Galton-Watson过程中极限鞅密度函数的Lipschitz连续性

考虑上临界Galton-Watson过程中第n代粒子总数Z_n，令W表示鞅W_n=Z_n/mⁿ的极限.针对W的密度函数ω(x)的Lipschitz连续性问题，基于Kesten-Stigum定理，提出了更完善的证明方法和补充.同时进行了关于鞅极限性质的一系列讨论.首先修正了以往的证明方法，得到在δ≠1的情形下，ω(x)在［ε，∞)中是Lipschitz 连续的，阶为δ′=min(δ，1).在δ=1的时，ω(x)的Lipschitz连续性的阶为1/2，从而保证了结论的完整性.

Lipschitz Continuity of Martingale’s Limit Density Function in Galton-Watson Processes

Considering the total number Z_n of the n-th generation particles in the supercritical Galton-Watson process， let W denote the limit of martingale W_n=Z_n/mⁿ. Aiming at the Lipschitz continuity problem of the density function ω(x) of W， based on the Kesten-Stigum theorem， a more complete proof and supplement were proposed. A series of discussions on the limit properties of martingales were also conducted. First， the previous method of proof was modified， and it was obtained that in the case of δ≠1，ω(x) is Lipschitz continuous in ［ε，∞)， and the order is δ′=min(δ，1).When δ=1， the order of Lipschitz continuity of ω(x) is 1/2， thus ensuring the completeness of the conclusion.