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An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions

Part of the Lecture Notes in Mathematics book series (LNM,volume 1153)

Abstract

Let (X, α, P) be a probability space, x(1),x(2), ... coordinates for the countable product of copies of X, and ℱ ⊂ ℒ2 (X, α, P). Let vn be the normalized empirical measure n−1/2x(1)+...+δx(n)-nP). Let ℓ(ℱ) be the set of all bounded real functions G on ℱ with norm ‖G‖ := supfεℱ ¦G(f)¦. Let μ(f) := εfdμ for any measure μ. We are interested in central limit theorems where vn converges in law for ‖·‖. The limit is a Gaussian process Gp with mean 0 and covariance

$$EG_P (f)G_P (g) = P(fg) - P(f)P(g): = (f,g)_{P,O} ,$$

f,g ε ℱ. Say ℱ is GPBUC if GP can be chosen to have each sample function bounded and uniformly continuous for (.,.)P,0. Say the central limit theorem holds for ℱ if ℱ is GPBUC and we have convergence of upper integrals limn→∞ ∝*H(vn)dpn = EH(GP) for every bounded continuous real function H on ℓ(ℱ). (Thus, as noted by J. Hoffmann-Jørgensen, convergence "in law" does not require definition of laws.) In Sec. 4 an extended Wichura’s theorem is proved: given such convergence in law, there exist almost surely convergent realizations. Sec. 5 shows that the new definition of central limit theorem holding is equivalent to the previous definition of "functional Donsker class." Secs. 6–7 treat the case where ℱ={M1h≥M: 0<M<∞} for a random variable h. This case reduces to the study of weighted empirical distribution functions. Conditions for the central limit theorem are collected and made precise.

Keywords

  • Probability Space
  • Central Limit Theorem
  • Gaussian Process
  • Outer Probability
  • Empirical Process

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Partially supported by National Science Foundation Grant MCS-8202122

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Dudley, R.M. (1985). An extended Wichura theorem, definitions of Donsker class, and weighted empirical distributions. In: Beck, A., Dudley, R., Hahn, M., Kuelbs, J., Marcus, M. (eds) Probability in Banach Spaces V. Lecture Notes in Mathematics, vol 1153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074949

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  • DOI: https://doi.org/10.1007/BFb0074949

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