The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
Citations
125 citations
125 citations
120 citations
120 citations
118 citations
Cites background or methods from "The integral of a symmetric unimoda..."
...A Banach space E is said to have type p ∈ [1, 2] if there exists a constant Cp > 0 such that for all finite sequences x1, ....
[...]
...6(1) can be given for Banach spaces E having type p ∈ [1, 2]....
[...]
...Theorem 8.2 (Anderson)....
[...]
...As an application of Anderson’s inequality we have the following comparison result for E-valued Gaussian random variables (see Neidhardt [93, Lemma 28])....
[...]
...Hence, by Fubini’s theorem and Anderson’s inequality, P{X2 ∈ C} = P̃{X̃1 + X̃3 ∈ C} 6 P{X1 ∈ C}....
[...]
References
3,082 citations
"The integral of a symmetric unimoda..." refers background in this paper
...In Theorem 1 the equality in (1) holds for k<l if and only if, for every u, (E+y)r\Ku=Er\Ku-\-y....
[...]
...It will be noticed that we obtain strict inequality in (1) if and only if for at least one u, H(u)>H*(u) (because H(u) is continuous on the left)....
[...]
1,660 citations
927 citations
140 citations
"The integral of a symmetric unimoda..." refers background in this paper
...f ud[H*(u) - H(u)} = b[H*(b) - H(b)] - a[H*(a) - H(a)} (3) " + f [(H(u) - H*(u)]du....
[...]
...J a Since/(x) has a finite integral over E, bH(b)—>0 as b—>oo and hence also bH*(b)—>0 as b—*<x>; therefore the first term on the right in (3) can be made arbitrarily small in absolute value....
[...]