Limit Theorems Of Branching Random Walk And Multi-type Branching Processes

The study of specific models in random environments began in 1967 with Willkinso-n's study of the branching process in random environments.In recent years,it has become a hot research topic for more and more mathematics and related subjects.A series of results have been obtained about extinction probability,classification and limit property of that process.The research on the extended model such as branch random walk and Mandelbrot's cascade has also become a hot spot.The study of multi-type branching processes in random environment is in the initial stage.The research of these processes has a very good application background,such as genetic mutation or new types of species production,population growth and particle splitting,which are currently of particular concern.From a single-type branching process to a multi-type branching process,some of the results will encounter intrinsically difficult problems,and many new problems will emerge,and many new concepts and new methods are needed.Based on this,this paper further studies the Mandelbrot's cascade,branching random walk,and multi-type branching processes in a random environment on the basis of existing rcesearchThis article includes four chapters:The first chapter introduces the research background and domestic and internation-al research status of Mandelbrot's cascade,branching random walks and multi-type branching processes,and the main results of this article are summarizedIn the second chapter,we study the harmonic morments and the principle of large deviation and middle deviation of Mandelbrot's cascade in random environments,and find the critical value of harmonic moments of Mandelbrot martingale limit random variables,and also establish the principle of large deviation and medium deviation of free energy.The above conclusions apply to the related limit theorems of branching random walks in random environmentIn the third chapter,we study the Berry-Esseen bounds and the Cramer type large deviation of branching random walk in random environmentThe fourth chapter mainly studies the LP convergence of multi-type branching process in random environment where 1<p<2.