Properties of the random effect transformation

We analyze the influence of the random effect transformation on four regularity classes of distribution functions. We consider regularly varying, dominatedly varying, long-tailed, and generalized long-tailed distribution functions. We apply the random effect transformation to the primary distribution functions from these classes and investigate whether the resulting distribution function remains in the same class. We find that the random effect transformation has the greatest impact on regularly varying distribution functions. We illustrate the obtained theoretical results with several examples.