Perissinaki, Irini; (1998) Probabilistic inequalities for special convex bodies. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

It is proved that if the probability P is normalized Lebesgue measure on one of the lnp ball in Rn, then for any sequence t1,t2,....tn of positive numbers, the coordinate slabs {|xi|?ti} are subindependent, namely, [diagram] A consequence of this result is that the proportion of the volume of the unit ln1 ball which is inside the cube [-t,t]n is less than or equal to fn(t) = (1-(1-t)n)n. This estimate is remarkably accurate over most of the range of values of t. A reverse inequality, demonstrating this, is the second major result of this work. A similar phenomenon occurs for all lnp balls. A consequence of the subindependence of the coordinate slabs of the lnp balls, is a sort of Central Limit Theorem which is examined in the last chapter. This states that as n ? ?, the average (n - 1)-dimensional volume of the sections of the normalised lnp ball at distance t from the origin, tends to a Gaussian. In other words, if g0 is density of the marginal of the lnp ball, in direction 0, then [diagram] for each t, uniformaly in p.

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