AI 中文总结
该研究为有限马尔可夫模型的插件非参数最大似然估计器开发矩阵渐近微积分。从转移矩阵渐近分布出发，通过特定方式计算相关量，给出多种渐近公式，带来简化与计算优势，二阶项还能提供精细近似。
AI 中文摘要
在这项工作中，我们为有限马尔可夫模型中的插件非参数最大似然估计器开发了一种统一的矩阵级渐近微积分。从估计的转移矩阵的渐近分布开始，极限对象保持其作为高斯随机矩阵的自然矩阵形式，同时相应的行向量表示也可立即得到。要点在于转移矩阵的随机约束无需通过最小参数化去除：它们由切线方向和极限高斯矩阵的协方差结构承载，而相关微分直接在矩阵空间中计算。一个单一的随机微积分定理给出一阶极限分布、充分可微泛函的有限阶展开以及泛函为解析时的解析展开。这为矩阵幂、平稳特征、马尔可夫特征的有限维曲线、加性泛函方差、熵型量和可靠性指标的渐近公式提供了共同来源。所得协方差算子直接导致置信区间、置信区域、同时有限维带和 Wald 型检验。由于推导通过矩阵乘积和克罗内克表示而非坐标计算来表达，该方法还带来了显著简化，并且在许多情况下有计算优势。二阶项识别光滑泛函的曲率修正，并在高阶信息有用时提供精细近似。
英文摘要
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.
Comments30 pages, 5 figures
