^a{e}^{-\lambda y^b-\gamma y}\mathrm{d}y$ for all $x>x_{\ast }$. Here, $\ell$ is a slowly varying function, $m\mathbb{E}(e^{\gamma X})<1$, $b\in [0,1)$, and $a$ satisfies certain conditions.

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