Weak Limit Theorems For Multifractional Brownian Motion And Operator Self-similar Gaussian Processes

This dissertation consists of six Chapters.In Chapter 1, we introduce the background and our motivations. We mainly in-troduce fractional Brownian motion, multifractional Brownian motion, RL-fractional Brownian motion, RL-multifractional Brownian motion and briefly discuss the opera-tor self-similar Gaussian vector-valued processes. At the end, we will collect some facts about weak convergence.In Chapter 2, we discuss weak limit theorem for one-dimensional multifractional Brownian motion. We construct a sequence of stochastic processes from a standard Poisson process and show that the sequence converges in law to a mulitifractional Brow-nian motion in the Banach space C[0,1].In Chapter 3, we discuss weak limit theorem for generalized multifractional Brow-nian motion. Similarly as in Chapter 2, we will also construct a sequence of stochastic processes based on a standard Poisson process and show it approximates in law general-ized multifractional Brownian motion.In Chapter 4, we discuss weak limit theorems for one-dimensional multifractional Brownian motion of Riemann-Lioville type. We construct two sequences of stochastic processes, one of which is based on the Donsker's theorem, the other is based on a standard Poisson process. Furthermore, we prove that both these sequences converge in law to a multifractional Brownian motion of of Riemann-Lioville type in a class of Besov spaces.In Chapter 5, we discuss weak limit theorem for the RL-multifarctional Brownian sheet. A sequence of stochastic process based on the Donsker's theorem will be con-structed. Furthermore, we show the laws of this sequence converge weakly to the law of RL-multifractional sheet.In Chapter 6, we study operator-self-similar Gaussian vector-valued processes. First, we briefly discuss some properties of operator-self-similar processes. Next, we introduce the operator fractional Brownian motion and study two kinds of weak limit theorems for it, one of which is based on a stationary Rd-valued Gaussian sequences, the other is based on a standard Poisson process. Finally, we introduce the operator fractional Brownian motion of Riemann-Liouville type and study two kinds of weak limit theorems for it, one of which is based on a standard Poisson processes, the other is based on a I.I.D sequence of random variables.