【作者】 周红；

【导师】 石志岩；

【作者基本信息】 江苏大学 ， 统计学， 2021， 硕士

【摘要】 强极限定理作为概率论研究的中心问题之一,至今研究热度不减。强偏差定理是强极限定理的推广,是由不等式表示的一类强极限定理。随机环境中马氏链是在随机环境因素影响下的马氏链,弥补了经典马氏链的局限性。近年来国内外学者在经典马氏模型方面的研究成果颇丰,内容主要涉及中心极限定理、强大数定律、渐近等分性等极限理论。与经典马氏模型方面的研究成果相比,对于随机环境下马氏模型的研究仍有明显的欠缺。本文致力于将离散状态下马氏链的若干结果推广到随机环境下,主要研究了随机环境下树指标马氏模型和单无限马氏环境下的马氏模型。本文使用渐近对数似然比作为一种度量,通过构造非负鞅的方法,建立可列状态齐次马氏链的强偏差定理,最后得到了单无限马氏环境下可列齐次马氏链的强大数定律和渐近等分性;同时,建立了随机环境下双Cayley树上随机场泛函的一类强偏差定理,最后证明了有限状态空间里马氏环境下树指标马氏链的渐近等分性。更多还原

【Abstract】 As one of the central problems in probability theory,strong limit theorem is still attractive to many scholars.Strong deviation theorem is the extension of strong limit theorem,which is a kind of strong limit theorem expressed by inequality.The Markov chains in random environment are the Markov chains under the influence of random environment,which make up the limitation of classical Markov chains.In recent years,scholars at home and abroad have made a lot of research achievements in classical Markov model,mainly involving central limit theorem,the strong law of large numbers,Shannon-Mc Millan theorem and other limit theories.Compared with the research results of classical Markov model,the research of Markov model in random environment still has obvious deficiency.In this paper,we generalize some results of Markov chains in discrete state to random environment.We mainly study the tree-indexed Markov model in random environment and the Markov model in single-infinite in random environment.In this paper,the asymptotic likelihood ratio is used as a measure,and the strong deviation theorem of the listed homogeneous Markov chains is established by constructing the method of non-negative martingale.Finally,the strong law of large numbers and the asymptotic equipartition of the listed homogeneous Markov chains are obtained.At the same time,a class of strong deviation theorems for random field functional of double Cayley tree in random environment is established.Finally,the asymptotic equipartition property(AEP)of tree-indexed Markov chains in Markov environment in finite state space is proved.更多还原