Some Limit Theorems Of Random Sequences In Metric Spaces

The first part of this paper concerns some general laws of the iterated logarithm of independent orφ~* mixing random variables,geometrically weighted series and U-statistics in separable Banach spaces.The law of the iterated logarithm is well-known and important in probability and statistics so that it has been explored in many classical textbooks.It is well known that the classic laws of the iterated logarithm always require the finiteness of the second moments,and researchers always like to present the results under minimal conditions.Based on recent literatures,we prove some laws of the iterated logarithm in the case that the second,noments may be infinite.Some previous results are extcudcd.The second part of this paper concerns strong laws of large numbers for random compact sets and fuzzy random sets in separable Banach spaces.There are many things referring to random sets or fuzziness in many economic problems and our life.Hence,the asymptotic properties of random sets and fuzzy random sets are important and interesting to be revealed.However,researchers always pay their attention to an independent random sequence.Here we consider the array of independent as well as dependent random compact sets and fuzzy random sets,and also take into account of the sufficient(necessary) condition for the strong limit theorems of slowly varying weights random sets and fuzzy random sets.Strong approximation plays an important role in probability limit theorems, especially frequently used in statistical inference.The third part of this paper reduces to the Euclidean spaces-R~p.In this part,we obtain strong approximation theorems for trimmed sums of independent random variables and partial sums ofφ-mixing random vectors when their second moments may be infinite, respectively.At last,we established a general strong approximation theorem for L-statistics.The last part of this paper concerns functional data taking values in some metric space(e.g.,R~p space,Banach space,Hilbert space,C[0,1]space and so on).We propose the functional conditional U-statistics and the functional conditional hazard rate estimator,and then study their asymptotic properties, respectively.