Previous |  Up |  Next


Borel matrix; almost sure convergence; \text{\sl GB} and \text{\sl GC} sets; Gaussian processes
We study the Borel summation method. We obtain a general sufficient condition for a given matrix $A$ to have the Borel property. We deduce as corollaries, earlier results obtained by G. M"uller and J.D. Hill. Our result is expressed in terms belonging to the theory of Gaussian processes. We show that this result cannot be extended to the study of the Borel summation method on arbitrary dynamical systems. However, in the $L^p$-setting, we establish necessary conditions of the same kind by using Bourgain's entropy criterion.
[BC] Beck J., Chen W.: Irregularities of distribution. Cambridge Univ. Press, 1987. MR 0903025 | Zbl 1156.11029
[BZ] Beeckmann W., Zeller K.: Theorie der Limitierungsverfahren 2. Aufl. Erg. Math. Grenzeb., Springer Verlag, 1970. MR 0118990
[BL] Bellow A., Losert V.: On sequences of density zero in ergodic theory. Proc. Kakutani Conf., 1984. MR 0737387 | Zbl 0587.28013
[B] Bourgain J.: Almost sure convergence and bounded entropy. Israel J. Math. 63 (1988), 79-95. MR 0959049 | Zbl 0677.60042
[C] Conze J.P.: Convergence des moyennes ergodiques pour des sous-suites. Bull. Soc. Math. France 35 (1973), 7-15. MR 0453975 | Zbl 0285.28017
[Co] Cooke R.G.: Infinite matrices and sequence spaces. Macmillan, London, 1950. MR 0040451 | Zbl 0132.28901
[DR] Del Junco A., Rosenblatt J.: Counterexamples in Ergodic Theory and Number Theory. Math. Ann. 245 (1979), 185-197. MR 0553340 | Zbl 0398.28021
[D] Dudley R.M.: The size of compact subsets of Hilbert space and continuity of Gaussian processes. J. Functional Analysis 1 (1967), 290-330. MR 0220340
[GK] Grillenberger C., Krengel U.: On matrix summation and the pointwise ergodic theorem. Lecture Notes in Math., Springer 532 (1976), 113-124. MR 0486411 | Zbl 0331.28010
[GRR] Garsia A., Rodemich E. and Rumsey H. Jr.: A real variable lemma and continuity of paths of some Gaussian processes. Indiana Univ. Math. (1970), 565-578.
[H] Hill J.D.: Remarks on the Borel property. Pacific J. Math. 4 (1954), 227-242. MR 0062244 | Zbl 0057.29301
[K] Krengel U.: Ergodic Theorems. W. de Gruyter, 1989. MR 0797411 | Zbl 0649.47042
[KN] Kuipers L., Niederreiter H.: Uniform Distribution of Sequences. J. Wiley Ed., 1971. MR 0419394 | Zbl 0568.10001
[M] Müller G.: Sätze über Folgen auf kompakten Raümen. Monatsheft. Math. 67 (1963), 436-451. MR 0158199
[Pe] Peyerimhoff A.: Lectures on Summability. Lecture Notes in Math., Springer 107 (1969). MR 0463744 | Zbl 0182.08401
[Ph] Philipp W.: Über einen Satz von Davenport-Erdös-Le Veque. Monatsheft Math. 68 (1964), 52-58. MR 0162784
[SW] Schneider D., Weber M.: Une remarque sur un Théorème de Bourgain. Séminaire de Probabilités XXVII Lectures Notes in Math., Springer 1557 (1993), 202-206. MR 1308565 | Zbl 0799.60035
[T] Talagrand M.: Regularity of Gaussian processes. Acta. Math. 159 (1987), 99-149. MR 0906527 | Zbl 0712.60044
[W1] Weber M.: Une version fonctionnelle du Théorème ergodique ponctuel. Comptes Rendus Acad. Sci. Paris, Sér. I 311 (1990), 131-133. MR 1065444 | Zbl 0739.28007
[W2] Weber M.: Méthodes de sommation matricielles. Comptes Rendus Acad. Sci. Paris, Sér. I 315 (1992), 759-764. MR 1184897 | Zbl 0768.40002
[W3] Weber M.: GC sets, Stein's elements and matrix summation methods. Prépublication IRMA no 027, Université de Strasbourg, 1993.
[W4] Weber M.: GB and GC sets in ergodic theory. IXth Conference on Probability in Banach Spaces, Sandberg, August 1993, Denmark, Progress in Prob. V, Birkhauser, t. 35, 1994. MR 1308514 | Zbl 0808.28011
[W5] Weber M.: Coupling of the GB set property for ergodic averages. to appear in J. Theoretic. Prob. (1995), 1993. MR 1371072 | Zbl 0851.60037
Partner of
EuDML logo