The Central Limit Theorem and Poincare-Type Inequalities.
Abstract
We use Poincare type inequalities to prove the sufficiency and necessity of the Lindeberg condition in the central limit theorem. The central limit theorem is a fundamental theorem in probability and statistics. It states that the probability distribution of the sum of a large number of small and mutually independent random numerical observations approaches a normal distribution as the number of observations increases. The Lindeberg condition is a condition for which the central limit theorem holds. It has been proved to be both necessary and sufficient. A Poincare type inequality is an inequality which relates the integral of the square of a function to the integral of the square of its derivative. In this report we give a new proof of the central limit theorem by using Poincare type inequalities to prove both the necessity and sufficiency of the Lindeberg condition.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1986
- Accession Number
- ADA167534
Entities
People
- Louis H. Y. Chen
Organizations
- University of Wisconsin–Madison