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A generalization of the Men’shov-Rademacher theorem. (English. Russian original) Zbl 1181.60043
Math. Notes 86, No. 6, 861-872 (2009); translation from Mat. Zametki 86, No. 6, 925-937 (2009).
Summary: For a sequence \(\{X_n\}_{n\geq 1}\) of random variables with finite second moment and a sequence \(\{b_n\}_{n\geq 1}\) of positive constants, new sufficient conditions for the almost sure convergence of \(\Sigma _{n\geq 1} X_n /b_n\) are obtained and the strong law of large numbers, which states that lim\(_{n \rightarrow \infty} \sum_{k=1}^n X_k /b_n = 0\) almost surely, is proved. The results are shown to be optimal in a number of cases. In the theorems, assumptions have the form of conditions on \(\rho _n = \text{sup}_k\) (E\(X_k X_{k+n})^{+}\), \[ r_n = \underset {k} {\text{sup}} \frac {(\text EX_k X_{k+n})^+}{(\text EX_k^2)^{1/2}(\text EX_{k+n}^2)^{1/2}}, \] \(\text EX_n^2\), and \(b_n\), where \(x^+ = x\vee 0\) and \(n \in \mathbb N\).

60F15 Strong limit theorems
Full Text: DOI
[1] Sh. Levental’, Kh. Salekhi, and S. A. Chobanyan, ”General maximal inequalities related to the strong law of large numbers.” Mat. Zametki 81(1), 98–111 (2007) [Math. Notes 81 (1–2), 85–96 (2007)]. · doi:10.4213/mzm3520
[2] F. Móricz, ”The strong laws of large numbers for quasi-stationary sequences,” Z.Wahrsch. Verw.Gebiete 38(3), 223–236 (1977). · Zbl 0336.60027 · doi:10.1007/BF00537266
[3] V. F. Gaposhkin, ”Criteria for the strong law of large numbers for some classes of weakly stationary processes and homogeneous random fields,” Teor. Veroyatnost. Primenen. 22(2), 295–319 (1977).
[4] V. F. Gaposhkin, ”Convergence of series that are connected with stationary sequences,” Izv. Akad. Nauk SSSR Ser. Mat. 39(6), 1366–1392 (1975). · Zbl 0326.60038
[5] R. J. Serfling, ”On the strong law of large numbers and related results for quasistationary sequences,” Teor. Veroyatnost. Primenen. 25(1), 190–194 (1980). · Zbl 0423.60030
[6] V. F. Gaposhkin, ”On the growth order of partial sums of nonorthogonal series,” Anal.Math. 6(2), 105–119 (1980). · Zbl 0507.40001 · doi:10.1007/BF01919466
[7] T.-C. Hu, A. Rosalsky and A. Volodin, ”On convergence properties of sums of dependent random variables under second moment and covariance restrictions,” Statist. Probab. Lett. 78(14), 1999–2005 (2008). · Zbl 1283.60049 · doi:10.1016/j.spl.2008.01.073
[8] S. H. Sung, ”Maximal inequalities for dependent random variables and applications,” J. Inequal. Appl., Art. ID 598319 (2008). · Zbl 1145.60013
[9] H. Walk, ”Almost sure Cesàro and Euler summability of sequences of dependent random variables,” Arch. Math. (Basel) 89(5), 466–480 (2007). · Zbl 1132.60031 · doi:10.1007/s00013-007-2214-3
[10] F. Móricz, ”SLLN and convergence rates for nearly orthogonal sequences of random variables,” Proc. Amer. Math. Soc. 95(2), 287–294 (1985). · Zbl 0573.60026 · doi:10.2307/2044529
[11] I. Fazekas and O. Klesov, ”A general approach to the strong laws of large numbers,” Teor. Veroyatnost. Primenen. 45(3), 568–583 (2000) [Theory Probab. Appl. 45 (3), 436–449 (2002)]. · Zbl 0991.60021 · doi:10.4213/tvp486
[12] G. Cohen and M. Lin, ”Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory,” Israel J. Math. 148(1), 41–86 (2005). · Zbl 1086.60019 · doi:10.1007/BF02775432
[13] B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed. (Izd. AFTs, Moscow, 1999) [in Russian]. · Zbl 1188.42010
[14] F. Móricz and K. Tandori, ”Counterexamples in the theory of orthogonal series,” Acta Math. Hungar. 49(1–2), 283–290 (1987). · Zbl 0628.42010 · doi:10.1007/BF01956333
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