Exponential probability inequalities for WNOD random variables and their applications

Abstract

Some exponential probability inequalities for widely negative orthant dependent (WNOD, in short) random variables are established, which can be treated as very important roles to prove the strong law of large numbers among others in probability theory and mathematical statistics. By using the exponential probability inequalities, we study the complete convergence for arrays of rowwise WNOD random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers is obtained. In addition, the complete moment convergence for arrays of rowwise WNOD random variables is studied by using the exponential probability inequality and complete convergence that we established.Keywords Widely negative orthant dependent random variables · Complete convergence · Complete moment convergence · Marcinkiewicz-Zygmund type strong law of large numbers Mathematics Subject Classification 60F15 · 60E05

Strong laws of large numbers for arrays of row-wise extended negatively dependent random variables with applications

“…In fact, supposing {X n,k , 1 k n, n 1} and {a n,k , 1 k n, n 1} as in Theorem 2.1 of [16], and c n,k = n 1/p a n,k in Corollary 1 we get that n k=1 a n,k X n,k converges completely to zero provided only max 1 k n |a n,k | = O n −1/p , n → ∞. Furthermore, Corollary 1 still improves assumption (4.11) and the moment condition presented in Corollary 4.4 of [15].…”

“…an} + a n I {Xn>an} − a n I {Xn<−an} are widely orthant dependent with dominating sequence {M n , n 1} by Lemma 2.1 of[15] since the function T ℓ (t) = max(min(t, ℓ), −ℓ) is nondecreasing. Hence, the sequences {M n , n 1}, as they are nondecreasing transformations of widely orthant dependent random variables with the referred dominating sequence.…”

“…The random variablesX n I {|Xn| an} + a n I {Xn>an} − a n I {Xn<−an}are widely orthant dependent with dominating sequence {M n , n 1} by Lemma 2.1 of[15] since the function T ℓ (t) = max(min(t, ℓ), −ℓ) is nondecreasing. Hence, the sequences…”

Complete f-moment convergence for weighted sums of WOD arrays with statistical applications

“…The concept of WOD random variables was introduced by Wang et al [27]. Since then, many scholars studied the probability limit properties of WOD random variables and obtained some interesting results; Wang et al [37] presented some probability inequalities and moment inequalities for WOD random variables, further got the complete convergence for weighted sums of arrays of rowwise WOD random variables and gave some applications; Qiu and Hu [15] investigated the strong limit theorems for weighted sums of WOD random variables; Qiu and Chen [13] established a complete convergence result and a complete moment convergence result for weighted sums of WOD random variables under mild conditions; Shen et al [20] provided some exponential probability inequalities to get the complete convergence for arrays of rowwise WNOD random variables; Wu et al [32] investigated complete moment convergence for WOD random variables under some mild conditions; Li et al [11] established a Bernstein-type inequality for WOD random variables, and obtain the rates of strong convergence for kernel estimators of density and hazard functions under some suitable conditions; He [17]established the strong consistency and complete consistency of the Priestley-Chao estimator in nonparametric regression model with WOD errors under some general conditions and obtained the rates of strong consistency and complete consistency; Lu et al [14] studied the complete f -moment convergence for WOD random variables and gave some applications; Chen and Sung [3] obtained a Spitzer-type law of large numbers for WOD random variables. Shen and Wu [19] investigated the complete qth moment convergence and provided some sufficient conditions for sums of WOD random variables; Xi et al [28] presented some convergence properties for partial sums of WOD random variables and gave some applications; Lang et al [18] investigated the complete convergence for weighted sums of WOD random variables, and so on.…”

Complete Moment Convergence for Arrays of Rowwise Negatively Associated Random Variables and Its Application in Non-Parametric Regression Model

“…Thus, complete moment convergence is stronger than complete convergence. For more details about the complete moment convergence, we refer the readers to Wang and Hu [27], Liang, Li, and Rosalsky [17], Wu, Cabrea, and, Volodin [30], Guo and Zhu [7] and Shen, Xue, and Volodin [22], Shen et al [23] among others. Recently, Ko [13] extended Theorem A from complete convergence to complete moment convergence as follows.…”