We consider several types of scaling limits for the Wigner-Moyal equation of the parabolic waves in random media, the limiting cases of which include the standard radiative transfer limit, the geometrical-optics limit and the white-noise limit. We show under fairly general assumptions on the random refractive index field that sufficient amount of medium diversity (thus excluding the white-noise limit) leads to statistical stability or self-averaging in the sense that the limiting law is deterministic and is governed by one of the 6 different types of transport (Boltzmann or Fokker-Planck) equations depending on the specific scaling involved. We discuss the connection to the statistical stability of time-reversal procedure and the decoherence effect in quantum mechanics.