本论文研究了下临界分枝 Markov 链中的条件极限定理, 包括下临界分枝随机游动的条件中心极限定理, 下临界随机环境中的分枝随机游动的条件中心极限定理, 以及下临界分枝 Markov 链的 Yaglom 极限和拟平稳分布.

第一部分, 研究了下临界分枝随机游动的条件中心极限定理. 我们证明了在不灭绝的条件下, 落在 n^1/2x 之下的粒子个数依分布收敛, 其极限随机变量由下临界 Galton-Watson 过程的 Yaglom 极限以及标准正态随机变量所刻画.

第二部分, 研究了下临界随机环境中的分枝随机游动的条件中心极限定理. 在强 (strongly) 和中 (intermediately) 下临界分枝情形, 我们证明了在不灭绝的条件下, 存活粒子构成的点过程经过规范化后依分布收敛, 并且对极限分布给出了相应的刻画, 揭示了随机环境对极限行为的影响.

第三部分, 研究了下临界分枝 Markov 链的 Yaglom 极限和拟平稳分布问题. 我们利用矩方法证明了 Yaglom 极限的存在性并给出了相关的条件极限定理. 在此基础上, 我们利用概率的方法给出了所有拟平稳分布的统一积分刻画.

This thesis focuses on the conditional limit theorems for subcritical branching Markov chains, including the conditional central limit theorem for subcritical branching random walks, the conditional central limit theorem for the subcritical branching random walk in a random environment, and the Yaglom limit and quasi-stationary distributions for subcritical branching Markov chains.

In the first part, we study the conditional central limit theorem for subcritical branching random walks. We prove that conditioning on survival, the number of particles located in (−∞, n^1/2x) converges in law to a random variable, which is characterized by the Yaglom limit of the subcritical Galton-Watson process and the standard normal random variable.

In the second part, we study the conditional central limit theorem for the subcritical branching random walk in a random environment. In strongly and intermediately subcritical branching cases, we show that conditioned on non-extinction event, the point process can be rescaled so as to converge in law to a random variable, and give the corresponding characterization of the limit random variable, which reveals the influence of a random environment on the limit behavior.

In the final part, we study the Yaglom limit and quasi-stationary distributions for subcritical branching Markov chains. We prove the existence of Yaglom limit by using the moment approach and give some related conditional limit theorems. Based on Yaglom's theorem, we give explicit integral representations of all quasi-stationary distributions, the method of the proof is probabilistic.