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Analysis of multivariate symbol statistics in primitive rational models

Massimiliano Goldwurm, Claudio Macci, Marco Vignati, Elena Villa

TL;DR

This work analyzes multivariate symbol-count statistics ${\bf Y}_n$ for words of length $n$ generated by a rational stochastic model with a primitive transition-weights matrix. It develops a quasi-power framework for the moment generating function, yielding precise asymptotics for means and covariances, a multivariate Gaussian limit with covariance $\Gamma$, a large deviation principle with speed $n$, and a novel moderate-deviation principle bridging the two regimes via the Gärtner–Ellis theorem. The results extend known univariate findings to the multivariate setting and provide a rigorous description of the fluctuation and tail behavior of symbol-count vectors in regular languages. The methods rely on Perron–Frobenius theory, analytic perturbations of $M({\bf t})$, and convex-analytic large deviation techniques with explicit rate functions.

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 Gärtner-Ellis Theorem for both large and moderate deviations.

Analysis of multivariate symbol statistics in primitive rational models

TL;DR

This work analyzes multivariate symbol-count statistics for words of length generated by a rational stochastic model with a primitive transition-weights matrix. It develops a quasi-power framework for the moment generating function, yielding precise asymptotics for means and covariances, a multivariate Gaussian limit with covariance , a large deviation principle with speed , and a novel moderate-deviation principle bridging the two regimes via the Gärtner–Ellis theorem. The results extend known univariate findings to the multivariate setting and provide a rigorous description of the fluctuation and tail behavior of symbol-count vectors in regular languages. The methods rely on Perron–Frobenius theory, analytic perturbations of , and convex-analytic large deviation techniques with explicit rate functions.

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 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 , 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 Gärtner-Ellis Theorem for both large and moderate deviations.
Paper Structure (13 sections, 10 theorems, 65 equations)

This paper contains 13 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on large (and moderate) deviations
  3. Multivariate symbol statistics in rational models
  4. Analysis of the primitive case
  5. Mean values
  6. Covariances
  7. Multivariate Gaussian limit
  8. Large deviation results
  9. Moderate deviation results
  10. Conclusions
  11. Appendix: some known results
  12. Unique zero of a rate function
  13. Convergence to the unique zero of a rate function

Key Result

Theorem 1

Let $\{{\bf Z}_n\}_{n\geq 1}$ be a sequence of $\mathbb{R}^m$-valued random variables such that there exists the function $\Lambda:\mathbb{R}^m\to(-\infty,\infty]$ defined by Assume that the origin ${\bf 0}=(0,\ldots,0)\in\mathbb{R}^m$ belongs to the interior $\mathcal{D}_\Lambda^\circ$ and also that $\Lambda$ is essentially smooth and lower semi-continuous. Then $\{{\bf Z}_n\}_{n\geq 1}$ satisfi

Theorems & Definitions (14)

  • Theorem 1: Gärtner-Ellis Theorem
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 4
  • Proposition 1
  • ...and 4 more