Limit Theorems For Branching Process And Control Branching Processes In Random Environments

In this thesis, we study the limit theorems for branching process and controlled branching processes in random environments, the content are as follows:In the first chapter, we firstly introduce the background and development of the branching process and controlled branching processes in random environments; secondly, we introduce the definitions of these two branching processes; Finally, we give the main results of this thesis.In the second chapter, we study the large deviation for a supercritical branching process, and show that ?W-W_n?/W_n satisfies the large deviation principle in case p₁=0 and case p₁>0.In the third chapter, we discuss the limiting behaviors and extinction problems for controlled branching processes in random environments:firstly, under the normalization factor {S_n,n?0}, we study the normalization processes {???_n:n?N} and give the sufficient conditions of {???_n:n?N} a.s., L¹ and L² convergence; secondly, we get a necessary condition for {???n:n?N} convergence to a non-degenerate at 0 random variable; thirdly, under the normalization factor {I_n,n>0}, we discuss the normalization processes {???n:n?N} and get the sufficient conditions of {???n:n?N} a.s. and L¹ convergence; at last, we give the upper bound of the growth rate of processes {Z_n}_n?0 and a sufficient condition of the extinction of the processes in stationary ergodic environment.