zbMATH — the first resource for mathematics

Baum-Katz-Nagaev type results for martingales. (English) Zbl 1130.60020
In the present paper the author obtains a Baum-Katz-Nagaev type theorem for martingale difference sequences which are \(L^p\)-bounded for some \(p>2\). More precisely, let \(X_1,X_2,\dots\) be a sequence of random variables on some probability space \((\Omega,{\mathcal F},P)\) and let \({\mathcal F}_n:=\sigma\{X_1,\dots, X_n\}\) and \(S_n:=X_1+\cdots+ X_n\). \((X_n)\) \((n\geq 1)\) is called a martingale difference sequence (mds) if \((S_n)\) is a martingale w.r.t. \(({\mathcal F}_n)\) \((n \geq 0)\). (Here, \(S_0:=0\), and \({\mathcal F}_0\) is trivial.) Let \(p\geq 1\). The mds \((X_n)\) is called \(L^p\)-bounded if for some constant \(C\) we have \(\|X_n\|_p\leq C\) for all \(n\geq 1\). For \(\varepsilon>0\), \(p>0\) and \(0<r<2\) consider the series \[ \sum^\infty_{n=1}n^{p/r-2}P(|S_n|>\varepsilon n^{1/r}).\tag{+} \] The author obtains the following result:
Theorem: (i) Let \((X_n)\) be an \(L^p\)-bounded mds, and \(0<r<2<p\). Then the series (+) converges for all \(\varepsilon >0\).
(ii) There exists an \(L^q\)-bounded \((q<2)\) mds \((X_n)\) such that the series (+) diverges for \(p=2\), all \(0<r<2\) and \(\varepsilon >0\).
(iii) There exists an \(L^2\)-bounded mds \((X_n)\) such that the series (+) diverges for \(p=2\), \(r=1\) and all \(\varepsilon >0\).

60E15 Inequalities; stochastic orderings
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] Erdős, P., On a theorem of hsu and robbins, Ann. math. statist., 20, 286-291, (1949) · Zbl 0033.29001
[3] Erdős, P., Remark on my paper “on a theorem of hsu and robbins”, Ann. math. statist., 21, 138, (1950) · Zbl 0035.21403
[4] Hsu, P.L.; Robbins, H., Complete convergence and the law of large numbers, Proc. natl. acad. sci. USA, 33, 25-31, (1947) · Zbl 0030.20101
[5] Lesigne, E.; Volný, D., Large deviations for martingales, Stochastic process. appl., 96, 143-159, (2001) · Zbl 1059.60033
[6] Nagaev, S.V., Some limit theorems for large deviations, Theory probab. appl., 10, 214-235, (1965) · Zbl 0144.18704
[7] Spitzer, F., A combinatorial lemma and its applications to probability theory, Trans. amer. math. soc., 82, 323-339, (1956) · Zbl 0071.13003
[8] G. Stoica, The Baum-Katz theorem for bounded subsequences, Statist. Probab. Lett., in press · Zbl 1139.60315
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.