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Some strong limit theorems for \(\tilde \rho\)-mixing sequences of random variables. (English) Zbl 1148.60020
The authors extend classical strong limit theorems to the case of \(\tilde\rho\)-mixing random variables – a notion for weak dependence – without imposing any extra conditions. Essentially the strong limit theorems are the strong laws of large numbers of Kolmogorov and Marczinkiewicz, the three series theorem, and consequences of these are presented as corollaries.

60F15 Strong limit theorems
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[1] Bradley, R.C., On the spectral density and asymptotic normality of weakly dependent random fields, J. theoret. probab., 5, 355-373, (1992) · Zbl 0787.60059
[2] Bryc, W.; Smolenski, W., Moment conditions for almost sure convergence of weakly correlated random variables, Proc. amer. math. soc., 199, 2, 629-635, (1993) · Zbl 0785.60018
[3] Gan, Shixin, Almost sure convergence for \(\widetilde{\rho}\)-mixing random variable sequences, Statist. probab. lett., 67, 289-298, (2004) · Zbl 1043.60023
[4] Goldie, C.M.; Greenwood, P.E., Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes, Ann. probab., 14, 3, 817-839, (1986) · Zbl 0604.60032
[5] Peligrad, M.; Gut, A., Almost-sure results for a class of dependent random variables, J. theoret. probab., 12, 87-104, (1999) · Zbl 0928.60025
[6] Utev, Sergey; Peligrad, Magda, Maximal inequalities and an invariance principle for a class of weakly dependent random variables, J. theoret. probab., 16, 1, 101-115, (2003) · Zbl 1012.60022
[7] Wu, Qunying, Some convergence properties for \(\widetilde{\rho}\)-mixing sequences, J. engng. math., 18, 3, 58-64, (2001), (in Chinese) · Zbl 0992.60037
[8] Wu, Qunying, Convergence for weighted sums of \(\widetilde{\rho}\)-mixing random sequences, Math. appl., 15, 1, 1-4, (2002), (in Chinese) · Zbl 1009.60015
[9] Yang, Shanchao, Some moment inequalities for partial sums of random variables and their applications, Chinese sci. bull., 43, 17, 1823-1827, (1998)
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