Some Small Deviation Theorems Of Delayed Average Of Dependent Random Sequence

In this paper, we study the small deviation theorems （namely, the strong limittheorem expressed by inequalities）for delayed average of arbitrarily dependent discreterandom variables. The paper consists of four parts.In chapter1, we present the background and the recent development of limit theoryof probability theory. Secondly, we briefy introduce the basic ideas and methods of smalldeviation theorems of random variable proposed by Liuwen. Finally, we introduce somenew concepts of delayed average, the delayed likelihood ratio and the delayed relativeentropy of random variables.In chapter2, we use the delayed likelihood ratio and the delayed relative entropy asthe random measurement of deviation of the joint distribution of arbitrarily dependentrandom variables and the reference of independent geometric distributions. By con-structing, delayed likelihood ratio with parameter, and together with the Borel-Cantellilemma,and the pure analytical methods, we get a kind of small deviation theorems,which include many conclusions related to the limit properties such as the integer val-ued random variables, the delayed relative entropy density and the entropy functions ofgeometric distribution. Finally, we fnd that the delayed relative entropy density has thesimilar properties as the relative entropy density of information.In chapter3,let {ξn, n≥1} be a sequence of arbitrarily dependent random vari-ables under the measure P, and be i.i.d. under the reference measure Q.For eachk∈X, denoted by Sn,f（n）（k, ω） the frequencies of k in the sequence of ξ_n+1,···, ξ_n+f（n）,φ_n（k, ω）=S_n,f（n）（k, ω）-∑_i=n+1^n+f（n）qi（k）. We investigate some limit relationship betweenD（P∥Q） and φ_n（k, ω）.in some sense, where can be viewed as the inverse problem of thelaw of large numbers. Chapter4continues the research of the delayed likelihood ratio and the delayedrelative entropy. First, we use the truncation method to study the delayed average limitproperties of the dependent discrete random variables {ξ_n, n≥1}. Then, we give theChung-Teicher conditions of the small deviation theorems for delayed averages of randomsequence.