An explicit version of the Chebyshev-Markov-Stieltjes inequalities and its applications

Journal of inequalities and applications

Werner Hü¹ 

关键词: Hamburger moment problem;    algebraic moment problem;    orthogonal polynomials;    Laplace formula;    Christoffel-Darboux kernel;    Laguerre-Samuelson bound;    value-at-risk;    15A15;    15A24;    35F05;    42C05;    60E15;    65C60;    91B16;    91G10;   

DOI  :  10.1186/s13660-015-0709-1

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

Given is the Borel probability space on the set of real numbers. The algebraic-analytical structure of the set of all finite atomic random variables on it with a given even number of moments is determined. It is used to derive an explicit version of the Chebyshev-Markov-Stieltjes inequalities suitable for computation. These inequalities are based on the theory of orthogonal polynomials, linear algebra, and the polynomial majorant/minorant method. The result is used to derive generalized Laguerre-Samuelson bounds for finite real sequences and generalized Chebyshev-Markov value-at-risk bounds. A financial market case study illustrates how the upper value-at-risk bounds work in the real world.

【 授权许可】

CC BY