Mathematics – Probability
Scientific paper
2010-07-17
Mathematics
Probability
24 pages, submitted for publication
Scientific paper
Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate fi?nite sets of points at a prede?ned resolution gamma > 0. We establish a connection between the gap dimension of F and the uniform convergence of its sample averages under ergodic sampling. In particular, we show that if the gap dimension of F at resolution gamma > 0 is fi?nite, then for every ergodic process the sample averages of functions in F are eventually within 10 gamma of their limiting expectations uniformly over the class F. If the gap dimension of F is finite for every resolution gamma > 0 then the sample averages of functions in F converge uniformly to their limiting expectations. We assume only that F is uniformly bounded and countable (or countably approximable). No smoothness conditions are placed on F, and no assumptions beyond ergodicity are placed on the sampling processes. Our results extend existing work for i.i.d. processes.
Adams Terrence M.
Nobel Andrew B.
No associations
LandOfFree
The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-522777