Analysis of multivariate symbol statistics in primitive rational models
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912868828446720 |
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| author | Goldwurm, Massimiliano Macci, Claudio Vignati, Marco Villa, Elena |
| author_facet | Goldwurm, Massimiliano Macci, Claudio Vignati, Marco Villa, Elena |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_02391 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analysis of multivariate symbol statistics in primitive rational models Goldwurm, Massimiliano Macci, Claudio Vignati, Marco Villa, Elena Probability 60F10, 68Q45, 68Q87 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. |
| title | Analysis of multivariate symbol statistics in primitive rational models |
| topic | Probability 60F10, 68Q45, 68Q87 |
| url | https://arxiv.org/abs/2602.02391 |