Universal Persistence for Local Time of One-dimensional Random Walk

Abstract

We prove the power law decay $p(t,x)\sim t^{-\varphi (x,b)/2}$ in which $p(t,x)$ is the probability that the fraction of time up to $t$ in which a random walk $S$ of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds $x$ throughout $s\in [1,t]$. Here $\varphi (x,b)=\mathbb{P}(\text{Leˊvy}(1/2,\kappa (x,b))<0)$ for $\kappa (x,b)=\frac{\sqrt{1-x}b-\sqrt{1+x}}{\sqrt{1-x}b+\sqrt{1+x}}$ and $b=b_S\ge 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments).

11 pages

References in corpus (1)

• Persistence probabilities \& exponents