Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence

TL;DR

This paper extends classical limit theorems like WLLN, SLLN, and CLT to include deleting items partial sums, offering new insights into asymptotic bias estimators of sample expectation and variance.

Contribution

It develops deleting items theorems for WLLN, SLLN, and CLT, extending classical limit theory and enabling new bias estimation methods.

Findings

Derived deleting items theorems for WLLN, SLLN, and CLT.

Extended classical limit theory to deleting items sequences.

Constructed asymptotic bias estimators for sample expectation and variance.

Abstract

The deleting items theorems of weak law of large numbers (WLLN),strong law of large numbers (SLLN) and central limit theorem (CLT) are derived by substituting partial sum of random variable sequence with deleting items partial sum. We address the background of deleting items limit theory of random variable sequence, discuss the classical limit theory of Chebyshev WLLN, Bernoulli WLLN and Khinchine WLLN with standard mathematical analytical technique, then develop the deleting items theorems of WLLN, SLLN and CLT based on convergence theorems and Slutsky's theorem. Our theorems extend the classical limit theory of random variable sequence and provide the construction of some asymptotic bias estimators of sample expectation and variance.