变环境下两物种分枝过程灭绝时的极限定理

Asymptotics of extinction time of a 2-type branching process in a varyingg environment

考虑变环境下两物种分枝过程.假定系统中个体的后代分布为线性分式分布且后代分布的均值矩阵存在极限,设ν为过程的灭绝时,且对k1记M_(k)为第k-1代个体的后代均值矩阵.在一定条件下,本文证明当n→∞时,P(ν=n)和P(ν>n)渐近等价于后代均值矩阵谱半径乘积的一些函数,推广了Wang和Yao(2022)的结论.Wang和Yao(2022)需要一个特定的假设,即存在ε>0使得对于任意k1,有det(M_(k))<-ε.在该假设下,系统中的个体更倾向于产生另一个物种的子女,故而将一大类矩阵排除在均值矩阵之外.本文去除了这一额外的假设,研究了灭绝时的分布极限理论.此外,为证明灭绝时的极限定理,本文用谱半径乘积给出了非齐次2阶矩阵乘积内部元素极限行为的等价刻画,并得到渐近周期连分数一些精细的极限定理.这些结论在非齐次矩阵乘积的遍历理论及连分数的极限理论中也自有其意义.

In this paper,we study a 2-type linear-fractional branching process in varying environments with asymptotically constant mean matrices.Let v be the extinction time and for k≥1,let Mk be the mean matrix of offspring distribution of individuals of the(k-1)-th generation.Under certain conditions,we show that P(v=n)and P(v>n)are asymptotically equivalent to some functions of products of spectral radii of the mean matrices.We complement a former result of Wang and Yao(2022)which requires in addition a condition that ■k≥1,det(M_(k))<-ε for some ε>0.Such a condition requires that individuals in the system are more likely to produce children of another type so that it excludes a large class of mean matrices.However,our results do not need such an additional assumption.As byproducts,we also get some results on the asymptotics of products of nonhomogeneous matrices and limit periodic continued fractions that have their interests.