Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications.

*(English)*Zbl 1184.62099A sequence of random variables \(\{X_i, 1 \leq i \leq n \}\) is called negatively asociated (NA) if for every pair of disjoint subsets \( A \) and \( B\) of \(\{ 1, 2, \dots , n \}\),

\[ \text{Cov}(f(X_i, i \in A), g(X_j , j \in B)) \leq 0, \]

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

\[ \text{Cov}(f(X_n), g(X_{n+1}, \dots , X_{n+k} )) \leq q(n)(\text{Var}(f(X_n)) \text{Var}(g(X_{n+1}, \dots ,X_{n+k})))^{1/2}, \]

for all \( n, k \geq 1 \) 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 \(1 < p < 2\), 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.

\[ \text{Cov}(f(X_i, i \in A), g(X_j , j \in B)) \leq 0, \]

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

\[ \text{Cov}(f(X_n), g(X_{n+1}, \dots , X_{n+k} )) \leq q(n)(\text{Var}(f(X_n)) \text{Var}(g(X_{n+1}, \dots ,X_{n+k})))^{1/2}, \]

for all \( n, k \geq 1 \) 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 \(1 < p < 2\), 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.

Reviewer: Tae Il. Jeon (Taejon)

##### MSC:

62H20 | Measures of association (correlation, canonical correlation, etc.) |

60F15 | Strong limit theorems |

60E15 | Inequalities; stochastic orderings |

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

##### Keywords:

asymptotically almost negative association; residual Cesàro \(\alpha\)-integrability; strong residual Cesàro \(\alpha\)-integrability; \(L_{p}\)-convergence; complete convergence; Marcinkiewicz-Zygmand strong law
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\textit{D. Yuan} and \textit{J. An}, Sci. China, Ser. A 52, No. 9, 1887--1904 (2009; Zbl 1184.62099)

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