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 of sets AI ^d , upper and lower bounds are found for the probable size of sup {¦P _n –P) (A)¦ A }. If is the collection of lower layers in I ², or of convex sets in I ³, an asymptotic lower bound is ((log n)/n) ^1/2(log log n)^––1/2 for any >0. Thus the law of the iterated logarithm fails for these classes.If >0, is the greatest integer <, and 0, let be the class of all sets {x _d f(x₁,...,x _d-1)} where f has all its partial derivatives of orders bounded by K and those of order satisfy a uniform Hölder condition ¦D ^p (f(x)–f(y))¦K¦x –y¦ ^–. For 0<n^–/(d–1+) for a constant = (d,)>0. When = d-1 the same lower bound is obtained as for the lower layers in I ² or convex sets in I ³. For 0<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.This research was partially supported by National Science Foundation Grant MCS-79-04474