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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 \(\varepsilon >0\).

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