On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in rademacher type p Banach spaces

• Autorzy: Rosalsky A. , Thanh L. V.
• Języki publikacji: EN

Abstrakty

EN

For a sequence of random elements {V_n, n ≥1} taking values in a real separable Rademacher type p (1 ≤p ≤2) Banach space and positive constants b_n↑∞, conditions are provided for the strong law of large numbers ∑ⁿ_i=1V_i/b_n→0 almost surely. We treat the following cases: (i) {V_nn ≥1} is blockwise independent with EV_n=0, n≥1, and (ii) {V_n, n≥1} is blockwise p-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of Móricz [12], Móricz et al. [13], and Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative examples or counterexamples are provided.

Słowa kluczowe

EN

blockwise independent random elements; blockwise p-orthogonal random elements; strong law of large numbers; almost sure convergence; Rademacher type p Banach space

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2007
• Tom: Vol. 27, Fasc. 2
• Strony: 205--222
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Rosalsky A.

• Department of Statistics University of Florida Gainesville, Florida 32611, USA

autor

Thanh L. V.

• Department of Mathematics Vinh University Vinh City, Nghe An Province Vietnam

Bibliografia

• [1] A. Adler, A. Rosalsky and R. L. Taylor, A weak law for normed weighted sums of random elements in Rademacher type p Banach spaces, J. Multivariate Anal. 37 (1991), pp. 259-268.
• [2] A. Adler, A. Rosalsky and R. L. Taylor, Some strong laws of large numbers for sums of random elements, Bull. Inst. Math. Acad. Sinica 20 (1992), pp. 335-357.
• [3] A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York 1980.
• [4] A. Cantrell and A. Rosalsky, Some strong laws of large numbers for Banach space valued summands irrespective of their joint distributions, Stochastic Anal. Appl. 21 (2003), pp. 79-95.
• [5] S. Chobanyan, S. Levental and У. Mandrekar, Prokhorov blocks and strong law of large numbers under rearrangements, J. Theoret. Probab. 17 (2004), pp. 647-672.
• [6] Y. S. Chow and H. Teicher, Probability Theory: Independence, Inter changeability, Martin- gales, 3rd edition, Springer, New York 1997.
• [7] V. F. Gaposhkin, Series of block-orthogonal and block-independent systems (in Russian), Izv. Vyssh. Uchebn. Zaved. Mat. (1990), pp. 12-18. English translation in: Soviet Math. (Iz. VUZ Mat.) 34 (1990), pp. 13-20.
• [8] V. F. Gaposhkin, On the strong law of large numbers for blockwise independent and block- wise orthogonal random variables (in Russian), Teor. Veroyatnost. i Primenen. 39 (1994), pp. 804-812. English translation in: Theory Probab. Appl. 39 (1994), pp. 667-684.
• [9] J. Hoffmann-J0rgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probab. 4 (1976), pp. 587-599.
• [10] J. O. Howell and R. L. Taylor, Marcinkiewicz-Zygmund weak laws of large numbers for unconditional random elements in Banach spaces, in: Probability in Banach Spaces III. Proceedings of the Third International Conference on Probability in Banach Spaces Held at Tufts University, Medford USA, August 14-16, 1980, Lecture Notes in Math. No 860, Springer, Berlin 1981, pp. 219-230.
• [11] M. Loeve, Probability Theory, Yol. I, 4th edition, Springer, New York 1977.
• [12] F. Móricz, Strong limit theorems for blockwise m-dependent and blockwise quasiorthogonal sequences of random variables, Proc. Amer. Math. Soc. 101 (1987), pp. 709-715.
• [13] F. Móricz, K.-L. Su and R. L. Taylor, Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces, Acta Math. Hungar. 65 (1994), pp. 1-16.
• [14] Yu. V. Prohorov, On the strong law of large numbers (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 69 (1949), pp. 607-610.
• [15] Yu. V. Prohorov, On the strong law of large numbers (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 14 (1950), pp. 523-536.
• [16] P. Révész, The Laws of Large Numbers, Academic Press, New York 1968.
• [17] F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), pp. 347-374.
• [18] R. L. Taylor, Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture Notes in Math. No 672, Springer, Berlin 1978.
• [19] W. A. Woyczyński, Geometry and martingales in Banach spaces. Part II: Independent increments, in: Probability on Banach Spaces, J. Kuelbs (Ed.), Advances in Probability and Related Topics, P. Ney (Ed.), Vol. 4, Marcel Dekker, New York 1978, pp. 267-517.