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Register a new userExtension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence
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Authors
Jingwei Liu
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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 Slutskys 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.
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Based on deleting-item central limit theory, the classical Donskers theorem of partial-sum process of independent and identically distributed (i.i.d.) random variables is extended to incomplete partial-sum process. The incomplete partial-sum process Donskers invariance principles are constructed and derived for general partial-sum process of i.i.d random variables and empirical process respectively, they are not only the extension of functional central limit theory, but also the extension of deleting-item central limit theory. Our work enriches the random elements structure of weak convergence.
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