An explicit version of the ChebyshevMarkovStieltjes inequalities and its applications
 Werner Hürlimann^{1}Email author
https://doi.org/10.1186/s1366001507091
© Hürlimann 2015
Received: 16 December 2014
Accepted: 21 May 2015
Published: 13 June 2015
Abstract
Given is the Borel probability space on the set of real numbers. The algebraicanalytical 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 ChebyshevMarkovStieltjes 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 LaguerreSamuelson bounds for finite real sequences and generalized ChebyshevMarkov valueatrisk bounds. A financial market case study illustrates how the upper valueatrisk bounds work in the real world.
Keywords
Hamburger moment problem algebraic moment problem orthogonal polynomials Laplace formula ChristoffelDarboux kernel LaguerreSamuelson bound valueatriskMSC
15A15 15A24 35F05 42C05 60E15 65C60 91B16 91G101 Introduction
Let I be a real interval and consider probability measures μ on I with moments \(\mu_{k} = \int_{I} x^{k}\,d\mu (x)\), \(k = 0,1,2,\ldots\) , such that \(\mu_{0} = 1\). Given a sequence of real numbers \(\{ \mu_{k} \}_{k \ge 0}\), the moment problem on I consists to ask the following questions. Is there a μ on I with the given moments? If it exists, is the measure μ on I uniquely determined? If not, describe all μ on I with the given moments (see e.g. Kaz’min and Rehmann [1]).
There are essentially three different types of intervals. Either two endpoints are finite, typically \(I = [0,1]\), one endpoint is finite, typically \(I = [0,\infty)\), or no endpoints are finite, i.e. \(I = (  \infty,\infty )\). The corresponding moment problems are called Stieltjes moment problem if \(I = [0,\infty)\), the Hausdorff moment problem if \(I = [0,1]\), and the Hamburger moment problem if \(I = (  \infty,\infty )\). Besides this, the algebraic moment problem by Mammana [2] asks for the existence and construction of a finite atomic random variable with a given finite moment structure. 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.
In the present study, the focus is on the interval \(I = (  \infty,\infty )\). Motivated by a previous result from the author in a special case, we ask for a possibly explicit description of the set of all finite atomic random variables by given moment structure for an even arbitrary number of moments. Based on some basic results from the theory of orthogonal polynomials, as summarized in Section 2, we characterize in the main Theorem 3.1 these sets of finite atomic random variables. This constitutes a first specific answer to a research topic suggested by the author in the Preface of Hürlimann [3], namely the search for a general structure that belongs to the sets of finite atomic random variables by given moment structure. As an immediate application, we derive in Section 4 an explicit version of the ChebyshevMarkovStieltjes inequalities that is suitable for computation. It is used to derive generalized LaguerreSamuelson bounds for finite real sequences in Section 5, and generalized ChebyshevMarkov valueatrisk bounds in Section 6.
The historical origin of the ChebyshevMarkovStieltjes inequalities dates back to Chebyshev [4], who has first formulated this famous problem and has proposed also a solution without proof, however. Proofs were later given by Markov [5], Possé [6] and Stieltjes [7, 8]. Twentieth century developments include among others Uspensky [9], Shohat and Tamarkin [10], Royden [11], Krein [12], Akhiezer [13], Karlin and Studden [14], Freud [15] and Krein and Nudelman [16]. A short account is also found in Whittle [17], pp.110118. It seems that the ChebyshevMarkov inequalities have been stated in full generality for the first time by Zelen [18]. Explicit analytical results for moments up to order four have been given in particular by Zelen [18], Simpson and Welch [19], Kaas and Goovaerts [20], and Hürlimann [21] (see also Hürlimann [3]).
2 Basic classical results on orthogonal polynomials
Let \(( \Omega,A,P )\) be the Borel probability space on the set of real numbers such that Ω is the sample space, A is the σfield of events and P is the probability measure. For a measurable realvalued random variable X on this probability space, that is, a map \(X: \Omega \to R\), the probability distribution of X is defined and denoted by \(F_{X}(x) = P(X \le x)\).
