# Universal Persistence for Local Time of One-dimensional Random Walk

## TL;DR

This paper establishes a universal power law decay for the probability that a one-dimensional random walk's positive time fraction exceeds a threshold, revealing a deep connection with Lévy distributions and walk asymmetry.

## Contribution

It proves a universal power law decay in the local time of one-dimensional random walks, linking it to Lévy distributions and asymmetry parameters, extending prior results.

## Key findings

- Power law decay of probability with exponent depending on local time.
- Explicit formula for decay exponent involving Lévy distribution.
- Universal behavior independent of specific walk details.

## Abstract

We prove the power law decay $p(t,x) \sim t^{-\phi(x,b)/2}$ 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 $\phi(x,b)= \mathbb{P}(\text{L\'evy}(1/2,\kappa(x,b))<0)$ for $\kappa(x,b) = \frac{\sqrt{1-x} b - \sqrt{1+x}}{\sqrt{1-x} b + \sqrt{1+x}}$ and $b=b_S \geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments).