By Alexanra Bellow, Alexandra Bellow, Roger L. Jones

Nearly far and wide Convergence II comprises chosen papers shape the second one overseas convention on nearly in every single place convergence in likelihood and Ergodic concept, which came about on October 16-20, 1989 at Northwestern collage. It comprises survey papers and significant unique contributions and may attract researchers within the quarter of ergodic idea and harmonic research in addition to to graduate scholars getting into those fields of research

**Read or Download Almost Everywhere Convergence II. Proceedings of the International Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, Illinois, October 16–20, 1989 PDF**

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**Additional info for Almost Everywhere Convergence II. Proceedings of the International Conference on Almost Everywhere Convergence in Probability and Ergodic Theory, Evanston, Illinois, October 16–20, 1989**

**Example text**

The indicator function f = 1» will be constructed in a sequence of steps. ,n m such that sup l < m T n ,l * is large on Ω. Start with a large integer K = Kj. Define c = ix = 1/K, Let α > 0 be an irrational number. β = βχ = 1/K2. /2. As the sequence {na} is dense in [0,1) (mod 1) we can inductively find integers K < nt < n2 < ··· < n m such that (1) nka € ( - a t l - a t + ß/A) (mod 1). Call N = n m . /(4N)f and let (3) c = [K/ 7 ] be the integral part of K/7. Note that since K/7 — 1 < c < K/7 we have (4) K - l < K - 7 < c7 < K and hence (5) /3cT < i K = i = e.

Dtsch. Math. Verein. 46, (1936), 179-195. 47 A Note on the Strong Law of Large Numbers for Partial Sums of Independent Random Vectors * Erich Berger Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13, D-3400 Göttingen, Germany A b s t r a c t . 's. This exponential estimate, in conjunction with a new majorization technique, was recently employed by the author [3] for proving a very general SLLN in the vector space setting. It is the aim of this note to demonstrate how this last mentioned result can in a convenient way be exploited for deriving more tractable (but still rather general) sufficient criteria for the SLLN.

Then Kolmogorov's theorem is a direct consequence, by the same path used in [3] to prove that (3) above implies (1) above. The same arguments apply to any lacunary sequences as well as 2 r and 2"r; details may be seen in [19]. e. ) 29 THEOREM £ [12]: There is a constant C such that |O0)£ < C\K\2L2 for all *eL2[0, 1) . Proo/. Let n(f>) = ^ ώ JP^M^' *> that n Then 4(l)r"fi,*= f f J 0 J 0 (l-* r (0)A(0«- 2 ""'"«di We also have the two easy estimates (>) |i-*»(o| < rin Σ l l - e "