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.
Publié le : 1980-02-14
Classification:
Invariance principles,
domains of attraction,
stable laws,
Banach space valued random variables,
mixing sequences of random variables,
60F05,
60B10
@article{1176994825,
author = {Philipp, Walter},
title = {Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 68-82},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994825}
}
Philipp, Walter. Weak and $L^p$-Invariance Principles for Sums of $B$-Valued Random Variables. Ann. Probab., Tome 8 (1980) no. 6, pp. 68-82. http://gdmltest.u-ga.fr/item/1176994825/