Definition 2.1
The terminology ‘orthogonal’ refers to the scalar product induced by the expectation operator \(\langle X,Y \rangle = E[XY]\), where X, Y are random variables for which this quantity exists. An orthogonal polynomial is uniquely defined if either its leading coefficient is one (socalled monic polynomial) or \(\langle p_{n}(X),p_{n}(X) \rangle = 1\) (socalled orthonormal property). A monic orthogonal polynomial is throughout denoted by \(q_{n}(x)\) while an orthogonal polynomial with the orthonormal property is denoted by \(P_{n}(x)\). Some few classical results are required.
Lemma 2.1
(Chebyshev determinant representation)
The monic orthogonal polynomials form an orthogonal system of functions.
Lemma 2.2
(Orthogonality relations)
Lemma 2.3
(Christoffel [23] and Darboux [24])
Proof
This is shown in Akhiezer [13], Chapter 1, Section 2, p.9. □
Note that Brezinski [25] has shown that the ChristoffelDarboux formula is equivalent to the wellknown three term recurrence relation for orthogonal polynomials. For further information on orthogonal polynomials consult the recent introduction by Koornwinder [26] as well as several books by Szegö [27], Freud [15], Chihara [28], Nevai [29], etc.
3 An explicit solution to the algebraic moment problem
Lemma 3.1
(Mammana [2])
Theorem 3.1
(Discrete random variables on \(( \infty,\infty)\) with an even number of given moments \(\{ \mu_{k} \}_{k = 0,1,\ldots,2n}\))
Remark 3.1
Before proceeding with the proof, it is important to note that Theorem 3.1 is a generalization of the special case \(n = 2\) in Hürlimann [21], Proposition II.2.
Proof
4 An explicit version of the ChebyshevMarkovStieltjes inequalities
As a main application of Theorem 3.1, we get a completely explicit version of the ChebyshevMarkovStieltjes inequalities, which go back to Chebyshev, Markov, Possé and Stieltjes, and have been stated and studied at many places. Our formulation corresponds essentially to the inequalities found in the appendix of Zelen [18] for the infinite interval \(( \infty,\infty)\), and find herewith an implementation suitable for computation, as will be demonstrated in the next two sections. The probabilistic bounds (4.1) below are nothing else than the special case \(f(x) = 1\) of (5.10), Section 1.5, in Freud [15]. For the interested reader the essential steps of the modern probabilistic proof are sketched.
Theorem 4.1
(ChebyshevMarkovStieltjes inequalities)
Proof
5 Generalized LaguerreSamuelson bounds
5.1 The theoretical bounds
Theorem 5.1
(Generalized LaguerreSamuelson \((LS)\) bounds)
5.2 The applied bounds
It is interesting to observe that by increasing order of approximation the generalized LS bounds seem to converge to the true empirical bound. Attained bounds (up to four significant figures) are marked in bold face. For completely symmetric Cauchy sequences the analysis of this property has been suggested to us by corresponding simulations for the order \(m=2\). One can ask for a rigorous proof (or disproof) of it.
In Applied Statistics one does not usually encounter completely symmetric sequences. In empirical studies or simulations, the property \(s_{2k1}\neq 0\), \(k=1,\ldots,m\), is rather the rule than the exception. It is therefore worthwhile to present some simulations for arbitrary sequences stemming from distributions defined on the whole real line, whether symmetric or not.
