Analysis of multivariate symbol statistics in primitive rational models

TL;DR

This paper investigates the long-term behavior of multivariate symbol count statistics in primitive rational models, deriving asymptotic means, covariances, large deviations, Gaussian limits, and a new moderate deviation result.

Contribution

It introduces a comprehensive asymptotic analysis framework for multivariate symbol statistics in rational models, including novel moderate deviation results.

Findings

Asymptotic expressions for means and covariances

Large deviation principle with speed n

Multivariate Gaussian limit and moderate deviation result

Abstract

We study the asymptotic behaviour of sequences of multivariate random variables representing the number of occurrences of a given set of symbols in a word of length $n$ generated at random according to a rational stochastic model. Assuming primitive the matrix of the total weights of transitions of the model, we first determine asymptotic expressions for the mean values and the covariances of such statistics. Then we establish two asymptotic results that generalize known univariate cases to different regimes: a large deviation principle with speed $n$, implying almost sure convergence, and a multivariate Gaussian limit. Additionally, we introduce a novel moderate deviation result as a bridge between these regimes. Central to our proofs is a quasi-power property for the moment generating function of the statistics, allowing us to employ the G\"artner-Ellis Theorem for both large and moderate deviations.