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 $\{x(t)\}$, $0\le t\le \tau$, 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: $⟨\tau ⟩\sim \ln N$ 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.

arXiv comment6 pages, 6 figures

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• Correspondence between noisy sample space reducing process and records in correlated random events
• On multidimensional record patterns
• Scaling in simple continued fraction