Study On Decay Parameters Of General Birth And Death Process

For the study of the decay parameter of the general process of birth and death,Var.Doorn gave some important conclusions about the quasi-stationary distribution in the process of birth and death in 1986,and also gave the estimation form of the decay parameter.Chen Mufa also gives an expression for calculating the decay parameter,but it is very difficult to calculate for some examples.As is known,for the general process of birth and death,if the probability 1is extinct,when < +?,the quasi-stationary distribution exists and is unique.When = +?,if the decay parameter is greater than zero,there is a family of quasi-stationary distributions;if the attenuation parameter is equal to zero,the quasi-stationary distribution does not exist.Therefore,this has caused us trouble,how to effectively determine whether the attenuation parameter is greater than zero.Then this paper mainly uses the conclusions in a paper published by Sean P.Meyn and R.L Tweedie in 1993,and gives the Lyapunov function and satisfies the theorem(3.0.6),while giving When = +?,the attenuation condition is greater than zero,and the conditions are relatively simple compared to the conditions of Var.Doorn and Chen Mu.In the end,the existence of quasi-stationary distribution in the process of quadratic progressive symmetric birth and death is studied,and the sufficient conditions for the existence of quasi-stationary distribution are given.At the same time,several examples are given to illustrate the differences between the conditions given in this paper and the Var.Doorn method and the Chen Mu method.The first chapter is the introduction part,which mainly introduces the research background and significance of the quasi-stationary distribution of the general birth and death process,the research history and the status quo.The second chapter is preliminary knowledge,introducing the Markov chain,the general process of birth and death,the process of dual birth and death,and the related knowledge of exponential ergodicity and quasi-stationary distribution,and the related theorems used in later proofs.The third chapter mainly discusses the condition that the decay parameter of the general birth and death process is greater than zero.The sufficient condition that the attenuation parameter is positive is given by the theorem(3.1.1).The fourth chapter gives the conditions for the existence of quasi-stationary distribution in the process of quadratic progressive symmetric birth and death and gives an example to compare with the method of Var.Doorn and the method of Chen Mufa.Finally,the main work and conclusions of this paper are summarized,and the related problems are not solved.