Some finite sample properties of negatively dependent random variables. (English) Zbl 1199.60048

Farcomeni, Alessio

Teor. Jmovirn. Mat. Stat. 77, 141-148 (2007) and Theory Probab. Math. Stat. 77, 155-163 (2008).

A sequence of random variables \(X_i\) is said to be negatively (positively) dependent if \[ P\left(\cap_{i=1}^n\{X_i\leq z_i\}\right)\leq(\geq)\prod_{i=1}^n P(X_i\leq z_i) \] and \[ P\left(\cap_{i=1}^n\{X_i>z_i\}\right)\leq(\geq)\prod_{i=1}^n P(X_i>z_i) \] for \(z_i\in\mathbb R,i=1,\dots,n\). For the review of concepts of negative dependence see the paper by H. W. Block, Thomas H. Savits and M. Shaked [Ann. Probab. 10, 765–772 (1982; Zbl 0501.62037)]. The author discusses some finite sample properties of vectors of negatively dependent random variables. Some inequalities, widely used for independent random variables, are proved. Some basic tools such as the symmetrization lemma, are extended to the case of negatively dependent random variables.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60E15	Inequalities; stochastic orderings
47N30	Applications of operator theory in probability theory and statistics

Keywords:

negatively dependent random variables; inequalities

Citations:

Zbl 0501.62037