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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable measures and central limit theorems in spaces of stable type


Authors: Michael B. Marcus and Wojbor A. Woyczyński
Journal: Trans. Amer. Math. Soc. 251 (1979), 71-102
MSC: Primary 60B12; Secondary 60E07
MathSciNet review: 531970
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0531970-2
PII: S 0002-9947(1979)0531970-2
Keywords: Stable measures, domains of attraction, weak law of large numbers, random integral, quasi-normed space, space of stable type p
Article copyright: © Copyright 1979 American Mathematical Society