Abstract
Let {Y, Y i , −∞ < i < ∞} be a doubly infinite sequence of identically distributed and asymptotically linear negative quadrant dependence random variables, {a i , −∞ < i < ∞} an absolutely summable sequence of real numbers. We are inspired by Wang et al. (Econometric Theory 18:119–139, 2002) and Salvadori (Stoch Environ Res Risk Assess 17:116–140, 2003). And Salvadori (Stoch Environ Res Risk Assess 17:116–140, 2003) have obtained Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions. In this paper, we prove the complete convergence of \({\left\{ {{\sum\nolimits_{k = 1}^n {{\sum\nolimits_{i = - \infty }^\infty {a_{{i + k}} Y_{i} /n^{{1/t}} } }} },n \geq 1} \right\}}\) under some suitable conditions. The results obtained improve and generalize the results of Li et al. (1992) and Zhang (1996). The results obtained extend those for negative associated sequences and ρ*-mixing sequences.
Similar content being viewed by others
References
Burton RM, Dehling H (1990) Large deviations for some weakly dependent random process [J]. Statist Probab Lett 9:397–401
Ibragimov IA (1962) Some limit theorems for stationary process [J]. Theory Probab Appl 7:349–382
Li DL, Rao MB, Wang XC (1992) Complete convergence of moving average processes [J]. Statist Probab Lett 14:111–114
Peligrad M, Gut A (1999) Almost sure results for a class of dependent random variables [J]. J Theoret Probab 12:87–104
Phillips P (1987) Time series regression with a unit root [J]. Econometrica 55:277–302
Salvadori G (2003) Linear combinations of order statistics to estimate the quantiles of generalized pareto and extreme values distributions[J]. Stoch Environ Res Risk Assess 17:116–140
Shao QM (1989) On complete convergence for a ρ-mixing sequence [J] (in Chinese). Acta Math Sinica 32:377–393
Wang QY, Lin YX, Gulati CM (2002) The invariance principle for linear processes with applications [J]. Econometric Theory 18:119–139
Zhang LX (1996) Complete convergence of moving average processes under dependence assumptions [J]. Statist Probab Lett 30:165–170
Zhang LX (2000a) A functional central limit theorem for asymptotically negative dependence random fields [J]. Acta Math Hung 86(3):237–259
Zhang LX (2000b) Convergence rates in the strong laws of nonstationary ρ*-mixing random fields [J] (in Chinese). Acta Math Scientia 20:303–312
Zhang LX, Wen JW (2001) A weak convergence for negatively associated fields [J]. Statist Probab Lett 53:259–267
Author information
Authors and Affiliations
Corresponding author
Additional information
CIC Number O211, AMS (2000) Subject Classification 60F15, 60G50
Research supported by National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Cai, Gh., Wu, H. Moment inequality and complete convergence of moving average processes under asymptotically linear negative quadrant dependence assumptions. Stoch Environ Res Ris Assess 20, 1–5 (2006). https://doi.org/10.1007/s00477-005-0241-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00477-005-0241-9