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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.

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