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

Abstract

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 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.