Some Convergence Properties About φ-mixing Sequence And Its’ Precise Asymptotics,Strong Consistency

The concept of (φ-Mixing Sequences was introduced by Dobrushin when he was researching the process of Markov.The exactly definition of φ-Mixing Sequences is that:Let {xn,n≥1}is an random variable sequences. MarkFnm=σ(x1,n≤i≤m), define the mixing coefficient φ(n)={sup sup (|P(B|A)-P(B)|);k≥1,A∈F1k,B∈Fk+n∞,P(A)≠0}. When n�'∞, if φ(n)�'0,we said that{xn,n≥1} is φ-Mixing Sequences.Because φ-Mixing Sequences is taking an much important part in many branch subjects(For example in Markov-Process, Random Field and Time Series Analysis and so on). Rencently, the development of φ-Mixing Sequences in limit theory is becoming more and more fast, and that had got many results which were similar to Independent Sequences. Howere, the Independent Sequences are inconsisdent with many phenomenones in our life, and the φ-Mixing Random Sequences are the Mixing Sequences whose sample datas are fit for a relationship, which in comparison with Independent Sequences are more coincide to actual situation, and that have supplied an wonderful stage for academician to researching actual situations in mathmatical way.This paper are divided into six chapters and the last four is the chief section.The third chapter had discussed the covergence of almost everywhere aboutt φ-Mixing Sequences, and had got the Three Series Theorem which is similar to the situation of Independent Sequences, the result about covergence of almost everywhere had extended the corresponding literatures to an weak moment condition.The fourth one was in the light of the most general weighted sums convergence and had discussed the complete convergence of weighted product sums for φ-Mixing random sequences. Extended the corresponding literatures to an more general condition.The fifth chapter had discussed the strong stationary φ-Mixing Sequences’precise asymptotics.The sixth had discussed φ sample regression functions in strong consistency.