, Volume 107, Issue 1, pp 289-299

The subindependence of coordinate slabs inl p n balls

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Abstract

It is proved that if the probabilityP is normalised Lebesgue measure on one of thel p n balls in R n , then for any sequencet 1 , t 2 , …, t n of positive numbers, the coordinate slabs {|x i |≤t i } are subindependent, namely, \(P\left( {\mathop \cap \limits_1^n \{ \left| {x_i } \right| \leqslant t_i \} } \right) \leqslant \prod\limits_1^n {P(\{ \left| {x_i } \right| \leqslant t_i \} )} \) . A consequence of this result is that the proportion of the volume of thel 1 n ball which is inside the cube[−1, t] n is less than or equal tof n (t)=(1−(1−t) n ) n . It turns out that this estimate is remarkably accurate over most of the range of values oft. A reverse inequality, demonstrating this, is the second major result of the article.

Supported in part by NSF DMS-9257020.
Supported by a grant from Public Benefit Foundation Alexander S. Onassis. This work will form part of a Ph.D. thesis written by the second-named author.