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February, 1980 Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables
Walter Philipp
Ann. Probab. 8(1): 68-82 (February, 1980). DOI: 10.1214/aop/1176994825


Suppose that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index $\alpha$. Then without changing its distribution, one can redefine the sequence on a new probability space such that these partial sums converge in probability and consequently even in $L^p (p < \alpha)$ to the corresponding stable process. This provides a new method to prove functional central limit theorems and related results. A similar theorem holds for stationary $\phi$-mixing sequences of random variables.


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Walter Philipp. "Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables." Ann. Probab. 8 (1) 68 - 82, February, 1980.


Published: February, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0426.60033
MathSciNet: MR556415
Digital Object Identifier: 10.1214/aop/1176994825

Primary: 60F05
Secondary: 60B10

Keywords: Banach space valued random variables , domains of attraction , Invariance principles , mixing sequences of random variables , Stable laws

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 1 • February, 1980
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