Continuous sample space reducing stochastic process

TL;DR

This paper introduces a continuous sample space reducing stochastic process model, analyzing its survival time and size distribution, and generalizes it to noisy processes with tunable parameters, revealing power-law behaviors.

Contribution

The paper presents a novel continuous version of the SSR process and extends it to noisy cases with tunable parameters, providing new insights into its statistical properties.

Findings

Survival time statistics differ subtly from the discrete case.

The model explains noisy SSR processes with a tunable parameter.

The size distribution follows a power-law with a nontrivial exponent.

Abstract

We propose a simple model for sample space reducing (SSR) stochastic process, where the dynamical variable denoting the size of the state space is continuous. In general, one can view the model as a multiplicative stochastic process, with a constraint that the size of the state space cannot be smaller than a visibility parameter $\epsilon$. We study the survival time statistics that reveal a subtle difference from the discrete version of the process. A straightforward generalization can explain the noisy SSR process, characterized by a tunable parameter $\lambda \in [0, 1]$. We also examine the statistics of the size of the state space that follows a power-law distributed probability $\mathbb{P}_{\epsilon}(z\le \epsilon) \sim z^{-\alpha}$, with a nontrivial value of the exponent as a function of the tunable parameter $\alpha = 1+\lambda$.