Small Value Probability For Supercritical Multitype Branching Process

Small value probability of branching process is an important subject for research about the limit properties of stochastic processes.The rate of decay of small value probability is closely related to its branching mechanism,which is generally divided into Schr(?)der case and B(?)ttcher case.In the single-type case,the classification of Schr(?)der case and B(?)ttcher case is determined by probability of producing one descendant.According to Jones(2004)[20],the small value of supercritical multitype branching process is also determined by branching mechanism which is much more complicated than that in the single-type case.we obtained more concrete classification condition for the supercritical multitype branching process by analyzing the spectral radius and power convergence of the matrix A and the minimal descendants matrix B in[20],and obtain a lot of interesting conclusions,such as the existence of neither Schr(?)der case nor B(?)ttcher case.Beyond that,we improved the method in[30]and give a prove of the upper and lower bound of the small value probability for supercritical multitype branching processes.