Empirical and Poisson Processes on Classes of Sets or Functions Too Large for Central Limit Theorems

Summary

Let P be the uniform probability law on the unit cube I ^d in d dimensions, and P _n the corresponding empirical measure. For various classes С of sets A ⊂ I ^d, upper and lower bounds are found for the probable size of sup {|P _n − P)(A)|: A ∈ С}. If С is the collection of lower layers in I ², or of convex sets in I ³, an asymptotic lower bound is

$$\left( {{{\left( {\log n} \right)} \mathord{\left/ {\vphantom {{\left( {\log n} \right)} n}} \right.} n}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)\left( {\log \log n} \right)^{ - \delta - {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}} \quad {\rm{for}}\,{\rm{any}}\,\delta > 0.$$

Thus the law of the iterated logarithm fails for these classes. If α > 0, β is the greatest integer <α, and 0 < K < ∞, let С be the class of all sets $\{ x_d \underline \le \,f\,(x_1,\, \ldots,\,x_{d - 1} )\}$ where f has all its partial derivatives of orders $\underline \le \,\beta$ bounded by K and those of order β satisfy a uniform Hölder condition $|D^p (f(x)\, - \,f(y))|\,|\underline \le K|x\, - \,y|^{\alpha - \beta }$. For 0 < α < d − 1 one gets a universal lower bound δn ^{−α/(d − 1 + α)}, for a constant δ = δ(d, α) > 0. When α = d − 1 the same lower bound is obtained as for the lower layers in I ² or convex sets in I ³. For $0 < \,\alpha \,\underline \le \,d\, - \,1$ there is also an upper bound equal to a power of log n times the lower bound, so the powers of n are sharp.

The research was partially supparted by National Science Foundation Grants MCS-79-04474