Abstract
We show that linear combinations of independent and identically distributed strictly stable variables with positive random coefficients is equal in distribution to a function of these random coefficients times a random variable from the same stable distribution. Furthermore, this result is used to show that a random linear combination of independent standard Wiener processes has the distribution of a function of these random coefficients times one standard Wiener process. In opposition to the central limit theorem, this result does not require a large number of terms but it holds with two or more terms. This has implications to simplify stochastic differential equations with a finite number of noises with random coefficients that can be used in modeling anomalous diffusion.
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