## 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

### Keywords:

Baum-Katz-Nagaev theorem; Martingale difference
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### References:

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