A universal result in almost sure central limit theory. (English) Zbl 1053.60022

Berkes, István; Csáki, Endre

Stochastic Processes Appl. 94, No. 1, 105-134 (2001).

The paper presents results generalizing the almost sure central limit theorem in several ways. The weights in formula are relaxed, the considered functions of observations are allowed general such that sum and/or maximum became examples, only, and an arbitrary distribution function can serve as the limit instead of Gaussian one. The paper finishes with interesting consequences for random walks, U-statistics, Kolmogorov-Smirnov statistics, etc.

Reviewer: Petr Lachout (Praha)

MSC:

60F15	Strong limit theorems
60F05	Central limit and other weak theorems

Keywords:

almost sure central limit theorem; logarithmic averages; summation methods

References:

[1]	Atlagh, M., Théorème central limite presque sûr et loi du logarithme itéré pour des sommes de variables aléatoires indépendantes, C. R. Acad. Sci. Paris Sér. I., 316, 929-933 (1993) · Zbl 0776.60022
[2]	Atlagh, M.; Weber, M., Un théorème central limite presque sûr relatif à des sous-suites, C. R. Acad. Sci. Paris Sér. I, 315, 203-206 (1992) · Zbl 0757.60018
[3]	Atlagh, M.; Weber, M., Une nouvelle loi forte des grandes nombres, (Bergelson, V.; March, P.; Rosenblatt, J., Convergence in Ergodic Theory and Probability. (1996), W. de Gruyter: W. de Gruyter Berlin), 41-62 · Zbl 0866.60028
[4]	Berkes, I., On the almost sure central limit theorem and domains of attraction, Probab. Theory Related Fields, 102, 1-18 (1995) · Zbl 0821.60030
[5]	Berkes, I., Results and problems related to the pointwise central limit theorem, (Szyszkowicz, B., Asymptotic Results in Probability and Statistics (a volume in honour of Miklós Csörgő). (1998), Elsevier: Elsevier Amsterdam), 59-96 · Zbl 0979.60014
[6]	Berkes, I.; Csáki, E.; Csörgő, S., Almost sure limit theorems for the St. Petersburg game, Statist. Probab. Lett., 45, 23-30 (1999) · Zbl 0943.60026
[7]	Berkes, I.; Dehling, H., Some limit theorems in log density, Ann. Probab., 21, 1640-1670 (1993) · Zbl 0785.60014
[8]	Berkes, I.; Dehling, H., On the almost sure central limit theorem for random variables with infinite variance, J. Theoret. Probab., 7, 667-680 (1994) · Zbl 0804.60021
[9]	Berkes, I.; Dehling, H.; Móri, T. F., Counterexamples related to the a.s. central limit theorem, Studia Sci. Math. Hungar., 26, 153-164 (1991) · Zbl 0765.60018
[10]	Bingham, N. H.; Rogers, L. C.G., Summability methods and almost sure convergence, (Bellow, A.; Jones, R., Almost Everywhere Convergence II. (1991), Academic Press: Academic Press New York), 69-83 · Zbl 0745.40003
[11]	Brosamler, G., An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc., 104, 561-574 (1988) · Zbl 0668.60029
[12]	Chandrasekharan, K.; Minakshisundaram, S., Typical Means. (1952), Oxford University Press: Oxford University Press Oxford · Zbl 0047.29901
[13]	Cheng, S.; Peng, L.; Qi, Y., Almost sure convergence in extreme value theory, Math. Nachr., 190, 43-50 (1998) · Zbl 0932.60028
[14]	Csáki, E.; Földes, A., On the logarithmic average of additive functionals, Statist. Probab. Lett., 22, 261-268 (1995) · Zbl 0816.60031
[15]	Csörgő, M.; Horváth, L., Invariance principles for logarithmic averages, Math. Proc. Cambridge Phil. Soc., 112, 195-205 (1992) · Zbl 0766.60038
[16]	Darling, D.; Erdős, P., A limit theorem for the maximum of normalized sums of independent random variables, Duke Math. J., 23, 143-155 (1956) · Zbl 0070.13806
[17]	Darling, D.; Kac, M., On occupation times for Markoff processes, Trans. Amer. Math. Soc., 84, 444-458 (1957) · Zbl 0078.32005
[18]	Denker, M., Asymptotic Distribution Theory in Nonparametric Statistics. (1985), F. Vieweg & Sohn: F. Vieweg & Sohn Braunschweig and Wiesbaden · Zbl 0619.62019
[19]	Dvoretzky, A., Erdős, P., 1951. Some problems on random walk in space. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, pp. 353-367.; Dvoretzky, A., Erdős, P., 1951. Some problems on random walk in space. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, pp. 353-367. · Zbl 0044.14001
[20]	Dvoretzky, A.; Kiefer, J.; Wolfowitz, J., Asymptotic minimax character of the sample distribution function and the classical multinomial estimator, Ann. Math. Statist., 27, 642-649 (1956) · Zbl 0073.14603
[21]	Einmahl, U., Strong invariance principles for partial sums of independent random vectors, Ann. Probab., 15, 1419-1440 (1987) · Zbl 0637.60041
[22]	Einmahl, U., The Darling-Erdős theorem for sums of i.i.d. random variables, Probab. Theory Related Fields, 82, 241-257 (1989) · Zbl 0687.60032
[23]	Fahrner, I.; Stadtmüller, U., On almost sure max-limit theorems, Statist. Probab. Lett., 37, 229-236 (1998) · Zbl 1246.60034
