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Central limit theorems for sub-linear expectation under the Lindeberg condition.

Cheng Hu1

  • 1School of Mathematics and Statistics, Shandong Normal University, Jinan, China.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

This study explores central limit theorems for sub-linear expectation, establishing key findings for independent, non-identically distributed random variables. The research provides a distance bound crucial for deriving these theorems under the Lindeberg condition.

Keywords:
CapacityCentral limit theoremG-normal distributionLindeberg conditionSub-linear expectation

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Area of Science:

  • Probability Theory
  • Mathematical Statistics

Background:

  • Sub-linear expectation provides a more general framework than traditional linear expectation.
  • Central limit theorems are fundamental in probability and statistics, describing the convergence of sums of random variables.

Purpose of the Study:

  • To investigate central limit theorems for sub-linear expectation with independent, non-identically distributed random variables.
  • To establish a distance bound between normalized sums and G-normal distributions.
  • To derive central limit theorems for capacity and summability methods under the Lindeberg condition.

Main Methods:

  • Derivation of a distance bound using properties of sub-linear expectation.
  • Application of the Lindeberg condition for convergence proofs.
  • Analysis of capacity and summability methods within the sub-linear expectation framework.

Main Results:

  • A novel distance bound is established for normalized sums of independent, non-identically distributed random variables under sub-linear expectation.
  • The central limit theorem for sub-linear expectation is derived under the Lindeberg condition.
  • The central limit theorem for capacity and capacity for summability methods are also obtained under the Lindeberg condition.

Conclusions:

  • The findings extend the applicability of central limit theorems to more general expectation frameworks.
  • The established bound and derived theorems offer valuable tools for analyzing random variables without identical distributions.
  • This work contributes to the theoretical understanding of probability and statistics in non-linear expectation spaces.