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

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 $$\lim_{x \to +\infty} \frac{e^{\lambda x^b}}{\ell(x) x^a } \, \mathbb{P}(M > x) = \frac{1 - p_0}{1 - m}$$ 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 $$\lim_{x \to +\infty} \frac{e^{\lambda x^b+\gamma x}}{\ell(x) x^a } \, \mathbb{P}(M > x) \quad \text{exists and belongs to } (0, \infty)$$ provided that $\sum_{k=1}^\infty k (\log k) p_k < \infty$ and for some $x_*>0$, $\mathbb{P}(X > x) = \int_x^\infty \ell(y) y^a e^{-\lambda y^b - \gamma y} \, \mathrm{d}y$ for all $x > x_*$. Here, $\ell$ is a slowly varying function, $m \mathbb{E}(e^{\gamma X}) < 1$, $b \in [0,1)$, and $a$ satisfies certain conditions.