含瞬时态、具有突变率的广义生-灭Q-矩阵(I)

AN EXTENDED BIRTH-DEATH Q-MATRIX WITH INSTANTANEOUS STATE AND CATASTROPHES (I)

研究含瞬时态、具有突变率的广义生-灭拟q-矩阵,给出易于验证的Q过程存在性准则,并构造出全部Q过程和全部诚实Q过程,证明了不需附加任何条件,所有诚实Q过程都是常返的,给出诚实Q过程是遍历的充要条件,并求出其遍历测度,以及证明了不存在可配称Q过程.最后给出两个例子以说明我们的结果易于验证.

A new structure with the special property that instantaneous state and catastrophes is imposed to ordinary Birth-Death processes is considered. The authors give easy-checking existence criteria for such Markov processes. All the Q-processes and the honest Q-processes are explicitly constructed. Recurrent and ergodicity properties for the honest Q-processes are investigated. Surprisingly, it can be proved that all the honest Q-processes are recurrent without necessarily imposing any extra conditions. Ergodicity of such processes is also investigated and solved. Equilibrium distributions are then established. All the Q-processes are not symmetric. Two examples are provided to illustrate our results.