Limit theorems for iterated random topical operators

TL;DR

This paper establishes limit theorems such as the CLT, local limit, renewal, and large deviations for iterated random topical operators, modeling complex systems like task graphs and digital circuits.

Contribution

It introduces spectral gap methods to prove these theorems for systems with memory loss, providing algebraic conditions for variance positivity and generic results for matrix-defined operators.

Findings

Proves CLT, local limit, renewal, and large deviations for iterated topical operators.

Provides algebraic conditions for variance positivity in the CLT.

Shows that conditions and variance positivity are generic for matrix-based operators.

Abstract

Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n,x_0)=A(n)x(n-1,x_0)$. This can modelize a wide range of systems including, task graphs, train networks, Job-Shop, timed digital circuits or parallel processing systems. When A(n) has the memory loss property, we use the spectral gap method to prove limit theorems for $x(n,x_0)$. Roughly speaking, we show that $x(n,x_0)$ behaves like a sum of i.i.d. real variables. Precisely, we show that with suitable additional conditions, it satisfies a central limit theorem with rate, a local limit theorem, a renewal theorem and a large deviations principle, and we give an algebraic condition to ensure the positivity of the variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we give more effective statements and show that the additional conditions and the positivity of the variance in the CLT are generic.