On sequences of the white noises

• Autorzy: Beśka M. , Ciesielski Z.
• Języki publikacji: EN

Abstrakty

EN

The aim of the paper is to prove the strong law of large numbers for Gaussian functionals (Theorem 3.1). The functionals are of the form f (X_i), where / is integrable with respect to the Gaussian noise and the random vectors X_i are coordinatewise suitable correlated. In the last section we comment on the possibility of building noise analysis corresponding to the Legendre orthogonal polynomials analogous to the Wiener white noise theory based on Hermite orthogonal polynomials (Mehler’s kernel).

Słowa kluczowe

EN

Gebelein’s inequality; Hermite polynomials; Legendre polynomials; white noise; Wiener decomposition

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2006
• Tom: Vol. 26, Fasc. 1
• Strony: 201--209
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Beśka M.

• Faculty of Applied Mathematics, Gdańsk University of Technology, Narutowicza 12/12, 80-952 Gdańsk, Poland

autor

Ciesielski Z.

• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-125 Sopot, Poland

Bibliografia

• [1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge
• [2] M. Beśka and Z. Ciesielski, Ergodic type theorem for Gaussian systems, Proc. Steklov Inst. Math. 248 (2005), pp. 40-45.
• [3] M. Beśka and Z. Ciesielski, Gebelein’s inequality and its consequences, Banach Center Publ. 72 (2006), pp. 11-23.
• [4] H. Dym and H. P. McKean, Gaussian Processes, Function Theory and Inverse Spectral Problem, Academic Press, 1976.
• [5] N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 (1) (1981), pp. 119-122.
• [6] N. Etemadi, On the laws of large numbers for nonnegative random variables, J. Multivariate Anal. 13 (1983), pp. 187-193.
• [7] H. Gebeiein, Das statistische Problem der Korrelation als Variations und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung, Z. Angew. Math. Mech. 21 (1941), pp. 364-379.