Simulation of generalized LS bounds for completely symmetric sequences
Normal distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  3.531  2.744  2.523  2.451  2.427  2.418  2.416  2.415  2.414  2.414 
10^{3}  31.61  6.774  4.478  3.829  3.574  3.467  3.425  3.408  3.400  3.397  3.395 
10^{4}  100  11.94  6.308  4.806  4.204  3.911  3.752  3.660  3.604  3.569  3.507 
10^{5}  316.2  21.15  9.247  6.491  5.579  5.261  5.158  5.125  5.112  5.106  5.103 
Gamma distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  4.125  3.204  2.947  2.869  2.848  2.842  2.841  2.840  2.840  2.840 
10^{3}  31.61  7.869  5.302  4.549  4.268  4.164  4.128  4.116  4.111  4.110  4.109 
10^{4}  100  14.61  8.468  6.863  6.310  6.101  6.007  5.948  5.901  5.863  5.792 
10^{5}  316.2  26.85  13.97  11.26  10.43  10.16  10.06  10.03  10.01  10.01  10.01 
Lognormal distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  4.376  3.483  3.184  3.073  3.017  2.983  2.968  2.960  2.953  2.925 
10^{3}  31.61  15.04  13.36  13.27  13.26  13.26  13.26  13.26  13.26  13.26  13.26 
10^{4}  100  25.98  20.90  20.16  20.10  20.09  20.09  20.09  20.09  20.09  20.09 
10^{5}  316.2  44.26  30.35  27.01  26.12  25.90  25.84  25.82  25.82  25.82  25.82 
Cauchy distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  6.643  6.438  6.423  6.423  6.423  6.423  6.423  6.423  6.423  6.423 
10^{3}  31.61  17.33  14.69  14.07  13.93  13.92  13.92  13.92  13.92  13.92  13.92 
10^{4}  100  57.78  52.70  52.10  52.08  52.08  52.08  52.08  52.08  52.08  52.08 
10^{5}  316.2  197.1  186.7  186.2  186.2  186.2  186.2  186.2  186.2  186.2  186.2 
Simulation of generalized LS bounds for arbitrary sequences
Normal distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  3.607  2.780  2.590  2.520  2.479  2.454  2.438  2.432  2.431  2.431 
10^{3}  31.61  6.712  4.346  3.687  3.472  3.411  3.394  3.391  3.390  3.390  3.390 
10^{4}  100  11.91  6.327  4.838  4.263  4.023  3.926  3.889  3.874  3.868  3.865 
10^{5}  316.2  21.18  9.256  6.462  5.510  5.202  5.119  5.100  5.095  5.094  5.094 
Laplace distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  4.302  3.739  3.677  3.670  3.670  3.670  3.670  3.670  3.670  3.670 
10^{3}  31.61  8.897  6.661  6.063  5.809  5.693  5.633  5.597  5.574  5.557  5.542 
10^{4}  100  14.74  8.720  7.196  6.687  6.528  6.483  6.469  6.465  6.464  6.463 
10^{5}  316.2  26.74  13.54  10.49  9.355  8.849  8.602  8.479  8.411  8.373  8.333 
Pearson type VII distribution ( α = 3)  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  7.665  7.646  7.646  7.646  7.646  7.646  7.646  7.646  7.646  7.646 
10^{3}  31.61  22.16  21.80  21.79  21.79  21.79  21.79  21.79  21.79  21.79  21.79 
10^{4}  100  88.94  88.80  88.80  88.80  88.80  88.80  88.80  88.80  88.80  88.80 
10^{5}  316.2  227.6  225.2  225.2  225.2  225.2  225.2  225.2  225.2  225.2  225.2 
Bowers distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  6.291  6.063  6.050  6.050  6.050  6.050  6.050  6.050  6.050  6.050 
10^{3}  31.61  13.71  11.21  10.36  10.04  9.930  9.911  9.907  9.907  9.907  9.907 
10^{4}  100  38.67  34.27  33.36  33.26  33.25  33.25  33.25  33.25  33.25  33.25 
10^{5}  316.2  118.1  103.5  100.5  100.0  99.93  99.92  99.92  99.92  99.92  99.92 
Cauchy distribution  

n  Order of approximation  \(\boldsymbol {\max\frac{x_{i}\mu}{\sigma}}\)  
1  2  3  4  5  6  7  8  9  10  
10^{2}  9.950  9.269  9.260  9.260  9.260  9.260  9.260  9.260  9.260  9.260  9.260 
10^{3}  31.61  30.08  30.07  30.07  30.07  30.07  30.07  30.07  30.07  30.07  30.07 
10^{4}  100  95.31  95.28  95.28  95.28  95.28  95.28  95.28  95.28  95.28  95.28 
10^{5}  316.2  291.6  291.1  291.1  291.1  291.1  291.1  291.1  291.1  291.1  291.1 
6 Generalized ChebyshevMarkov VaR bounds
An improved version, which takes into account the sample moments of order three and four, has been derived in Hürlimann [21], Theorem 4.1, Case 1 (see also the monograph Hürlimann [3] for a more general exposition). This approach has been further investigated in Hürlimann [37–39]. Based on Theorem 4.1, it is shown how a sequence of increasingly accurate generalized ChebyshevMarkov (CM) VaR bounds can be constructed. Based on the first 10 order of approximations, we demonstrate then the use of the obtained algorithm through a reallife case study. A comparison with the estimated VaR of some important return distributions is instructive and justifies the application and further development of the CM VaR bounds. Similar generalizations to bounds for the conditional valueatrisk measure (CVaR) can also be obtained (see Hürlimann [21], Theorem 4.2, for the special case \(m = 2\)).
