The generalization of the Kac-Bernstein Theorem

• Autorzy: Quine M. P. , Seneta E.
• Języki publikacji: EN

Abstrakty

EN

The Skitovich-Darmois Theorem of the early 1950's establishes the normality of independent X₁, X₂,…, X_n 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 n = 2, which is generally called Bernstein's Theorem, and also adapt an early little known argument of Kac to provide a direct simple proof when n = 2. A large bibliography is provided.

Słowa kluczowe

EN

independence; characterization; normality; Bernstein's theorem; Cramér's theorem; Marcinkiewicz's theorem; characteristic function; Laplace transform; real-variable; real function; moments; cumulants

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 1999
• Tom: Vol. 19, Fasc. 2
• Strony: 441--452
• Opis fizyczny: Bibliogr. 40 poz.

Twórcy

autor

Quine M. P.

• School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

autor

Seneta E.

• School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Bibliografia

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