|
Complete moment convergence for weighted sums of widely orthant-dependent random variables and its application in nonparametric regression models
Lu CHENG, Junjun LANG, Yan SHEN, Xuejun WANG
Front. Math. China. 2022, 17 (4): 571-590.
https://doi.org/10.1007/s11464-021-0915-8
We establish some results on the complete moment convergence for weighted sums of widely orthant-dependent (WOD) random variables, which improve and extend the corresponding results of Y. F. Wu, M. G. Zhai, and J. Y. Peng [J. Math. Inequal., 2019, 13(1): 251–260]. As an application of the main results, we investigate the complete consistency for the estimator in a nonparametric regression model based on WOD errors and provide some simulations to verify our theoretical results.
References |
Related Articles |
Metrics
|
|
Pseudo-orthogonality for graph 1-Laplacian eigenvectors and applications to higher Cheeger constants and data clustering
Antonio Corbo ESPOSITO, Gianpaolo PISCITELLI
Front. Math. China. 2022, 17 (4): 591-623.
https://doi.org/10.1007/s11464-021-0961-2
The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is an NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the eigenvalues of graph 1-Laplacian. In this paper, we first give new notations to describe the paths, among critical eigenvectors of the graph 1-Laplacian, realizing sets with prescribed genus. We introduce the pseudo-orthogonality to characterize m3(G), a special eigenvalue for the graph 1-Laplacian. Furthermore, we use it to give an upper bound for the third graph Cheeger constant h3(G), that is, h3(G) 6 m3(G). This is a first step for proving that the k-th Cheeger constant is the minimum of the 1-Laplacian Raylegh quotient among vectors that are pseudo-orthogonal to the vectors realizing the previous k - 1 Cheeger constants. Eventually, we apply these results to give a method and a numerical algorithm to compute m3(G), based on a generalized inverse power method.
References |
Related Articles |
Metrics
|
|
Distance signless Laplacian spectrum of a graph
Huicai JIA, Wai Chee SHIU
Front. Math. China. 2022, 17 (4): 653-672.
https://doi.org/10.1007/s11464-021-0986-6
Let G be a simple connected graph with n vertices. The transmission Tv of a vertex v is defined to be the sum of the distances from v to all other vertices in G, that is, Tv = Σu∈V duv, where duv denotes the distance between u and v. Let T1, ..., Tn be the transmission sequence of G. Let = (dij)n×n be the distance matrix of G, and be the transmission diagonal matrix diag(T1, ..., Tn). The matrix is called the distance signless Laplacian of G. In this paper, we provide the distance signless Laplacian spectrum of complete k-partite graph, and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius q(G).
References |
Related Articles |
Metrics
|
14 articles
|