Abstract: 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.

Key words: branching process, supercritical, martingale convergence, Kesten-Stigum theorem, Lipschitz continuous