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 $m:={\sum }_{k=1}^{\mathrm{\infty }}k{p}_k\in (0,1)$. 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 $$\underset{x\to +\mathrm{\infty }}{\lim }\frac{e^{\lambda x^b}}{\mathrm{\ell }(x)x^a}\mathbb{P}(M>x)=\frac{1-p_0}{1-m}$$ whenever $\mathbb{P}(X>x)=\mathrm{\ell }(x)x^a{e}^{-\lambda x^b}$ for some slowly varying function $\mathrm{\ell }$, $b\in [0,1)$, and under further assumptions on $a$. Next, we prove that $$\underset{x\to +\mathrm{\infty }}{\lim }\frac{e^{\lambda x^b+\gamma x}}{\mathrm{\ell }(x)x^a}\mathbb{P}(M>x)\text{exists and belongs to }(0,\mathrm{\infty })$$ provided that ${\sum }_{k=1}^{\mathrm{\infty }}k(\log k)p_k<\mathrm{\infty }$ and for some $x_{\ast }>0$, $\mathbb{P}(X>x)={\int }_x^{\mathrm{\infty }}\mathrm{\ell }(y)y^a{e}^{-\lambda y^b-\gamma y}\mathrm{d}y$ for all $x>x_{\ast }$. Here, $\mathrm{\ell }$ is a slowly varying function, $m\mathbb{E}(e^{\gamma X})<1$, $b\in [0,1)$, and $a$ satisfies certain conditions.