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


Authors: Niels T. Andersen, Evarist Giné and Joel Zinn
Journal: Trans. Amer. Math. Soc. 308 (1988), 603-635
MSC: Primary 60F17; Secondary 60F05
DOI: https://doi.org/10.1090/S0002-9947-1988-0930076-3
MathSciNet review: 930076
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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.


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  • [1] A. de Acosta (1981), Inequalities for $ B$-valued random vectors with applications to strong laws of large numbers, Ann. Probab. 9, 157-161. MR 606806 (83c:60009)
  • [2] A. de Acosta, A. Araujo and E. Giné (1978), On Poisson measures, Gaussian measures and the central limit theorem in Banach spaes, Advances in Probability and Related Topics, Vol. IV (J. Kuelbs, ed.), Dekker, New York, pp. 1-68. MR 515429 (80h:60009)
  • [3] N. T. Andersen and V. Dobrić (1987), The central limit theorem for stochastic processes, Ann. Probab. 15, 164-177. MR 877596 (88f:60009)
  • [4] N. T. Andersen, E. Giné, M. Ossiander and J. Zinn (1988), The central limit theorem and the law of iterated logarithm for empirical processes under local conditions, Probability Theory and Related Fields, 77, 271-308. MR 927241 (89g:60009)
  • [5] G. W. Bennett (1962), Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 57, 33-45.
  • [6] M. Csörgö, S. Csörgö, L. Horvath and D. M. Mason (1986), Normal and stable convergence of integral functions of the empirical distribution function, Ann. Probab. 14, 86-118. MR 815961 (87d:60036)
  • [7] W. Feller (1971), An introduction to probability theory and its applications, Vol. II, 2nd ed., Wiley, New York. MR 0270403 (42:5292)
  • [8] E. Giné and J. Zinn (1983), Central limit theorems and weak laws of large numbers in certain Banach spaces, Z. Wahrsch. Verw. Gebiete 62, 323-354. MR 688642 (84j:60017)
  • [9] E. Giné and J. Zinn (1986), Empirical processes indexed by Lipschitz functions, Ann. Probab. 14, 1329-1338. MR 866353 (88i:60064)
  • [10] E. Giné and J. Zinn (1984), Some limit theorems for empirical processes, Ann. Probab. 12, 929-989. MR 757767 (86f:60028)
  • [11] E. Giné and J. Zinn (1986), Lectures on the central limit theorem for empirical processes, Lecture Notes in Math., vol. 1221, Springer-Verlag, New York and Berlin, pp. 50-113. MR 875007 (88i:60063)
  • [12] B. Heinkel (1983), Majorizing measures and limit theorems for $ {c_0}$-valued random variables, Lecture Notes in Math., vol. 990, Springer-Verlag, New York and Berlin, pp. 136-149. MR 707514 (85e:60006)
  • [13] -, (1987), Some exponential inequalities with applications to the central limit theorem in $ C[0,\,1]$ (to appear).
  • [14] J. Høffmann-Jorgensen (1984), Stochastic processes on Polish spaces (to appear).
  • [15] N. Jain and M. Marcus (1975), Central limit theorem for $ C(S)$-valued random variables, J. Funct. Anal. 19, 216-231. MR 0385994 (52:6853)
  • [16] D. Juknevičienè (1986), On the central limit theorem in the space $ c(s)$ and majorizing measures, Lietuvos Mat. Rink. 26, 362-373. MR 862754 (89a:60025)
  • [17] J. Kuelbs and J. Zinn (1979), Some stability results for vector valued random variables, Ann. Probab. 7, 75-84. MR 515814 (80h:60014)
  • [18] M. Ledoux (1982) Loi du logarithme itéré dans $ C(S)$ et fonction characteristique empirique, Z. Wahrsch. Verw. Gebiete 60, 425-435. MR 664427 (83k:60035)
  • [19] M. Ledoux and M. Talagrand (1986), Characterization of the law of the iterated logarithm in Banach spaces, Ann. Probab. (to appear). MR 942766 (89i:60016)
  • [20] V. Mandrekar and J. Zinn (1980), Central limit problem for symmetric case: Convergence to non-Gaussian laws, Studia Math. 67, 279-296. MR 592390 (82f:60026)
  • [21] M. B. Marcus (1987) $ \xi $-radial processes and random Fourier series, Mem. Amer. Math. Soc., vol. 68, no. 368. MR 897272 (88k:60071)
  • [22] M. Marcus and G. Pisier (1984), Characterization of almost surely continuous $ p$-stable random Fourier series and strongly stationary processes, Acta Math. 152, 245-301. MR 741056 (86b:60069)
  • [23] M. Marcus and G. Pisier (1984), Some results on the continuity of stable processes and the domain of attraction of continuous stable processes, Ann. Inst. H. Poincaré 20, 177-199. MR 749623 (86b:60070)
  • [24] V. Panlauskas and A. Račkauskas (1984), On operators of state type, Lietuvos Mat. Rink. 24, 145-159. MR 773603 (86m:60011)
  • [25] G. Pisier (1984), Remarques sur les classes de Vapnik-Červonenkis, Ann. Inst. H. Poincaré Probab. Statist. 20, 287-298. MR 771890 (86h:60010)
  • [26] W. Stout (1974), Almost sure convergence, Academic Press, New York. MR 0455094 (56:13334)
  • [27] M. Talagrand (1987), Regularity of Gaussian processes, Acta Math. 159, 99-149. MR 906527 (89b:60106)
  • [28] -, (1986), Necessary conditions for sample boundedness of $ p$-stable processes (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0930076-3
Keywords: Central limit theorems, law of the iterated logarithm, empirical processes, Marcinkiewicz laws of large numbers, bracketing conditions, majorizing measures
Article copyright: © Copyright 1988 American Mathematical Society

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