Convergence And Harmonic Moments For Bisexual Branching Processes In A Random Environment

As an important extension of classical bisexual branching processes and branch-ing processes in random environments,the study of the model of bisexual branching processes in a random environment is interesting both in theory and in applications.A series of results have been obtained in extinction and asymptotic properties,but the theoretical system is not perfect enough,and some limit results have not been discussed,for the study of harmonic moment and convergence rate,there is basically no involving.In order to complete some limit properties,this paper mainly studies L?-convergence,(??1),of different normalization processes,harmonic moment of the normalized processes by factor(Sn)and a.s.convergence rate of the normalized processes by factor(In).This paper is divided into four chapters:The first chapter introduces the models and definitions of bisexual branching pro-cesses in a random environment and varying environment.Secondly,it introduces the current situation and development trend,and lists main conclusions of this paper.In the second chapter,we give a sufficient condition and a necessary condition for L?-convergence,(?? 1),of the normalization sequences(Wn)to a finite and non-degeterate random variable;A sufficient condition for L2-convergence of sequences(Wn)is established by applying Doob's martingales theorem;We discuss a sufficient condition for L?-convergence,??1,of normalization sequences(Wn)to a finite and non-degenerate random variable,and the necessary condition for L2-convergence of(Wn)is obtained.In the third chapter,we obtain the critical value for the existence of the annealed harmonic moment of W by applying decomposition of female process and Laplace transform.In the fourth chapter,we study the exponential and geometric convergence rates of(Wn)to its limit W in a.s.by applying Squeeze Theorem,similar conclusions hold for a bisexual branching process in a varying environment.