Baum-Katz-Nagaev type results for martingales. (English) Zbl 1130.60020

Stoica, George

J. Math. Anal. Appl. 336, No. 2, 1489-1492 (2007).

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 \(\varepsilon >0\).

Reviewer: Klaus Schürger (Bonn)

MSC:

60E15	Inequalities; stochastic orderings
60G42	Martingales with discrete parameter

Keywords:

Baum-Katz-Nagaev theorem; Martingale difference

References:

[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