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Conditional weak laws in Banach spaces

Author: Ana Meda
Journal: Proc. Amer. Math. Soc. 131 (2003), 2597-2609
MSC (2000): Primary 60F10, 60B10, 60F05, 60G50
Published electronically: November 27, 2002
MathSciNet review: 1974661
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Abstract: Let $(B,\Vert\cdot\Vert) $ be a separable Banach space. Let $Y, Y_1, Y_2, \ldots$ be centered i.i.d. random vectors taking values on $B$ with law $\mu$, $\mu (\cdot )=P(Y\in\cdot )$, and let $S_n =\sum_{i=1}^n Y_i.$ Under suitable conditions it is shown for every open and convex set $0 \notin D\subset B$ that $P\left( \Vert{\frac{{\displaystyle S_n}}{\displaystyle n}} - v_d \Vert> \varepsilon \Big\vert\frac{{\displaystyle S_n}}{\displaystyle n}\in D\right)$ converges to zero (exponentially), where $v_d$ is the dominating point of $D.$ As applications we give a different conditional weak law of large numbers, and prove a limiting aposteriori structure to a specific Gibbs twisted measure (in the direction determined solely by the same dominating point).

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Additional Information

Ana Meda
Affiliation: Departmento de Matemáticas, Cub. 132, Facultad de Ciencias, UNAM, Circuito Exterior s/n, Ciudad Universitaria, Coyoacán 04510, México D. F., México

Keywords: Conditional laws, dominating point, large deviations, Banach spaces
Received by editor(s): July 16, 2000
Received by editor(s) in revised form: March 21, 2002
Published electronically: November 27, 2002
Additional Notes: The author was supported in part by Grant PAPIIT-DGAPA IN115799 of UNAM, and the final version was written while holding a Postdoctoral position at IMP, México
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2002 American Mathematical Society

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