Investigation Of Some Properties Of The L(?)distribution Class

In the study of probability theory,the distribution function is the most basic research object,and the tail of the distribution function is also very important.In 1964,Chistyakov first proposed the concepts of heavy-tailed distribution cluster,long-tailed distribution cluster L and subexponential distribution cluster and studied their properties.The L(?)distribution cluster is an extension of long-tailed distribution cluster L,which was proposed by Chover when he studied the branching process in 1973.In 1986,Cline provides an close relationship between the L(?)distribution clusters and Regular variation functions.In 2011,According to Foss,slow-varying functions are closely related to long-tail distribution clusters.Later,many scholars further studied the properties of the L(?)distributed clusters and gave many applications.In this paper,we will study new proofs or new properties of uniform convergence,equivalence and stationarity of sequences of the L(?)distribution clusters on the positive real axis.The first chapter of this paper is the introduction,which introduces the historical background of the L(?)distribution clusters and some existing achievements,as well as the main research contents and innovation points of this paper.The second chapter introduces some related distribution families,slow variable functions,regular functions,and the theorems and lemmas used in the subsequent proofs.In the third chapter,we perfect the theorem of uniform convergence of the L(?)distribution clusters and give several proofs of uniform convergence of the L(?)distribution clusters on positive real axis by the relation between slow variable function and long-tail distribution.In the fourth chapter,a new equivalent theorem of long-tail distribution clustering and a new equivalent theorem of the L(?)cluster distribution are obtained by means of the related theorems of slow variable functions.On the basis of some scholars' research,a new equivalent theorem of fifth cluster distribution and its proof are obtained by weakening some conditions.The fifth chapter the canonical transformation of random variables sequences and a column with heavy-tailed random sequence inspired the research results,the smoothness of the stationarity of the L(?)cluster distribution sequence and their proofs are given.