Some Limit Theorems Of Linear Processes And Large Deviations For Empirical Measure Sequences

Theory of Probability is a science of quantitatively studying regularity of random,phenomena, which is extensively applied in natural science, technological science, and managerial science etc. Hence, it has been developing rapidly since 1930' s and many new branches have emerged from time to time. Limit Theory is one of the important branches and also an essential theoritical basis of science of Probability and Statistics. As stated in the classical book "Limit Distributions for Sums of Independent Random Variables" (1954) by Gendenko and Kolmogrov. " The epistemological value of the theory of probability is revealed only by limit theorems. Without limit theorems it is impossible to understand the real contentof the primary concept of all our sciences-the concept of probability." The classical limit theorems of probability theory for independent random, variables had been developed successfully in 1930 s and 1940's, and they are the significant, achievements in the progress of Probability. The basic results were summed up in Gendenko and Kolmogrov s monograph " Limit Distributions for Sums of IndependentRandom Variables" (1954) and Petrov s monograph "Sums of Independent Random Variables" (1975).The linear processes are one of the most representative models in time series.Studying various limiting properties of linear processes is one of orientationsof the current study of Limit Theory. The linear processes are of special importantance in time series analysis and they arise in a wide variety of contexts.Applications to economics, engineering and physical, sciences are extremely broad and a vast amount of literature is devoted to the study of the limiting theorems for the linear processes under various conditions on errors.For example,under the martingle difference assumption on error.under the strong mixing condition on error and under LPQD condition on error, the central limit theorem(CLT)and the functional central limit theorem, (FCLT)of the linear processes are proved. Lu(2003) gave the invariance principle for linear process generated by a negatively associated sequence. Under some suitable conditions, other limitingresults have been obtained for the linear processes. For example, Philipps and Solo(1992) proved the strong law of large numbers and the law of the iteratedlogarithm. Wang et al(2003) proved strong approximation for long memory processes, Lu(2004) gave strong approximation for linear process by Wiener processes.Burton and Dehling(1990) obtained a large deviation principle for the linear processes. Yang (1996) established CLT and the law of the iterated logarithm,Li et al. (1992a). Zhang (1996). Yu and Wang(2002) obtained the results on the complete convergence etc.Large deviations for empirical measure sequences and some significant results of the linear processes about weak limit properties and strong limit properties have been reached through deep research in this dissertation. This paper has five-parts, in the first chapter, we discuss large deviations principle(LDP) of empirical measure. By the relation between the open set under the weak convergence topology and the open sphere under the metric 0. we prove the following the large deviation principle of empirical measure of stationary NA sequences under the condition (?)₁>0.Theorem 1.1.1 Suppose {X_i,i∈N} is a stationary NA random variablessequence such that (?)₁ > 0. Define the empirical measureδ_n =1/n sum from i=1 to nδ_Xin>>>>> 1, and then {P{δ_n,∈·},n→∞} satisfies the large deviation principle on (M₁(R),(?)), i.e. there is a convex lower semicontinuous function I : M₁(R)→[0.∞] with compact level sets I^-1[0.a].a≥0. and such that for each closed set F (?) M₁{R), we havefor each open set G (?) M₁(R). we haveMoreover, for (?)α∈M₁(R)where...