Exact solution for a sample space reducing stochastic process

TL;DR

This paper presents an exact analytical solution for a sample-space reducing stochastic process, revealing Gaussian behavior in survival times with mean and variance scaling logarithmically with system size, and links to record statistics.

Contribution

It provides the first exact solution for the survival time distribution in a sample-space reducing process, connecting it to record statistics and scale-invariant features.

Findings

Survival time distribution is exactly solvable and asymptotically Gaussian.

Mean and variance of survival time scale logarithmically with system size.

Connection established between SSR process survival times and record statistics.

Abstract

Stochastic processes wherein the size of the state space is changing as a function of time offer models for the emergence of scale-invariant features observed in complex systems. I consider such a sample-space reducing (SSR) stochastic process that results in a random sequence of strictly decreasing integers $\{x(t)\}$, $0\le t \le \tau$, with boundary conditions $x(0) = N$ and $x(\tau)$ = 1. This model is shown to be exactly solvable: $\mathcal{P}_N(\tau)$, the probability that the process survives for time $\tau$ is analytically evaluated. In the limit of large $N$, the asymptotic form of this probability distribution is Gaussian, with mean and variance both varying logarithmically with system size: $\langle \tau \rangle \sim \ln N$ and $\sigma_{\tau}^{2} \sim \ln N$. Correspondence can be made between survival time statistics in the SSR process and record statistics of i.i.d. random variables.