The law of the iterated logarithm for LNQD sequences

Yong Zhang

doi:10.1186/s13660-017-1607-5

Abstract

Let {xi_{i},iin{mathbb{Z}}} be a stationary LNQD sequence of random variables with zero means and finite variance. In this paper, by the Kolmogorov type maximal inequality and Stein’s method, we establish the result of the law of the iterated logarithm for LNQD sequence with less restriction of moment conditions. We also prove the law of the iterated logarithm for a linear process generated by an LNQD sequence with the coefficients satisfying sum_{i=-infty}^{infty}|a_{i}|<infty by a Beveridge and Nelson decomposition.

Highlights

Two random variables X and Y are said to be negatively quadrant dependent (NQD, for short), if P(X ≤ x, Y ≤ y) – P(X ≤ x)P(Y ≤ y) ≤ 0 for all x, y ∈ R
A much stronger concept than LNQD was introduced by Joag-Dev and Proschan [3]: for a finite index set I, the r.v.s. {Xi, i ∈ I} are said to be negatively associated (NA, for short), if for any disjoint nonempty subsets A and B of I, and any coordinatewise nondecreasing function G and H with G : RA → R and H : RB → R and EG2(Xi, i ∈ A) < ∞, EH2(Xj, j ∈ B) < ∞, we have Cov(G(Xi, i ∈ A), H(Xj, j ∈ B)) ≤ 0
Zhang Journal of Inequalities and Applications (2018) 2018:11 results and a uniform LIL for partial sums of an LNQD sequence, Wang and Wu [12] obtained the strong laws of large numbers for arrays of rowwise NA and LNQD random variables, Wang and Wu [13] established the central limit theorem for stationary linear processes generated by LNQD sequence, Li et al [14] established some inequalities for LNQD sequence, Shen et al [15] proved the complete convergence for weighted sums of LNQD sequence, and so forth

Summary

Introduction

Two random variables X and Y are said to be negatively quadrant dependent (NQD, for short), if P(X ≤ x, Y ≤ y) – P(X ≤ x)P(Y ≤ y) ≤ 0 for all x, y ∈ R. A sequence {Xk, k ∈ Z} is said to be linear negatively quadrant dependent (LNQD, for short) if for any disjoint finite subsets A, B ⊂ Z and any positive real numbers rj, i∈A riXi and j∈B rjXj are NQD. Zhang Journal of Inequalities and Applications (2018) 2018:11 results and a uniform LIL for partial sums of an LNQD sequence, Wang and Wu [12] obtained the strong laws of large numbers for arrays of rowwise NA and LNQD random variables, Wang and Wu [13] established the central limit theorem for stationary linear processes generated by LNQD sequence, Li et al [14] established some inequalities for LNQD sequence, Shen et al [15] proved the complete convergence for weighted sums of LNQD sequence, and so forth.

Main results

Conclusions