Rosenthal inequality for NOD sequences and its applications

Abstract

Rosenthal inequality for NOD (negatively orthant dependent) random variable sequences is established. As its applications, two theorems of complete convergence of weighted sums for arrays of NOD random variables are given, which extend the corresponding known results.

Whole‐brain mapping of monosynaptic afferent inputs to the CRH neurons in the medial prefrontal cortex of mice

“…The hippocampus, as part of the limbic system of the brain, plays an important role in long-term memory and spatial navigation (Cossart & Khazipov, 2022). The CPu is involved in the origin of epileptic focus and is related to the reestablishment of limbic epileptic networks, which may be responsible for the targeted behavioral seizures (Gan, 2004).…”

General Theorems on Exponential and Rosenthal's Inequalities

“…The following exponential inequality was presented for negatively orthant dependent r.v. 's in Lemma 3 of [3] and for extended negatively dependent r.v. 's in Lemma 1.2 of [11].…”

“…The proof applies the classical ideas of Fuk and Nagaev [2] (see also [6]). The same method was used in the proofs of Lemma 3 in [3] and Lemma 1.2 in [11]. Now we shall study Hoeffding's inequality.…”

Moment inequality of the minimum for nonnegative negatively orthant dependent random variables

“…A number of limit theorems for NOD random variables have been established by many authors. We refer to Volodin [25] for the Kolmogorov exponential inequality, Asadian et al [5] and Gan et al [7] for the Rosenthals type inequality, Kim [12] for Hájek-Rényi type inequality, Amini et al [1,3], Ko and Kim [10], Klesov et al [13], Wu and Zhu [33], Wu [28], Shen [19,21] and Wu et al [31] for almost sure convergence, Wu and Jiang [32] for the strong consistency of M estimator in a linear model, Kuczmaszewska [14], Taylor et al [24], Wang et al [27] and Sung [22] for exponential inequalities, Amini and Bozorgnia [2], Wu [29,30], Qiu et al [18], Zarei and Jabbari [34], Sung [23], Wang et al [26] and Shen [20] for complete convergence, and so forth.…”