This paper studies the propagation characteristics related to higher-order moments of decentered annular beams through non-Kolmogorov turbulence. The analytical expressions for the mean-squared beam width , the skewness parameter , and the kurtosis parameter are derived. The analytical expression for the non-Kolmogorov turbulence parameter is also derived, and the differences between two non-Kolmogorov turbulence parameters and are examined. It is shown that depends on both and , but and only depend on . decreases monotonically as the spectral power law exponent increases, but there exist a maximum of and a minimum of when . When propagation distance is long enough, reaches zero, i.e., the intensity distribution is perfectly symmetric about the centroid position axis. In free space, both and may appear on propagation. However, it is always or on propagation when turbulence is not weak. On the other hand, in turbulence, the maximum of increases as the decentered parameter increases and the obscure ratio decreases. In particular, when is long enough, the beam spot is elliptical in free space, but it becomes circular in turbulence.
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