THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS

Title & Authors
THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS
Bae, Jong-Sig; Kim, Sung-Yeun;

Abstract
Let $\small{Z_n(s,\;f)=n^{-1}\;{\sum}^{ns}_{i=1}(f(X_i)-Pf)}$ be the sequential empirical process based on the independent and identically distributed random variables. We prove that convergence problems of $\small{sup_{(s,\;f)}|Z_n(s,\;f)|}$ to zero boil down to those of $\small{sup_f|Z_n(1,\;f)|}$. We employ Ottaviani's inequality and the complete convergence to establish, under bracketing entropy with the second moment, the almost sure convergence of $\small{sup_{(s,\;f)}|Z_n(s,\;f)|}$ to zero.
Keywords
sequential Glivenko-Cantelli class;Ottaviani's inequality;complete convergence;almost sure convergence;uniform law of large numbers;
Language
English
Cited by
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