Central Limit Theorems for Weighted Sums of Dependent Random Vectors in Hilbert Spaces via the Theory of the Regular Variation

2021 ◽  
Author(s):  
Ta Cong Son ◽  
Le Van Dung
Keyword(s):  
Central Limit ◽  
Hilbert Spaces ◽  
Regular Variation ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Weighted Sums ◽  
Random Vectors

scholarly journals Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhicheng Chen ◽  
Xinsheng Liu
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Random Vectors ◽  
Standard Normal ◽  
Almost Sure Limit Theorems ◽  
General Standard

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

CENTRAL LIMIT THEOREMS FOR WEIGHTED SUMS OF LINEAR PROCESSES: LP -APPROXIMABILITY VERSUS BROWNIAN MOTION

2009 ◽  
Vol 25 (3) ◽  
pp. 748-763 ◽  
Author(s):  
Kairat T. Mynbaev
Keyword(s):  
Brownian Motion ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
Limit Theorems ◽  
Stochastic Calculus ◽  
Central Limit Theorems ◽  
Weighted Sums ◽  
Linear Processes

Standardized slowly varying regressors are shown to be Lp-approximable. This fact allows us to provide alternative proofs of asymptotic expansions of nonstochastic quantities and central limit results due to P.C.B. Phillips, under a less stringent assumption on linear processes. The recourse to stochastic calculus related to Brownian motion can be completely dispensed with.

scholarly journals High-dimensional limit theorems for random vectors in ℓpn-balls

2019 ◽  
Vol 21 (01) ◽  
pp. 1750092 ◽  
Author(s):  
Zakhar Kabluchko ◽  
Joscha Prochno ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Large Deviation ◽  
Rate Function ◽  
Central Limit Theorems ◽  
High Dimensional ◽  
Random Vectors ◽  
Random Direction

In this paper, we prove a multivariate central limit theorem for [Formula: see text]-norms of high-dimensional random vectors that are chosen uniformly at random in an [Formula: see text]-ball. As a consequence, we provide several applications on the intersections of [Formula: see text]-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an [Formula: see text]-ball onto a line spanned by a random direction [Formula: see text]. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime [Formula: see text] this displays in speed and rate function deviations of the [Formula: see text]-norm on an [Formula: see text]-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

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