Some Strong Limit Theorems Of Moving Average Of Weighted Sums For Random Sequence

Limit theorem is one of the most important research contents of the modern proba-bility theory, which also plays an important role in applications. There are large amountof literatures about strong limit theorem in the world. Enlightened by the ideas in [8] and[17], we introduce the notion of random sequence moving likelihood ratio and movingrelative entropy concept as the random measure of random sequence joint distributionand the deviation of the reference product. A subset in the sample space is also givenby means of moving relative entropy, in which we get a kind of strong limit theoremsfor discrete random sequence weighted sums expressed by inequality§namely small de-viation theorems. Moving likelihood ratio with the parameter is constructed and almosteverywhere convergence of the random sequence is proved by the classical Borel-Cantellilemma. Diferent from the tedious segmentation interval method in [17], the proof withthis method is more concise and the application is wider. As its application, the rela-tionship between moving relative entropy of discrete information source and strong lawof large numbers are discussed.In Chapter2, We use Borel-Cantelli lemma to give a short proof of the strong law oflarge number for the moving average Bernoulli random sequence with analysis method.In the third chapter, the notions of moving likelihood ratio and moving relativeentropy of arbitrary binary random sequence (about the measure μ) relative to anotherprobability measure μ in space (, F) are introduced. And the moving relative entropyis used as a random measure of the deviation of arbitrary binary random sequence aboutmeasure μ and μ (independent case). Thus, the weighted sums for the sequences ofbinary random variable and moving average small deviation theorem are put forward onthe set H(c). In the fourth chapter, we use the results obtained in Chapter3to investigate thelimit property of moving relative entropy of arbitrary binary information source. A seriesof meaningful results are obtained which generalize some relevant conclusions in [17].In the last chapter, a small deviation theorem for the moving relative entropy ofarbitrary dependent discrete random variable sequence is established by using the relativeentropy of arbitrary discrete random sequence’s joint distribution respect to referenceproduct distributionan+nqk(xk).