Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 61, Issue 3, pp 355–368

Empirical and Poisson processes on classes of sets or functions too large for central limit theorems


  • R. M. Dudley
    • Dept. of MathematicsMassachusetts Institute of Technology

DOI: 10.1007/BF00539835

Cite this article as:
Dudley, R.M. Z. Wahrscheinlichkeitstheorie verw. Gebiete (1982) 61: 355. doi:10.1007/BF00539835


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 A⊂I d , upper and lower bounds are found for the probable size of sup {¦P n −P) (A)¦∶ A ε ∉}. 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)−δ−1/2 for any δ>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 ∉ be the class of all sets {x d ≦f(x1,...,x d-1)} where f has all its partial derivatives of orders ≦ β bounded by K and those of order β satisfy a uniform Hölder condition ¦D p (f(x)−f(y))¦≦K¦x −y¦ α−β. For 0<α<d−1 one gets a universal lower bound δnα/(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<α≦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.

Copyright information

© Springer-Verlag 1982