Abstract
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.
Highlights
Let I be a real interval and consider probability measures μ on I with moments μk = I xk dμ(x), k =, . . . , such that μ =
It is well known that the latter plays a crucial role in the construction of explicit bounds to probability measures and integrals given a fixed number of moments
An improved version, which takes into account the sample moments of order three and four, has been derived in Hürlimann [ ]
Summary
For computational purposes it suffices to know that if a solution exists, the atoms of the random variable solving AMP(r) must be identical with the distinct real zeros of the orthogonal polynomial qr(x) of degree r = n + , as shown by the following precise recipe. R – , for a random variable Y with the ‘moments’ {μk}k= , ,..., n+ , and plays the role of the ‘non-existent’ orthogonal polynomial of degree r = n + How, in this modified setting, does the solution of the system AMP(r), as generated by qr(y), looks like?. (Chebyshev-Markov-Stieltjes inequalities) Let FX(x) be the distribution function of an arbitrary random variable X defined on (–∞, ∞) with the moments {μk}k= , ,..., n.
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