On a Class of Affine Invariant Dependent Random Variables Related to Characterization Results

Copyright © 2014 Werner Hürlimann. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions under which a naive algebraic solution to a random affine stochastic equation yields a genuine stochastic solution with a given independence property are derived. The well-known example by Letac is characterized and generalized to a non-trivial dependence structure. As by-products, a series representation of the exact probability density of a sum of independent gamma random variables is recovered, and a new elementary proof of Lukacs’ characterization of the gamma distribution is added