Stable measures and central limit theorems in spaces of stable type

Marcus, Michael B.; Woyczyński, Wojbor A.

doi:10.1090/S0002-9947-1979-0531970-2

Stable measures and central limit theorems in spaces of stable type
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by Michael B. Marcus and Wojbor A. Woyczyński PDF

Trans. Amer. Math. Soc. 251 (1979), 71-102 Request permission

Abstract:

Let X be a symmetric random variable with values in a quasinormed linear space E. X satisfies the central limit theorem on E with index p, $0 < p \leqslant 2$, if $\mathcal {L}{n^{ - 1/p}}({X_1} + \cdots + {{\text {X}}_n}))$ converges weakly to some probability measure on E. Hoffman-Jorgensen and Pisier have shown that Banach spaces of stable type 2 provide a natural environment for the central limit theorem with index $p = 2$. In this paper we show that, for $0 < p < 2$, quasi-normed linear spaces of stable type p provide a natural environment for the central limit theorem with index p. A similar result holds also for the weak law of large numbers with index p.

References

Anatole Beck, A convexity condition in Banach spaces and the strong law of large numbers, Proc. Amer. Math. Soc. 13 (1962), 329–334. MR 133857, DOI 10.1090/S0002-9939-1962-0133857-9
Jean Bretagnolle, Didier Dacunha-Castelle, and Jean-Louis Krivine, Lois stables et espaces $L^{p}$, Ann. Inst. H. Poincaré Sect. B (N.S.) 2 (1965/1966), 231–259 (French). MR 0203757
I. Csiszár and Balram S. Rajput, A convergence of types theorem for probability measures on topological vector spaces with applications to stable laws, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 1, 1–7. MR 420761, DOI 10.1007/BF00533204
J. L. Doob, Stochastic processes, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. Reprint of the 1953 original; A Wiley-Interscience Publication. MR 1038526
R. M. Dudley and Marek Kanter, Zero-one laws for stable measures, Proc. Amer. Math. Soc. 45 (1974), 245–252. MR 370675, DOI 10.1090/S0002-9939-1974-0370675-9
B. Gnedenko, To the theory of the domains of attraction of stable laws, Uchenye Zapiski Moskov. Gos. Univ. Matematika 30 (1939), 61–81 (Russian, with English summary). MR 0002042

Sums of independent Banach space valued random variables

15

Jørgen Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), 159–186. MR 356155, DOI 10.4064/sm-52-2-159-186
J. Hoffmann-Jørgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probability 4 (1976), no. 4, 587–599. MR 423451, DOI 10.1214/aop/1176996029
Naresh C. Jain and Michael B. Marcus, Integrability of infinite sums of independent vector-valued random variables, Trans. Amer. Math. Soc. 212 (1975), 1–36. MR 385995, DOI 10.1090/S0002-9947-1975-0385995-7
Michael B. Marcus, Uniform convergence of random Fourier series, Ark. Mat. 13 (1975), 107–122. MR 372994, DOI 10.1007/BF02386200
Jean-Pierre Kahane, Some random series of functions, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0254888
M. Kłosowska, The domain of attraction of a non-Gaussian stable distribution in a Hilbert space, Colloq. Math. 32 (1974), 127–136. MR 365646, DOI 10.4064/cm-32-1-127-136

Bemerkungen zu meiner Arbeit “Über die Summen zufälliger Grössen"

102

J. L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1976), no. 1, 1–29. MR 407568, DOI 10.2307/1971054
J. Kuelbs and V. Mandrekar, Harmonic analysis on certain vector spaces, Trans. Amer. Math. Soc. 149 (1970), 213–231. MR 301162, DOI 10.1090/S0002-9947-1970-0301162-2
J. Kuelbs and V. Mandrekar, Domains of attraction of stable measures on a Hilbert space, Studia Math. 50 (1974), 149–162. MR 345155, DOI 10.4064/sm-50-2-149-162
S. Kwapień, Sums of independent Banach space valued random variables (after J. Hoffmann-Jørgensen), Séminaire Maurey-Schwartz Année 1972–1973: Espaces $L^{p}$ et applications radonifiantes, Exp. No. 6, Centre de Math., École Polytech., Paris, 1973, pp. 6. MR 0436246
Michael B. Marcus, Some new results on central limit theorems for $C(S)$-valued random variables, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 526, Springer, Berlin, 1976, pp. 167–186. MR 0451329
Michael B. Marcus and Wojbor A. Woyczyński, Domaine d’attraction normal dans les espaces de type stable $p$, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 14, A915–A917 (French, with English summary). MR 451330

Espaces de cotype p

Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90

Sur l’existence d’une topologie du type de Sazonov sur une espace de Banach

V. V. Petrov, Summy nezavisimykh sluchaĭ nykh velichin, Izdat. “Nauka”, Moscow, 1972 (Russian). MR 0322927
G. Pisier, Une propriété du type $p$-stable, Séminaire Maurey-Schwartz 1973–1974: Espaces $L^{p}$, applications radonifiantes et gèmétrie des espaces de Banach, Exp. No. 8, Centre de Math., École Polytech., Paris, 1974, pp. 10 pp. (errata, p. E.1) (French). MR 0399811

Sur les espaces qui ne contiennent pas de

uniformement

277

E. J. G. Pitman, Some theorems on characteristic functions of probability distributions. , Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 393–402. MR 0132568
E. L. Rvačeva, On domains of attraction of multi-dimensional distributions, Select. Transl. Math. Statist. and Probability, Vol. 2, American Mathematical Society, Providence, R.I., 1962, pp. 183–205. MR 0150795
A. Tortrat, Lois et $(\lambda )$ dans les espaces vectoriels et lois stables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 37 (1976/77), no. 2, 175–182. MR 428371, DOI 10.1007/BF00536779
K. Urbanik and W. A. Woyczyński, A random integral and Orlicz spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 161–169 (English, with Russian summary). MR 215329
W. A. Woyczyński, Geometry and martingales in Banach spaces, Probability—Winter School (Proc. Fourth Winter School, Karpacz, 1975) Lecture Notes in Math., Vol. 472, Springer, Berlin, 1975, pp. 229–275. MR 0394131
W. A. Woyczyński, Weak convergence to a Gaussian measure of martingales in Banach spaces, Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), Academic Press, London, 1977, pp. 319–331. MR 0482990

Functional analysis

Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
Stefan Rolewicz, Metric linear spaces, 2nd ed., PWN—Polish Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984. MR 802450