# THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS

• Bae, Jong-Sig (Department of Mathematics and Institute of Basic Science, Sungkyunkwan University) ;
• Kim, Sung-Yeun (Department of Mathematics and Institute of Basic Science, Sungkyunkwan University)
• Published : 2006.08.01

#### Abstract

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

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