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A strong approximation for logarithmic averages of partial sums of random variables. (English) Zbl 0846.60029

Summary: Let \(S_n\) be the partial sums of \(\rho\)-mixing stationary random variables and let \(f(x)\) be a real function. We give sufficient conditions under which the logarithmic average of \(f(S_n/ \sigma_n)\) converges almost surely to \(\int^\infty_{- \infty} f(x)d \Phi (x)\). We also obtain strong approximation for \[ H(n) = \sum^n_{k = 1} k^{-1} f(S_k/ \sigma_k) - \log n \int^\infty_{- \infty} f(x)d \Phi (x) \] which will imply the asymptotic normality of \(H(n)/ \log^{1/2} n\). But for partial sums of i.i.d. random variables our results will be proved under a weaker moment condition than assumed for \(\rho\)-mixing random variables.

MSC:

60F15 Strong limit theorems
60F05 Central limit and other weak theorems
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[1] I. Berkes andH. Dehling, Some limit theorems in log density.Ann. Probab. 21 (1993), 1640–1670. · Zbl 0785.60014 · doi:10.1214/aop/1176989135
[2] R.C. Bradley, A sufficient condition for linear growth of variances in a stationary random variables,Proc. Amer. Math. Soc. 83 (1981), 586–589. · Zbl 0476.60037 · doi:10.1090/S0002-9939-1981-0627698-5
[3] G.A. Brosamler, An almost everywhere central limit theorem,Math. Proc. Camb. Phil. Soc. 104 (1988), 561–574. · Zbl 0668.60029 · doi:10.1017/S0305004100065750
[4] M. Csörgo andL. Horváth, Invariance principles for logarithmic averages,Math. Proc. Camb. Phil. Soc. 112 (1992), 195–205. · Zbl 0766.60038 · doi:10.1017/S0305004100070870
[5] M. Csörgo andP. Révész,Strong Approximation in Probability and Statistics, New York, Academic Press, 1981.
[6] U. Einmahl, Strong invariance principles for partial sums of independent random variables,Ann. Probab. 15 (1987), 1419–1440. · Zbl 0637.60041 · doi:10.1214/aop/1176991985
[7] A. Fisher A pathwise central limit theorem for random walks,preprint (1989).
[8] D.L. Hanson andP. Russo, Some results on increments of the Wiener process with applications to lag sums of i.i.d. random variables,Ann. Probab. 11 (1983), 609–623. · Zbl 0519.60030 · doi:10.1214/aop/1176993505
[9] L. Horváth andD. Khoshnevisan, Strong approximation in almost sure local central limit theorem,preprint (1994).
[10] L. Horváth andD. Khoshnevisan, Strong approximation for logarithmic averages, to appear inStud. Sci. Math. Hung. · Zbl 0851.60027
[11] G. Hurelbaatar, On the almost sure central limit theorem for -mixing random variables, to appear inStud. Sci. Math. Hung. · Zbl 0849.60017
[12] G. Hurelbaatar, On the almost sure local and global central limit theorem for weakly dependent random variables, to appear inAnnales Univ. Sci. Budapest, Sect. Math. · Zbl 0847.60019
[13] K. Ito andH.P. McKean,Diffusion process and their sample paths, Springer-Verlag, Berlin, Heidelberg, New York, 1974. · Zbl 0285.60063
[14] M.T. Lacey andW. Philipp, A note on the almost sure central limit theorem,Stat. Prob. Letters 9 (1990), 201–205. · Zbl 0691.60016 · doi:10.1016/0167-7152(90)90056-D
[15] P. Major, An approximation of partial sums of independent random variables,Z. Wahrsch. Verw. Gebiete 35 (1976), 213–220. · Zbl 0338.60031 · doi:10.1007/BF00532673
[16] P. Mandl,Analytical Treatment of One-Dimensional Markov Processes, New York, Springer-Verlag, 1968. · Zbl 0179.47802
[17] M. Peligrad, Invariance principles for mixing sequences of random variables,Ann. Probab. 10 (1982), 968–981. · Zbl 0503.60044 · doi:10.1214/aop/1176993718
[18] P. Schatte, On strong versions of the central limit theorem,Math. Nachr. 137 (1988), 249–256. · Zbl 0661.60031 · doi:10.1002/mana.19881370117
[19] P. Schatte, On the central limit theorem with almost sure convergence,Probab. Math. Stat. 11 (1991), 237–246. · Zbl 0741.60019
[20] Q.M. Shao, On the invariance principle for -mixing sequences of random variables,Chin. Ann. of Math. 10 (1989), 427–433. · Zbl 0683.60023
[21] Q.M. Shao, Almost sure invariance principles for mixing sequences of random variables,Stoch. Proc. Appl. 48 (1993), 319–334. · Zbl 0793.60038 · doi:10.1016/0304-4149(93)90051-5
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