Exact solution for a sample space reducing stochastic process

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 $\{x(t)\}$, $0\le t \le τ$, with boundary conditions $x(0) = N$ and $x(τ)$ = 1. This model is shown to be exactly solvable: $\mathcal{P}_N(τ)$, the probability that the process survives for time $τ$ 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: $\langle τ\rangle \sim \ln N$ and $σ_τ^{2} \sim \ln N$. Correspondence can be made between survival time statistics in the SSR process and record statistics of i.i.d. random variables.