The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component

Andersen, Niels T.; Giné, Evarist; Zinn, Joel

doi:10.1090/S0002-9947-1988-0930076-3

The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component
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Trans. Amer. Math. Soc. 308 (1988), 603-635 Request permission

Abstract:

Weak convergence results are obtained for empirical processes indexed by classes $\mathcal {F}$ of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes $\{ f(X): f \in \mathcal {F}\}$ ("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in ${\mathbf {R}}$ and its associated empirical measures.

References

Alejandro de Acosta, Inequalities for $B$-valued random vectors with applications to the strong law of large numbers, Ann. Probab. 9 (1981), no. 1, 157–161. MR 606806
Alejandro de Acosta, Aloisio Araujo, and Evarist Giné, On Poisson measures, Gaussian measures and the central limit theorem in Banach spaces, Probability on Banach spaces, Adv. Probab. Related Topics, vol. 4, Dekker, New York, 1978, pp. 1–68. MR 515429
N. T. Andersen and V. Dobrić, The central limit theorem for stochastic processes, Ann. Probab. 15 (1987), no. 1, 164–177. MR 877596
Niels T. Andersen, Evarist Giné, Mina Ossiander, and Joel Zinn, The central limit theorem and the law of iterated logarithm for empirical processes under local conditions, Probab. Theory Related Fields 77 (1988), no. 2, 271–305. MR 927241, DOI 10.1007/BF00334041

Probability inequalities for sums of bounded random variables

57

Miklós Csörgő, Sándor Csörgő, Lajos Horváth, and David M. Mason, Normal and stable convergence of integral functions of the empirical distribution function, Ann. Probab. 14 (1986), no. 1, 86–118. MR 815961
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
Evarist Giné and Joel Zinn, Central limit theorems and weak laws of large numbers in certain Banach spaces, Z. Wahrsch. Verw. Gebiete 62 (1983), no. 3, 323–354. MR 688642, DOI 10.1007/BF00535258
Evarist Giné and Joel Zinn, Empirical processes indexed by Lipschitz functions, Ann. Probab. 14 (1986), no. 4, 1329–1338. MR 866353
Evarist Giné and Joel Zinn, Some limit theorems for empirical processes, Ann. Probab. 12 (1984), no. 4, 929–998. With discussion. MR 757767
Evarist Giné and Joel Zinn, Lectures on the central limit theorem for empirical processes, Probability and Banach spaces (Zaragoza, 1985) Lecture Notes in Math., vol. 1221, Springer, Berlin, 1986, pp. 50–113. MR 875007, DOI 10.1007/BFb0099111
Bernard Heinkel, Majorizing measures and limit theorems for $c_{0}$-valued random variables, Probability in Banach spaces, IV (Oberwolfach, 1982) Lecture Notes in Math., vol. 990, Springer, Berlin, 1983, pp. 136–149. MR 707514, DOI 10.1007/BFb0064268

Some exponential inequalities with applications to the central limit theorem in

Stochastic processes on Polish spaces

Naresh C. Jain and Michael B. Marcus, Central limit theorems for $C(S)$-valued random variables, J. Functional Analysis 19 (1975), 216–231. MR 0385994, DOI 10.1016/0022-1236(75)90056-7
D. Yuknyavichene, The central limit theorem in the space $C(S)$ and majorizing measures, Litovsk. Mat. Sb. 26 (1986), no. 2, 362–373 (Russian, with English and Lithuanian summaries). MR 862754
J. Kuelbs and Joel Zinn, Some stability results for vector valued random variables, Ann. Probab. 7 (1979), no. 1, 75–84. MR 515814
Michel Ledoux, Loi du logarithme itéré dans ${\cal C}(S)$ et fonction caractéristique empirique, Z. Wahrsch. Verw. Gebiete 60 (1982), no. 3, 425–435 (French). MR 664427, DOI 10.1007/BF00535725
M. Ledoux and M. Talagrand, Characterization of the law of the iterated logarithm in Banach spaces, Ann. Probab. 16 (1988), no. 3, 1242–1264. MR 942766
V. Mandrekar and J. Zinn, Central limit problem for symmetric case: convergence to non-Gaussian laws, Studia Math. 67 (1980), no. 3, 279–296. MR 592390, DOI 10.4064/sm-67-3-279-296
Michael B. Marcus, $\xi$-radial processes and random Fourier series, Mem. Amer. Math. Soc. 68 (1987), no. 368, viii+181. MR 897272, DOI 10.1090/memo/0368
M. B. Marcus and G. Pisier, Characterizations of almost surely continuous $p$-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), no. 3-4, 245–301. MR 741056, DOI 10.1007/BF02392199
M. B. Marcus and G. Pisier, Some results on the continuity of stable processes and the domain of attraction of continuous stable processes, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 2, 177–199 (English, with French summary). MR 749623
Vygantas Paulauskas and A. Yu. Rachkauskas, Operators of stable type, Litovsk. Mat. Sb. 24 (1984), no. 2, 145–159 (Russian, with English and Lithuanian summaries). MR 773603
Gilles Pisier, Remarques sur les classes de Vapnik-Červonenkis, Ann. Inst. H. Poincaré Probab. Statist. 20 (1984), no. 4, 287–298 (French, with English summary). MR 771890
William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0455094
Michel Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), no. 1-2, 99–149. MR 906527, DOI 10.1007/BF02392556

Necessary conditions for sample boundedness of

stable processes