arXiv subjects

Sung Won Ahn

Publications and source records attributed to Sung Won Ahn.

3 recordsLinked to original sources

arXiv2021

Optimal rates of convergence for quenched central limit theorem rates for hitting times of one-dimensional random walks in random environments

We consider the rates of convergence of the quenched central limit theorem for hitting times of one-dimensional random walks in a random environment. Previous results had identified polynomial upper bounds for the rates of decay which are sometimes slower than $n^{-1/2}$ (the optimal rate in the classical Berry-Esseen estimates). Here we prove that the previous upper bounds are in fact the best possible polynomial rates for the quenched CLT.

math.PR↗

arXiv2017

Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments

Unlike classical simple random walks, one-dimensional random walks in random environments (RWRE) are known to have a wide array of potential limiting distributions. Under certain assumptions, however, it is known that CLT-like limiting distributions hold for the walk under both the quenched and averaged measures. We give upper bounds on the rates of convergence for the quenched central limit theorems for both the hitting time and position of the RWRE with polynomial rates of convergence that depend on the distribution on environments.

math.PR↗

arXiv2015

Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim_{n\to\infty}\frac{X_n}{n}=v_α>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_ω(X_n < xn)$ with $x \in (0,v_α)$ decay approximately like $\exp\{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-γ} \log P_ω( X_n < x n)$ converges to $0$ or $-\infty$ depending on whether $γ> 1-1/s$ or $γ< 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_ω( X_n < x n)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.

math.PR↗