On the reversed sub-martingale property of empirical discrepancies in arbitrary sample spaces
- 33 Downloads
The empirical discrepancy is defined as a supremum over a class of functions of a collection of centered sample averages. For uncountable classes the discrepancy need not be measurable, and distributional assertions can become dependent on the structure of the underlying probability space. This paper shows that one such assertion—the reversed sub-martingale property—is valid when interpreted in terms of measurable cover functions for the canonical model, but that it can fail in other constructions of the underlying model.
Key WordsEmpirical discrepancies reversed sub-martingale property canonical model measurable cover function
Unable to display preview. Download preview PDF.
- 3.Gaenssler, P. (1983).Empirical Processes. IMS Lecture Notes—Monograph Series, Vol. 3, Institute of Mathematical Statistics, Hayward, California.Google Scholar
- 4.Halmos, P. R. (1950).Measure Theory, Van Nostrand, Princeton.Google Scholar
- 5.Pollard, D. (1984).Convergence of Stochastic Processes, Springer, New York.Google Scholar