[24]	Feller, W., Generalization of a probability limit theorem of Cramér, Trans. Amer. Math. Soc., 54, 361-372 (1943) · Zbl 0063.01342
[25]	Fisher, A., 1989. A pathwise central limit theorem for random walks. Preprint.; Fisher, A., 1989. A pathwise central limit theorem for random walks. Preprint.
[26]	Giné, E.; Zinn, J., A remark on convergence in distribution of U-statistics, Ann. Probab., 22, 117-125 (1994) · Zbl 0801.60015
[27]	Horváth, L.; Khoshnevisan, D., Weight functions and pathwise local central limit theorems, Stochastic Process. Appl., 59, 105-123 (1995) · Zbl 0841.60018
[28]	Ibragimov, I.A., 1996. On almost sure versions of limit theorems. Dokl. Akad. Nauk 350, 301-303 (in Russian).; Ibragimov, I.A., 1996. On almost sure versions of limit theorems. Dokl. Akad. Nauk 350, 301-303 (in Russian).
[29]	Ibragimov, I. A.; Lifshits, M., On the convergence of generalized moments in almost sure central limit theorem, Statist. Probab. Lett., 40, 343-351 (1998) · Zbl 0933.60017
[30]	Ibragimov, I. A.; Lifshits, M., On almost sure limit theorems, Theory Probab. Appl., 44, 254-272 (1999) · Zbl 0970.60032
[31]	Kesten, H.; Spitzer, F., A limit theorem related to a new class of self similar processes, Z. Wahrsch. Verw. Gebiete, 50, 5-25 (1979) · Zbl 0396.60037
[32]	Koroljuk, V. S.; Borovskich, Yu. V., Theory of U-statistics (1994), Kluwer: Kluwer Dordrecht · Zbl 0785.60015
[33]	Lacey, M.; Philipp, W., A note on the almost everywhere central limit theorem, Statist. Probab. Lett., 9, 201-205 (1990) · Zbl 0691.60016
[34]	Lifshits, M., 2000a. On the difference between CLT and ASCLT. Zapiski Seminarov POMI 260, 186-201 (in Russian).; Lifshits, M., 2000a. On the difference between CLT and ASCLT. Zapiski Seminarov POMI 260, 186-201 (in Russian).
[35]	Lifshits, M., 2000b. An almost sure limit theorem for sums of random vectors. Preprint.; Lifshits, M., 2000b. An almost sure limit theorem for sums of random vectors. Preprint.
[36]	Marcus, M.; Rosen, J., Logarithmic averages for the local time of recurrent random walks and Levy processes, Stochastic Process. Appl., 59, 175-184 (1995) · Zbl 0853.60059
[37]	Móri, T. F., On the strong law of large numbers for logarithmically weighted sums, Ann. Univ. Sci. Budapest. Sect. Math., 36, 35-46 (1993) · Zbl 0791.60020
[38]	Oodaira, H., 1976. Some limit theorems for the maximum of normalized sums of weakly dependent random variables. Proceedings of Third Japan-USSR Symposium on Probability Theory, Lecture Notes in Mathematics, Vol. 550, Springer, Berlin, pp. 467-474.; Oodaira, H., 1976. Some limit theorems for the maximum of normalized sums of weakly dependent random variables. Proceedings of Third Japan-USSR Symposium on Probability Theory, Lecture Notes in Mathematics, Vol. 550, Springer, Berlin, pp. 467-474. · Zbl 0355.60022
[39]	Peligrad, M.; Révész, P., On the almost sure central limit theorem., (Bellow, A.; Jones, R., Almost Everywhere Convergence II. (1991), Academic Press: Academic Press New York), 209-225 · Zbl 0751.60022
[40]	Philipp, W., Stout, W., 1975. Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the AMS, No. 161.; Philipp, W., Stout, W., 1975. Almost sure invariance principles for partial sums of weakly dependent random variables. Memoirs of the AMS, No. 161. · Zbl 0361.60007
[41]	Rodzik, B.; Rychlik, Z., An almost sure central limit theorem for independent random variables, Ann. Inst. H. Poincaré, 30, 1-11 (1994) · Zbl 0793.60023
[42]	Rodzik, B.; Rychlik, Z., On the central limit theorem for independent random variables with almost sure convergence, Probab. Math. Statist., 12, 299-309 (1996) · Zbl 0874.60022
[43]	Schatte, P., On strong versions of the central limit theorem, Math. Nachr., 137, 249-256 (1988) · Zbl 0661.60031
[44]	Serfling, R., Approximation Theorems of Mathematical Statistics. (1980), Wiley: Wiley New York · Zbl 0538.62002
[45]	Shorack, G., Extension of the Darling and Erdös theorem on the maximum of normalized sums, Ann. Probab., 7, 1092-1096 (1979) · Zbl 0421.60046
[46]	Strassen, V., 1967. Almost sure behavior of sums of independent random variables and martingales. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol II, Part I, Univ. of California Press, pp. 315-343.; Strassen, V., 1967. Almost sure behavior of sums of independent random variables and martingales. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol II, Part I, Univ. of California Press, pp. 315-343. · Zbl 0201.49903
[47]	Weigl, A., 1989. Zwei Sätze über die Belegungszeit beim Random Walk. Diplomarbeit, TU Wien.; Weigl, A., 1989. Zwei Sätze über die Belegungszeit beim Random Walk. Diplomarbeit, TU Wien.