The Branching Processes With Migration In Random Environment

It is well known, Branching process with migration in random environments that developed in recent years is one of most important stochastic processes. its application spreads domain and so on industry, agriculture, economy, insurance, biology, medicine, project technology and social sciences.The key question for the influence of a branching process with migration is the transition matrix, also called it the environment of the branching process with migration. As the direct positive natural environment has randomness, also may classical the branching process with migration in environment randomization, obtains the branching process with migration in random environment, the abbreviation is BPMRE, according to the environment relies on the time ,the space, the time and space BPMRE also may divide into the BPMRE relies on time , BPMRE relies on space, BPMRE relies on the time and space.This article first studies into the branching process with migration, later studies the branching process with immigration,(BPRME),finally discussed the time sequence estimator for the BPIRE. After the environment randomization, very more results in classics environment may be made simple promoted, some actually needs to do massive works. This paper mainly is summarizes the work of the aspect predecessor. the paper is divided into three parts:The first part introduced the develop history and the research in present situation for the BPMRE and the BPIME.The second part studied the property of the BPMRE: the Markov property, moment estimator, the limit property and the extinction.The third part uses the assistance structure method, structure a stochastic process which has the same distribution with the BPIRE; discussed the correlated probability characteristic function; later obtains the moment estimate and its limit distribution for the immigration in stationary ergodic environment ;finally, have the limit theorem with stationary ergodic environment; and the convergence rates for the sub critical BPIRE with i.i.d. envrionment .