General characterization theorems for the Weibull and the stable distributions

Let X's be i.i.d. random variables with common distribution F and let a's be a set of random variables independent of the X's and satisfying with probability one, where a is a positive constant. The class of distributions F with the property that the variable has the same distribution F as is determined. In particular, it is shown under some additional conditions that the above property characterizes the stable distribution with characteristic exponent a. The result follows from the solution to a general integral functional equation, which also provides some other characterization theorems. An analogous characterization theorem for the Weibull distribution is obtained using the property that has the same distribution F as .