The subindependence of coordinate slabs inl ⁿ_p balls

Abstract

It is proved that if the probabilityP is normalised Lebesgue measure on one of thel ⁿ_p balls in Rⁿ, 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 ⁿ₁ 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.