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Let F be a family of Borel measurable functions on a complete separablemetric space. The gap (or fat-shattering) dimension of F is a combinatorialquantity that measures the extent to which functions f in F can separate finitesets of points at a predefined resolution gamma > 0. We establish a connectionbetween the gap dimension of F and the uniform convergence of its sampleaverages under ergodic sampling. In particular, we show that if the gapdimension of F at resolution gamma > 0 is finite, then for every ergodicprocess the sample averages of functions in F are eventually within 10 gamma oftheir limiting expectations uniformly over the class F. If the gap dimension ofF is finite for every resolution gamma > 0 then the sample averages offunctions in F converge uniformly to their limiting expectations. We assumeonly that F is uniformly bounded and countable (or countably approximable). Nosmoothness conditions are placed on F, and no assumptions beyond ergodicity areplaced on the sampling processes. Our results extend existing work for i.i.d.processes.

The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes