Sentence examples for extend the complete from inspiring English sources

The main purpose of this paper is to extend the complete convergence for weighted sums of random variables to the case of -mixing random variables.

In this paper we extend the complete convergence for weighted sums of dependent random fields to the case that different indices have different powers in the normalization.

for all ϵ > 0, where S j = ∑ i ≤ j X i. Next we extend the complete convergence for weighted sums of ρ ∗ -mixing random field to the case that indices have different powers in the normalization.

In this paper, based on Gut and Stadtmüller (2011 [1]) and Qiu and Chen (2014 [2]), we extend the complete convergence and complete moment theorems for independent random variables to the negatively associated sequences of random variables without necessarily imposing any extra conditions, which extend the corresponding results of Gut and Stadtmüller (2011 [1]) and Qiu and Chen (2014 [2]).

The age at which children entered primary school was lowered from 7 to 6 years, thus extending the complete course of general-secondary schooling from 10 to 11 years.

In a word, it is meaningful that this paper extends the complete convergence and complete moment convergence under sub-linear expectation.

Baum and Katz [11] extended the complete convergence of Hsu and Robbins [10], Chow [12] first investigated the complete moment convergence.

Recently, many authors extended the complete convergence for an array of rowwise independent random variables to the cases of dependent random variables.

The result extends the complete moment convergence in (Ko in J. Inequal. Appl. 2016:131, 2016) to Hilbert spaces as well as generalizes the Baum-Katz type theorem in (Huan et al. in Acta Math. Hung. 144(1):1320149, 2014) to the complete moment convergence.

The result extends the complete moment convergence for NA random variables in (mathbb{R}^{1}) (the main result in Ko [1]) to a Hilbert space as well as generalizes the Baum-Katz type theorem (Theorem 2.1 in Huan et al. [2]) for CNA random vectors in a Hilbert space to the complete moment convergence in a Hilbert space.

In February 1853, the Commonwealth of Virginia had chartered the Covington and Ohio Railroad to extend the line completed by the Virginia Central westward across the Alleghany Mountains to the Ohio River.