A note on the rate of convergence in the strong law of large numbers for martingales. (English) Zbl 1237.60025

The study of the convergence of the series \[ \sum_{n=1}^\infty n^{-2+p/r} \text{P}(|S_n|n^{-1/r} >\varepsilon),\;\varepsilon>0 \] for a martingale \((S_n)_{n}\) with \(S_n = \sum_{i=1}^n X_i\) in terms of boundedness conditions of the martingale difference sequence \((X_n)_n\) for different parameter \(0 < p, r \) has a long history since it goes back to Kolmogorov’s strong law of large numbers for martingales (cf. [J. Elton, Ann. Probab. 9, 405–412 (1981; Zbl 0463.60039)]). For the history of the problem, see [L. E. Baum and M. Katz, Trans. Am. Math. Soc. 120, 108–123 (1965; Zbl 0142.14802)], [S. V. Nagaev, Select. Transl. Math. Stat. Probab. 5, 240–250 (1965; Zbl 0202.48303)], [Z. A. Łagodowski and Z. Rychlik, Probab. Theory Relat. Fields 71, 467–476 (1986; Zbl 0555.60021)], [the author, Stat. Probab. Lett. 78, No. 7, 924–926 (2008; Zbl 1139.60315); J. Math. Anal. Appl. 336, No. 2, 1489–1492 (2007; Zbl 1130.60020)] and [J. Dedecker and F. Merlevède, Theory Probab. Appl. 52, No. 3, 416–438 (2008) and Teor. Veroyatn. Primen. 52, No. 3, 562–587 (2007; Zbl 1158.60009)].
In these notes, the author shows for the so far missing cases \(1\leq r \leq p<2\) and \(p>1\) that the \(L^p\)-boundedness of \((X_n)_n\) implies the convergence of the series. In the limiting case \(p = r =1\), he shows that for \((X_n)_n\) to be bounded in \(L \log L\) turns out to be sufficient.


60F15 Strong limit theorems
60F25 \(L^p\)-limit theorems
60G42 Martingales with discrete parameter
Full Text: DOI


[1] Baum, L. E.; Katz, M., Convergence rates in the law of large numbers, Trans. Amer. Math. Soc., 120, 108-123 (1965) · Zbl 0142.14802
[2] Christofides, T. C., Maximal inequalities for demimartingales and a strong law of large numbers, Statist. Probab. Lett., 50, 357-363 (2000) · Zbl 0966.60013
[3] Dedecker, J.; Merlevède, F., Inequalities for partial sums of Hilbert valued dependent sequences and applications, Math. Methods Statist., 15, 176-206 (2006)
[4] Dedecker, J.; Merlevède, F., Convergence rates in the law of large numbers for Banach-valued dependent variables, Theory Probab. Appl., 52, 416-438 (2008) · Zbl 1158.60009
[5] Elton, J., A law of large numbers for identically distributed martingale differences, Ann. Probab., 9, 405-412 (1981) · Zbl 0463.60039
[6] Fazekas, I.; Klesov, O., A general approach to the strong law of large numbers, Theory Probab. Appl., 45, 436-449 (2000) · Zbl 0991.60021
[7] Hu, S.; Chen, G.; Wang, X., On extending the Brunk-Prokhorov strong law of large numbers for martingale differences, Statist. Probab. Lett., 78, 3187-3194 (2008) · Zbl 1152.60317
[8] Hu, S.; Hu, M., A general approach rate to the strong law of large numbers, Statist. Probab. Lett., 76, 843-851 (2006) · Zbl 1090.60030
[9] Kovalʼ, V. A., On a sufficient condition for the validity of the strong law of large numbers for martingales, Ukrainian Math. J., 52, 1554-1560 (2000) · Zbl 0979.60018
[10] Łagodowski, Z. A.; Rychlik, Z., Rate of convergence in the strong law of large numbers for martingales, Probab. Theory Related Fields, 71, 467-476 (1986) · Zbl 0555.60021
[11] Marcinkiewicz, J.; Zygmund, A., Sur les fonctions indépendantes, Fund. Math., 29, 60-90 (1937) · JFM 63.0946.02
[12] Nagaev, S. V., Some limit theorems for large deviations, Theory Probab. Appl., 10, 214-235 (1965) · Zbl 0144.18704
[13] Sheu, S. S.; Yao, Y. S., A strong law of large numbers for martingales, Proc. Amer. Math. Soc., 92, 283-287 (1984) · Zbl 0563.60031
[14] Stoica, G., The Baum-Katz theorem for bounded subsequences, Statist. Probab. Lett., 78, 924-926 (2008) · Zbl 1139.60315
[15] Stoica, G., Baum-Katz-Nagaev type results for martingales, J. Math. Anal. Appl., 336, 1489-1492 (2007) · Zbl 1130.60020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.