Boundedness Index of Sequences: A Statistical Approach
- 1Department of Engineering Fundamental Sciences, Sivas University of Science and Technology, Sivas 58140, Türkiye
Abstract
Statistical boundedness weakens ordinary boundedness by allowing it to fail on a set of density zero, and this notion can be graded by an order ρ∈(0,1]. For a real or complex sequence x=(xk), we introduce the statistical boundedness index η(x), defined as the infimum of all orders ρ∈(0,1] for which x is statistically bounded of order ρ; when no such order exists, x is called index-free. For each r∈[0,1], we study the space Xr of indexed sequences with η(x)≤r, and show that these spaces increase strictly with r, forming a continuous chain that interpolates between the bounded sequences and the statistically bounded sequences. The relationship between Xr and Sbr is also clarified, and we establish the main structural and duality properties of each Xr. These results yield a fine-grained classification of the asymptotic behaviour of real and complex sequences.
Related
- Weak type (1,1) boundedness of Bochner--Riesz means at the critical index on quantum tori
- Statistical Foundations for a Google Play User-Review Sentiment Index: Signal Fusion, Shrinkage, Distributional Validation, and Dynamic Smoothing
- Statistical Learning of Conditional Single-Index U-Processes Under Local Stationarity and Missing-At-Random Functional Responses
- Uniform Kreiss boundedness does not imply strong Cesàro boundedness
- A note on the boundedness of the multidimensional Katugampola operator in Campanato spaces
BibTeX
@article{temizsu2026boundedness,
title = {{Boundedness Index of Sequences: A Statistical Approach}},
author = {Fatih Temizsu},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213938},
url = {https://www.mdpi.com/2227-7390/14/21/3938},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Boundedness Index of Sequences: A Statistical Approach
AU - Fatih Temizsu
PY - 2026
DA - 2026/10/28
JO - Mathematics
DO - 10.3390/math14213938
UR - https://www.mdpi.com/2227-7390/14/21/3938
AB - Statistical boundedness weakens ordinary boundedness by allowing it to fail on a set of density zero, and this notion can be graded by an order ρ∈(0,1]. For a real or complex sequence x=(xk), we introduce the statistical boundedness index η(x), defined as the infimum of all orders ρ∈(0,1] for which x is statistically bounded of order ρ; when no such order exists, x is called index-free. For each r∈[0,1], we study the space Xr of indexed sequences with η(x)≤r, and show that these spaces increase strictly with r, forming a continuous chain that interpolates between the bounded sequences and the statistically bounded sequences. The relationship between Xr and Sbr is also clarified, and we establish the main structural and duality properties of each Xr. These results yield a fine-grained classification of the asymptotic behaviour of real and complex sequences.
SN - 2227-7390
ER -