On a certain exponential inequality for Gaussian processes

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Integral inequalities for generalized concave or convex functions

AbstractLet ψ be convex with respect to ϑ, B a convex body in Rn and f a positive concave function on B. A well-known result by Berwald states that 1¦B¦∝B ψ(f(x)) dx ⩽ n ∝01 ψ(ξt)(1 − t)n − 1) dt (1) if ξ is chosen such that 1¦B¦∝B ϑ(f(x)) dx = n ∝01 ϑ(ξt)(1 − t)n − 1) dt.The main purpose in this paper is to characterize those functions f : B → R+ such that (1) holds

Minkowski sums and Brownian exit times

If C is a domain in R^n, the Brownian exit time of C is denoted by T^C. Given domains C and D in R^n this paper gives an upper bound of the distribution function of T^(C+D) when the distribution functions of T^C and T^D are known. The bound is sharp if C and D are parallel affine half-spaces. The paper also exhibits an extension of the Ehrhard inequality

Inequalities of the Brunn-Minkowski type for Gaussian measures

Let m 2 be an integer, let γ be the standard Gaussian measure on Rn}, and let Φ(t)=∫-∞}t exp (-s2/2)ds sqrt{2π}{small} -∞ t Le ∞. Given α 1}l̇, αm} ] 0,∞ this paper gives a necessary and sufficient condition such that the inequality Φ-1} (γ (α1}A1}+ċ+αm}A m} α1}Φ-1}(γA 1)+ċ+ αm}Φ-1}(γA m) is true for all Borel sets A 1,...,A m in hbfRn} of strictly positive γ-measure or all convex Borel sets A 1,...,A m in bfRn} of strictly positive γ-measure, respectively. In particular, the paper exhibits inequalities of the Brunn-Minkowski type for γ which are true for all convex sets but not for all measurable sets

Minkowski sums and Brownian exit times

If C is a domain in R^n, the Brownian exit time of C is denoted by T^C. Given domains C and D in R^n this paper gives an upper bound of the distribution function of T^(C+D) when the distribution functions of T^C and T^D are known. The bound is sharp if C and D are parallel affine half-spaces. The paper also exhibits an extension of the Ehrhard inequality