Limiting behaviour for arrays of rowwise widely orthant dependent random variables under conditions of R − h-integrability and its applications

ABSTRACT

In this paper, we mainly study the $L^r$ convergence, complete convergence and complete moment convergence for arrays of rowwise widely orthant dependent (WOD, for short) random variables under some conditions of R-h-integrability. The results in this paper generalize and improve the corresponding ones of Sung et al., Weak laws of large numbers for arrays under a condition of uniform integrability, Journal of the Korean Mathematical Society 45 (2008), pp. 289–300, Wang and Hu, Weak laws of large numbers for arrays of dependent random variables, Stochastics: An International Journal of Probability and Stochastic Processes 86 (2014), pp. 759–775, and Wu et al., Limiting behaviour for arrays of rowwise END random variables under conditions of h-integrability, Stochastics: An International Journal of Probability and Stochastic Processes 87 (2014), pp. 409–423. As applications of the complete convergence that we established, we present some results on complete consistency for the estimator in a nonparametric regression model based on WOD errors.

KEYWORDS:

• R−h-integrability
• $L^r$ convergence
• complete convergence
• complete moment convergence
• complete consistency
• nonparametric regression model
• widely orthant dependent random variables

AMS Classifications:

• 60F15
• 62G05

Acknowledgments

The authors are grateful to reviewer's valuable comments that significantly improved the presentation of our article.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Andrei Volodin  http://orcid.org/0000-0002-9771-846X

Additional information

Funding

The research of X.J. Wang was partially supported by the National Natural Science Foundation of China (11671012), the Natural Science Foundation of Anhui Province (1508085J06) and the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005). The research of T.-C. Hu was partially supported by the Ministry of Science and Technology, R.O.C. (MOST). The research of M. Ordóñez Cabrera was partially supported by the Plan Andaluz de Investigacion de la Junta de Andalucia FQM-127 Grant P08-FQM-03543 and by MEC Grant MTM2015-65242-C2-1-P. The research of A.Volodin has been partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).