Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes

Abstract

We prove precise large deviations for the symmetric elephant random walk with random step sizes. When the distribution of $|Z_1-1|$ is subexponential, we first show that, for each fixed$p\in[0,1]$, there exists a positive number sequence $b_n\to\infty$ such that \[ \mathbb P(S_n>x)\sim \mathbb P(S_n<-x)\sim \frac n2\overline F(x) \] uniformly for $x\ge b_n$, where $\overline F(x)=\mathbb P(|Z_1-1|>x)$. This existence result requires no moment assumptions and does not specify the growth rate of $b_n$. For $p\in[0,3/4]$, we further give explicit sufficient conditions on $b_n$ under Weibull-type and lognormal-type tail conditions with a regularly varying factor $x^γL(x)$, where $γ\in\mathbf R$ and $L$ is a positive measurable slowly varying function. Analogous results hold with $\mathbb P(|Z_1|>x)$ and non-strict tails.