Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains
Abstract
In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.
Related
- Asymptotic Theory for Restricted Maximum Likelihood Estimators in Generalized Linear Mixed Models
- Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in r-Largest Generalized Extreme-Value Models
- Marginal maximum likelihood estimation and asymptotic theory for latent variable models in high dimensions
- Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
- Identification problem and quasi-maximum likelihood estimation for matrix-variate CP-factor models
BibTeX
@misc{gavrilopoulos2026matrix,
title = {{Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains}},
author = {Georgios Gavrilopoulos and Samis Trevezas and Irene Votsi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.17161},
eprint = {2607.17161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17161},
}
RIS
TY - UNPB
TI - Matrix asymptotic calculus for plug-in maximum likelihood estimators in finite Markov chains
AU - Georgios Gavrilopoulos
AU - Samis Trevezas
AU - Irene Votsi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2607.17161
UR - https://arxiv.org/abs/2607.17161
AB - In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.
ER -