An exponential inequality for a NOD sequence and a strong law of large numbers

On an exponential inequality and a strong law of large numbers for monotone measures

“…When µ is a probability measure, Corollary 3.4 improves the corresponding results derived by many researchers [4,8,9,13,15,16] (see Table 1). Note that these results were obtained under different constraints.…”

“…Several statisticians attempts to find the fastest convergence rate in the strong law of large numbers on dependent random variables [5] has led to the creation of different types of exponential inequality. This motivates us to improve the inequalities which were obtained by many researchers [4,8,9,13,15,16]. On the other hand, some problems in mathematical economics, statistics, quantum mechanics and finance cannot be well analyzed by additive probabilities.…”

On the exponential inequalities for negatively dependent random variables

“…Recently, Wang et al [18] established an exponential inequality for identically distributed negatively dependent random variables with the finite Laplace transforms.…”

The consistency for estimator of nonparametric regression model based on NOD errors

“…They pointed out that NA random variables are NOD random variables, but neither NUOD nor NLOD implies NA. Various results and examples of NOD random variables can be found in Joag-Dev and Proschan [17], Bozorgnia et al [18], Asadian et al [19], Wang et al [20], Wu [21,22], Wang et al [23,24], Li et al [25] and Sung [26], etc. Obviously, by the definitions of NOD and pairwise NQD, NOD random variables are pairwise NQD random variables.…”