THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS

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

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