Generalized Mandelbrot Multiplicative Cascades And The Martingale Convergence Rate Of Super-critical Branching Process With Emigrate

This paper mainly concerned the central limit theorem for generalized Mandelbrot multiplicative cascades and the martingale convergence rate of the super-critical branching process with emigration, it consists of three chapters.Chapter one: Introduction. Firstly, we introduced the background and development status of generalized Mandelbrot multiplicative cascades and super-critical branching process with emigration, and then introduced the specific model, research ideas and main results of generalized Mandelbrot multiplicative cascades and super-critical branching process with emigration.Chapter two: Generalized Mandelbrot multiplicative cascades. This chapter focused on the central limit theorem for generalized Mandelbrot multiplicative cascades. Firstly, we verified that Y_n is a non-negative martingale, obtained the expressions of the expectation and variance of Y_n-Y_? as well as the properties of variance of Y_?, and then we applied the Lindeberg-Feller Center limit theorem to prove the central limit theorem of generalized Mandelbrot multiplicative cascades.Chapter three: Super-critical branching process with emigration. This chapter focused on super-critical branching process with emigration, we get some basic properties of the process,such as the expectation and variance of it. And then we consider the convergence rate of the normalized process.