On Some Strong Deviations Of Dependent Random Arrays

"Source" refers to the source of a message,which can be divided into discrete and continuous sources according to the distribution of the time and magnitude of the messages sent by the source.The message output from the source is in the form of symbols with discrete values,which can be finite.For example,the generalized source(random array)is an infinite sequence of n-dimensional random variables,in which each component is a random variable valued in the source table.Generalized information source can be understood as the divergence and propagation of signals,with more variables in the future.It is a generalization of random process.The purpose of this paper is to improve the existing strong limit theory proof method and apply it to the limit form of generalized information sources.For the complex information of human society or nature,it is of great significance and value to study the generalized information source.In this paper,we introduce the concept of asymptotic log-likelihood ratio(general information source)of samples of random array relative to rowwise-independent random array as a random measure to describe the dependence.By constructing random variables with one parameter which its expectation are bounded and using the Borel-Cantelli lemma,we obtain a class of strong limit theorems expressed by inequalities about the expectation of dependent random array relative to reference measure,namely,strong deviation theorems.In this paper,we prove the strong limit theorem of the geometric mean and the harmonic mean of the transition probability of the generalized information source sequence X={Xn}n?n.As an application we obtain the asymptotic equipartition theorem of the generalized information source.Finally,we discuss the basic properties of the upper and lower limit of the probability of random sequences.