Summary
Let
,n≧1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,
,P). Under weak metric entropy conditions on
,n≧1, and under growth conditions on
we show that there are non-zero numerical constantsC 1 andC 2 such that
where α(n) is a non-decreasing function ofn related to the metric entropy of
. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.
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Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany
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Yukich, J.E. Some limit theorems for the empirical process indexed by functions. Probab. Th. Rel. Fields 74, 71–90 (1987). https://doi.org/10.1007/BF01845640
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DOI: https://doi.org/10.1007/BF01845640
Keywords
- Entropy
- Growth Condition
- Stochastic Process
- Characteristic Function
- Probability Theory