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| ON THE NECESSARY AND SUFFICIENT CONDITION OFTHE EXISTENCE OF QUASI INVARIANT MEASURES |
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Citation： |
ZHANG YINNAN.ON THE NECESSARY AND SUFFICIENT CONDITION OFTHE EXISTENCE OF QUASI INVARIANT MEASURES[J].Chinese Annals of Mathematics B,1981,2(2):217~224 |
| Page view： 980
Net amount： 783 |
Authors： |
ZHANG YINNAN; |
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| Abstract： |
If E is a separable type-2 Banach space and Esub<0>sub is a linear subspace of E, then the following are equivalent:
(a) There exists a probability measure \[\mu \] on E, Which is \[{E_{\text{0}}}\]-quasi-invariant.
(b) There exists a sequence \[({X_n}) \subset E\] such that \[\sum {{e_n}(\omega ){X_n}} \] converges a.s., where \[{{e_n}(\omega )}\] are indepondend identically distributed symmetric stable random variables of
index 2,i,e.\[E(\exp (it{\kern 1pt} {\kern 1pt} {e_n}(\omega ))) = exp( - \frac{{{t^2}}}{2})\]for all real t, and
\[{E_{\text{0}}} \subset \{ x,x = \sum {{\lambda _n}{X_n}} ,\forall ({\lambda _n}) \in {l_2}\} \]
In this note we prove that \[\sum {{\lambda _n}{X_n}} \] is convergent. |
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