To study the relationship between the region range and the parameters of hollow beam in the non-Kolmogorov turbulence propagation path and the beam expansion in different regions, the expressions for the mean-squared width, Rayleigh range, and turbulence distance of hollow beams propagating through non-Kolmogorov turbulence were given by using the extended Huygens-Fresnel principle, and the propagation path was divided into three regions by using the turbulence distance for numerical analysis. The results show that the length of region Ⅰ and region Ⅱ and the starting point of region Ⅲ decrease first and then increase with the increasing of the turbulence generalized exponent parameter α (There is a minimal value, when α=3.11), and increase with the increasing of obscure ratio η and beam orders M(and N). When the value of M(and N)is small（M(and N)<3）, the effect of turbulence on beam spread in Rayleigh range can not be ignored. The larger M(and N)and η is, the easier it is to ignore the effect of turbulence on beam spread in Rayleigh range. In the transmission path, the beam enters area Ⅰ, area Ⅱ and area Ⅲ in turn, and then expands more and more violently. With the increasing of M(and N)and η, the length of region Ⅱ and the starting point of region Ⅲ increase more significantly than the length of region Ⅰ. The results provide a reference for the application of hollow beam propagation in turbulence.