Abstract
The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by Ko et al. (2007) and Jabbari et al. (2009). In addition, we also give the relevant precise asymptotics.
Highlights
Lehmann [1] introduced a natural definition of negative dependence: two random variables X and Y are said to be negative quadrant dependent (NQD, say) if P(X > x, Y > y) ≤ P(X > x)P(Y > y) for all x, y ∈ R
By (5), we can obtain that the strong convergence rate of ∑ni=1(Xi − EXi)/n is O(1)n−1/2log1/2n, which is obviously faster than the corresponding one n−1/2(pn log n)1/2 that Ko et al [13] obtained, where pn ≤ n/2 and pn → ∞ as n → ∞
By the analysis mentioned above, we know that the strong convergence rate O(1)n−1/2log1/2n of ∑ni=1(Xi − EXi)/n is much faster than the relevant one O(1)n−1/3(log n)2/3 Jabbari et al [7] obtained only for the special case of geometrically decreasing covariances
Summary
Lehmann [1] introduced a natural definition of negative dependence: two random variables X and Y are said to be negative quadrant dependent (NQD, say) if P(X > x, Y > y) ≤ P(X > x)P(Y > y) for all x, y ∈ R. By (5), we can obtain that the strong convergence rate of ∑ni=1(Xi − EXi)/n is O(1)n−1/2log1/2n, which is obviously faster than the corresponding one n−1/2(pn log n)1/2 that Ko et al [13] obtained, where pn ≤ n/2 and pn → ∞ as n → ∞. Theorem 2.1 in Jabbari et al [7] from strictly stationary negatively associated setting to weighted LNQD case.
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