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Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications. (English) Zbl 1184.62099

A sequence of random variables $\left\{{X}_{i},1\le i\le n\right\}$ is called negatively asociated (NA) if for every pair of disjoint subsets $A$ and $B$ of $\left\{1,2,\cdots ,n\right\}$,

$\text{Cov}\left(f\left({X}_{i},i\in A\right),g\left({X}_{j},j\in B\right)\right)\le 0,$

whenever $f$ and $g$ are coordinatewise nondecreasing and the covariance exists. A sequence of random variables $\left\{{X}_{n}$, $n\ge 1\right\}$ is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence $q\left(n\right)\to 0$ as $n\to \infty$ such that

$\text{Cov}\left(f\left({X}_{n}\right),g\left({X}_{n+1},\cdots ,{X}_{n+k}\right)\right)\le q\left(n\right){\left(\text{Var}\left(f\left({X}_{n}\right)\right)\text{Var}\left(g\left({X}_{n+1},\cdots ,{X}_{n+k}\right)\right)\right)}^{1/2},$

for all $n,k\ge 1$ and for all coordinatewise nondecreasing continuous functions $f$ and $g$ whenever the variances exist. For NA random variables a lot of sharp and elegant estimates are available. Some Rosenthal type moment inequalities are also introduced. For AANA random variables, some excellent results are also available. However, for AANA random variables, Rosenthal type inequalities are not yet available.

The authors establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively associated random variables, which extend the corresponding results for negatively associated random variables. As application of these inequalities, by employing the notions of residual Cesàro $\alpha$-integrability and strong residual Cesàro $\alpha$-integrability, they derive some results on ${L}_{p}$-convergence, where $1, and on complete convergence. In addition, they estimate the rate of convergence in the Marcinkiewicz-Zygmund strong law for partial sums of identically distributed random variables.

##### MSC:
 62H20 Statistical measures of associations 60F15 Strong limit theorems 60E15 Inequalities in probability theory; stochastic orderings 62H05 Characterization and structure theory (Multivariate analysis)
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