6.1 The theoretical bounds
Theorem 6.1
(Generalized CM VaR bound)
6.2 The applied bounds
 (i)Given a set of moments up to a fixed even order, one expects a single solution. This assertion may depend on the choice of a sufficiently small loss tolerance level, the properties of the largest real solution to the polynomial equation (6.4), as well as computational feasibility (representation of the higher order moments in machine precision). In this respect, the results of Table 3 are well behaved.Table 3
Generalized CM VaR bounds versus FFT VaR from best fitted return distributions
CM VaR bound of order m
SMI and SP500 data sets
SMI 3Y/1D
SP 3Y/1D
SMI 24Y/1D
SP 24Y/1D
SP 63Y/1M
1
14.10674
14.10674
14.10674
14.10674
14.10674
2
5.70929
5.80257
6.38478
6.57694
5.52054
3
4.63181
5.08737
5.85193
5.97425
5.05367
4
4.33603
5.08073
5.81340
5.97368
5.03519
5
4.24284
4.51830
5.15713
5.19075
4.07623
6
4.20294
4.04401
5.01443
4.88268
3.91300
7
4.18175
3.98179
4.91795
4.88033
3.89962
8
4.17181
3.97972
4.51596
4.56437
3.41525
9
4.16756
3.96797
4.44315
4.29779
3.31145
10
4.16568
3.45184
4.42918
4.28693
3.30855
maximum loss
4.20421
5.93094
9.39704
8.18808
5.95942
Normal approx.
2.57583
2.57583
2.57583
2.57583
2.57583
FFT VaR
SMI 3Y/1D
SP500 3Y/1D
SMI 24Y/1D
SP 24Y/1D
SP 63Y/1D
VG
3.18228
3.09191
3.55423
3.69317
2.65234
NVG
3.15521
3.08703
3.42190
3.52504
2.49884
TLFBG
3.46466
3.58737
3.83378
3.92492
3.57468
TLF
3.42602
3.57239
3.78536
3.71739
3.54963
NIG
3.43097
3.56194
3.72618
3.78626
3.58135
NTS
3.44828
3.60691
3.62529
3.72720
3.54761
best fit FFT
NVG
NVG
NTS
TLF
NTS
max. FFT VaR
TLFBG
NTS
TLF_BG
TLF_BG
NIG
 (ii)
The evaluation of the CM VaR bounds is easy to implement and its computation is fast. In this respect the method overcomes the lack of fast computation encountered with the historical VaR and the Monte Carlo VaR models.
 (iii)
The CM VaR bounds yield a sequence of increasingly accurate upper bounds to VaR that are consistent with the actuarial principle of safe or conservative pricing.

SMI 3Y/1D: 758 historic daily closing prices over 3 years from 04.01.2010 to 28.12.2012

SMI 24Y/1D: 6030 historic daily closing prices over 24 years from 03.01.1989 to 28.12.2012

SP500 3Y/1D: 754 historic daily closing prices over 3 years from 04.01.2010 to 31.12.2012

SP500 24Y/1D: 6049 historic daily closing prices over 24 years from 03.01.1989 to 31.12.2012

SP500 63Y/1M: 756 historic monthly closing prices over 63 years from Jan. 1950 to Dec. 2012
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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