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The sequential uniform law of large numbers
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Abstract
LetZn (s;f) = n1 P [ns ]i=1 (f(Xi) Pf) be the sequen-tial empirical process based on the independent and identically dis-tributed random variables. We prove that convergence problemsof sup(s;f) jZn (s;f)j to zero boil down to those of supf jZn (1;f)j.We employ Ottaviani's inequality and the complete convergence toestablish, under bracketing entropy with the second moment, thealmost sure convergence of sup(s;f) jZn (s;f)j to zero.
참고문헌 (8)
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- "SpringerTexts in Statistics" (1993), Springer-Verlag
- S. Shreve, "Stochastic calculus and finance, Springer Finance" (2004), Springer-Verlag
- "Empirical processes in M-estimation, Cambridge in Statististical and Probabilistic Mathematics" (2000), Cambridge University Press
- "Weak convergence and empirical processes with applications to statistics, Springer series in Statistics" (1996), Springer-Verlag
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