二次分支q—矩阵

R.R.Chen(1997)对连续时间分支过程进行了推广，其q-矩阵为（1．2），对这类q-矩阵，她得到了过程的唯一性准则，常返性和正常返性的条件。Zhang,Lin和Hou(1999)对转移函数的性质进行了讨论，得到了转移函数是随机单调的，强遍历，多项式一致收敛，指数遍历和Feller转移函数的条件，在R.R.Chen(1997)和Zhang,Lin和Hon(1999)的基础上本对二次分支过程进行了进一步的讨论，得到了过程的鞅性，同时还得到了大偏差上界估计成立的条件。

Quadratic Branching q-Processes

R.R.Chen (1997)studied an extended class of time-continuous branching processes with q-matrix as (1.1). The uniqueness .recurrence and positive recurrence criteria for the processes are presented in her paper. Zhang,Lin and Hou(1999) made a study to the property of transition function and obtained the conditions of transition function is stochastically monotone, strongly ergodic, polynomial uniformly convergent .exponentially and Feller transition function. On the basis of the papers of R. R. Chen (1997) and Zhang, Lin and Hou (1999). we make a further study to quadratic branching q - processes, we obtain martingale property of the process. In the meantime, we obtain the condition of the upper estimate of large deviations holds.