Analysis of multivariate symbol statistics in primitive rational models

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.

💡 Research Summary

The paper investigates the asymptotic behavior of multivariate symbol statistics generated by a rational stochastic model whose transition‑weight matrix is primitive. A finite alphabet Σ = {a₁,…,a_ℓ,b} is equipped with non‑negative matrices A₁,…,A_ℓ and B, together with positive vectors ξ and η, defining a linear representation (ξ, μ, η) of a rational formal series r. For each word w of length n, the probability of w is proportional to ξᵀ μ(w) η, which can be written as ξᵀ Mⁿ η with M = Σ A_i + B. The random vector Yₙ = (|w|{a₁},…,|w|{a_ℓ}) records the number of occurrences of the first ℓ symbols in a word drawn according to this distribution.

Assuming M is primitive, Perron–Frobenius theory yields a dominant eigenvalue λ>0 with strictly positive left and right eigenvectors vᵀ and u (normalized so that vᵀu = 1). Consequently, Mⁿ = λⁿ u vᵀ (1+O(εⁿ)) for some ε∈(0,1). The moment‑generating function of Yₙ is Ψₙ(t) = ξᵀ (A₁e^{t₁}+…+A_ℓe^{t_ℓ}+B)ⁿ η / ξᵀ Mⁿ η, and its logarithm Φₙ(t) = log Ψₙ(t) satisfies a quasi‑power property: (1/n) Φₙ(t) → Λ(t) = log y(t), where y(t) is the Perron–Frobenius eigenvalue of the perturbed matrix M(t)=A₁e^{t₁}+…+A_ℓe^{t_ℓ}+B. The function Λ is analytic on ℝ^ℓ, with gradient ∇Λ(0)=β and Hessian HΛ(0)=V. These give the first‑order approximations E