The Limit Theorems For Increments Of A Brownian Motion And Brownian Sheet

There are many profound results about the limit properties of Brownian motion and Brownian sheet.In this dissertation,we continue to discuss the limit properties of sample paths related to Brownian motion and Brownian sheet.This dissertation is based on previous studies,using the large deviation of Brownian motion as a tool,the local functional limit theorems for increments of Brownian motion and Brownian sheet are studied.The main contents of this dissertation are as follows:Firstly,the local functional law of the iterated logarithm for increments of a Brownian sheet is studied.Based on the results of the local functional limit theorems of increments for Brownian motion,the relevant results are extended to the local case,and the local functional law of the iterated logarithm for increments of a Brownian sheet is obtained.First,with the help of large deviation of the Brownian sheet,the local limit theorem for Brownian sheet is obtained.Then,using the large deviation for increment of the Brownian sheet as a tool,it is proved that the local limit theorem for increment of the Brownian sheet is obtained.In the process of proof,we have made an essential improvement on the previous method.Secondly,the local functional law of the triple logarithm law for increment of a Brownian motion is studied.On the basis of the results of the local iterated logarithm law of Brownian motion,the large deviation is used as a tool,and the condition is strengthened to prove that the local law of the triple logarithm is obtained,the result is different from the case of the local law of the iterated logarithm.Finally,a generalized local law of iterated logarithm for increment of Brownian sheet is studied.The condition of local law of iterated logarithm for increment of Brownian sheet is improved,and the theoretical result proves the local law of iterated logarithm for another case.This result is also correct generalization of the local limit theorem for increment of the Brownian sheet.