On some properties of quadratic stochastic processes
Abstract
In this paper we prove that every measurable quadratic stochastic process X : RN×Ω→R is continuous and has the form
X(x,·) = Σi,j=1NxixjYi,j(·) (a.e.),
where x = (x1,...,xN)∈RN and Yi,j: Ω→R are random variables. Moreover, we give a proof of the stability of the quadratic stochastic processes.
References
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2. P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76-86.
3. P.R. Halmos, Measure theory, Van Nostrand, New York, 1950.
4. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
5. S. Kurepa, On the quadratic functional, Publ. Inst. Math. (Beograd) 13 (1959), 57-72.
6. S. Kurepa, A cosine functional equation in Hilbert space, Canad. J. Math 12 (1960), 45-50.
7. B. Nagy, On a generalization of the Cauchy equation, Aequationes Math. 10 (1974), 165-171.
8. K. Nikodem, On convex stochastic processes, Aequationes Math. 20 (1980), 184-197.
9. K. Nikodem, On quadratic stochastic processes, Aequationes Math. 21 (1980), 192-199.
10. A. Ostrowski, Über die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichungen, Jahresber. Deutsch. Math.-Verein. 38 (1929), 54-62.
11. W. Sierpiński, Sur les fonctions convexes measurables, Fund. Math. 1 (1920), 125-128.
NikodemK. (1990). On some properties of quadratic stochastic processes. Annales Mathematicae Silesianae, 3, 58-69. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14303
Kazimierz Nikodem
Filia Politechniki Łódzkiej w Bielsku-Białej Poland
Filia Politechniki Łódzkiej w Bielsku-Białej Poland
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