The differential equations of birth-and-death processes, and the Stieltjes moment problem
The differential equations of birth-and-death processes, and the Stieltjes moment problem
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DOI：
10.1090/s0002-9947-1957-0091566-1
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发表时间：
1957-02
影响因子：
1.3
通讯作者：
S. Karlin;J. McGregor
中科院分区：
文献类型：
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作者：
S. Karlin;J. McGregor
PI，i+i(T)=Xit+o(T)，pi，i(T)=1(xi+，yi)t+0(T)，pi，i_i(T)=Pit+o(T)，as t-gt；O，其中，xi，pui是可以被认为是从状态i吸收到状态i+1，i-1的速率的常数。想象一个物质粒子从一个整数移动到相邻的整数是很有用的，路径函数X(T)是粒子在时间t的位置。在Feller的书[4，第17章]中，可以找到对这些过程的优雅描述和应用的概述。利用上述序条件和过程的马科夫性质，很容易证明无限矩阵P(T)=(Pij(T))，i，j=O，1，2，满足方程(1.1)P‘(T)=Ap(T)，t0，
Pi,i+i(t) = Xit + o(t), Pi,i(t) = 1 (Xi + ,Yi)t + 0(t), Pi,i_i(t) = pit + o(t), as t->O, where Xi, pui are constants which may be thought of as the rates of absorption from state i into states i+1, i-1. As a guide to one's intuition it is useful to think of a material particle which moves from integer to neighboring integer, the path function X(t) being the position of the particle at time t. An elegant description of these processes together with a survey of applications may be found in Feller's book [4, Chapter 17]. Using the above order conditions and the Markoffian nature of the process it is easy to show that the infinite matrix P(t) = (Pij(t)), i, j=O, 1, 2, satisfies the equation (1.1) P'(t) = AP(t), t 0,