THE GENERALIZATION OF THE KAC-BERNSTEIN THEOREM
Abstract: The Skitovich-Darmois Theorem of the early 1950’s establishes the
normality of independent from the independence of two linear
forms in these random variables. Existing proofs generally rely on the theorems of
Marcinkiewicz and Cramér, which are based on analytic function theory. We present a
self-contained real-variable proof of the essence of this theorem viewed as a generalization of
the case which is generally called Bernstein’s Theorem, and also adapt an early little
known argument of Kac to provide a direct simple proof when A large bibliography
is provided.
1991 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: independence; characterization; normality; Bernstein’s
theorem; Cramer’s theorem; Marcinkiewicz’s theorem; characteristic function; Laplace
transform; real-variable; real function; moments; cumulants.