# An Extended Wichura Theorem, Definitions of Donsker Class, and Weighted Empirical Distributions

• R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

## Abstract

Let $$({\text{X}}, a >,{\text{P}})$$ be a probability space, x(1),x(2),··· coordinates for the countable product of copies of X, and $$\mathcal{F} > \subset \mathcal{L} > ^2 ({\text{X}}, a >,{\text{P}})$$. Let ν n be the normalized empirical measure n−1/2x(1)+···+δx(n)−nP). Let $$\ell ^\infty ( \mathcal{F} > )$$ be the set of all bounded real functions G on $$\mathcal{F} >$$ with norm $$||{\text{G}}||_{ \mathcal{F} > } : = \sup _{{\text{f}}{\varepsilon } \mathcal{F} > } |{\text{G(f)|}}$$. Let μ(f) := ∫ fdμ for any measure μ. We are interested in central limit theorems where ν n converges in law for $$|| \cdot ||_{ \mathcal{F} > }$$. The limit is a Gaussian process GP with mean 0 and covariance
$${\text{EG}}_{\text{P}} ({\text{f}}){\text{G}}_{\text{P}} {\text{(g)}} = {\text{P(fg)}} - {\text{P(f)P(g)}}: = ({\text{f,g}})_{{\text{P}},0},$$

$${\text{f,g}} \in \mathcal{F} >$$. Say $$\mathcal{F} >$$ 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 $$\mathcal{F} >$$ if $$\mathcal{F} >$$ is GPBUC and we have convergence of upper integrals $$\lim _{{\text{n}} \to \infty } \int_{}^{\ast} {\text{H}} ( v > _{\text{n}} ){\text{d}}{\text{P}}^{\text{n}} = {\text{EH}}({\text{G}}_{\text{P}} )$$ for every bounded continuous real function H on $$\ell ^\infty ( \mathcal{F} > )$$. (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 $$\mathcal{F} > = \left\{ {{\text{M}}1_{{\text{n}} \geq {\text{M}}} :0 < {\text{M}} < \infty } \right\}$$ 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

Statistical Theory Probability Space Central Limit Theorem Gaussian Process Real Function
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.

## References

1. Chibisov, D. M. (1964). Some theorems on the limiting behavior of the empirical distribution function. Trudy Mat. Inst. Steklov (Moscow) 71 104–112; Selected Transls. Math. Statist. Prob. 6 147–156.
2. Cohn, Donald L. (1980). Measure Theory. Birkhäuser, Boston.
3. Csörgő, M. (1984). Review of Shorack and Wellner (1982). Math. Revs. 84f:60041.Google Scholar
4. Donsker, M. D. (1952). Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems, Ann. Math. Statist. 23 pp. 277–281.
5. Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals. North-Holland, Amsterdam.
6. Dudley, R. M. (1966). Weak convergence of probabilities on nonseparable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 109–126.
7. _____ (1967a). Measures on non-separable metric spaces. Ibid. 11 449–453.Google Scholar
8. _____ (1967b). The sizes of compact subsets of Hilbert spaces and continuity of Gaussian processes. J. Functional Analysis 1 290–330.Google Scholar
9. _____ (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563–1572.Google Scholar
10. _____ (1973). Sample functions of the Gaussian process. Ann. Probab. 1 66–103.Google Scholar
11. _____ (1978). Central limit theorems for empirical measures. Ibid. 6 899–929; Correction, ibid. 7 (1979) 909–911.Google Scholar
12. _____ (1984). A course on empirical processes. Ecole d’été de probabilités de St.-Flour, 1982. Lecture Notes in Math. 1097 2–142.Google Scholar
13. _____, and Walter Philipp (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. verw. Geb. 62 509–552.Google Scholar
14. Erdös, P. (1942). On the law of the iterated logarithm. Ann. Math. 43 419–436.
15. Feldman, Jacob (1971). Sets of boundedness and continuity for the canonical normal process. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 357–368. Univ. Calif. Press.Google Scholar
16. Fernandez, Pedro J. (1974). Almost surely convergent versions of sequences which converge weakly. Bol. Soc. Brasil. Math. 5 51–61.
17. Halmos, P. (1950). Measure Theory. Princeton, Van Nostrand. 2d. printing, Springer, N. Y. 1974.Google Scholar
18. Hoffmann-Jørgensen, J., and Niels Trolle Andersen (1984). Personal communication.Google Scholar
19. _____ (1984). Envelopes and perfect random variables (preprint, section of forthcoming book).Google Scholar
20. Itô, K., and H. P. McKean Jr. (1974). Diffusion processes and their sample paths. Springer, N. Y. (2d. printing, corrected).Google Scholar
21. Marczewski, E., and R. Sikorski (1948). Measures in nonseparable metric spaces. Colloq. Math. 1 133–139.
22. O’Reilly, N. E. (1974). On the weak convergence of empirical processes in sup-norm metrics. Ann. Probab. 2 642–651.
23. Petrovskii, I. G. (1935). Zur ersten Randwertaufgabe der Wärmeleitungagleichung. Compositio Math. 1 383–419.
24. Pollard, D. B. (1982). A central limit theorem for empirical processes. J. Austral. Math. Soc. Ser. A 33 235–248.
25. Pyke, R. (1969). Applications of almost surely convergent constructions of weakly convergent processes. Lecture Notes in Math. 89 187–200.
26. Shorack, G. R. (1979). Weak convergence of empirical and quantile processes in sup-norm metrics via KMT-constructions. Stochastic Processes Applics. 9 95–98.
27. _____, and J. Wellner (1982). Limit theorems and inequalities for the uniform empirical process indexed by intervals. Ann. Probab. 10 639–652.Google Scholar
28. Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theor. Prob. Appls. 1 261–290 (English), 289–319 (Russian).Google Scholar
29. Stute, W. (1982). The oscillation behavior of empirical processes, Ann. Probab. 10 86–107.
30. Talagrand, M. (1984). The Glivenko-Cantelli problem (preprint).Google Scholar
31. Wichura, M. J. (1970). On the construction of almost uniformly convergent random variables with given weakly convergent image laws. Ann. Math. Statist. 41 284–291.