×

Recent developments on the moment problem. (English) Zbl 1386.60058

Summary: We consider univariate distributions with finite moments of all positive orders. The moment problem is to determine whether or not a given distribution is uniquely determined by the sequence of its moments. There is a huge literature on this classical topic. In this survey, we will focus only on the recent developments on the checkable moment-(in)determinacy criteria including Cramér’s condition, Carleman’s condition, Hardy’s condition, Krein’s condition and the growth rate of moments, which help us solve the problem more easily. Both Hamburger and Stieltjes cases are investigated. The former is concerned with distributions on the whole real line, while the latter deals only with distributions on the right half-line. Some new results or new simple (direct) proofs of previous criteria are provided. Finally, we review the most recent moment problem for products of independent random variables with different distributions, which occur naturally in stochastic modelling of complex random phenomena.

MSC:

60E05 Probability distributions: general theory
44A60 Moment problems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Akhiezer, NI: The Classical Problem of Moments and Some Related Questions of Analysis. Oliver & Boyd, Edinburgh (1965). [Original Russian edition: Nauka, Moscow (1961)]. · Zbl 0135.33803
[2] Berg, C: The cube of a normal distribution is indeterminate. Ann. Probab. 16, 910-913 (1988). · Zbl 0645.60018 · doi:10.1214/aop/1176991795
[3] Berg, C: From discrete to absolutely continuous solutions of indeterminate moment problems. Arab. J. Math. Sci. 4, 1-18 (1998). · Zbl 0962.44012
[4] Berg, C, Chen, Y, Ismail, MEH: Small eigenvalues of large Hankel matrices: the indeterminate case. Math. Scand. 91, 67-81 (2002). · Zbl 1018.47020 · doi:10.7146/math.scand.a-14379
[5] Berg, C, Christensen, JPR: Density questions in the classical theory of moments. Ann. Inst. Fourier (Grenoble). 31, 99-114 (1981). · Zbl 0437.42007 · doi:10.5802/aif.840
[6] Billingsley, P: Probability and Measures. 3rd edn. Wiley, New York (1995). · Zbl 0822.60002
[7] Carleman, T: Les Fonctions Quasi-analytiques. Gauthier-Villars, Paris (1926). · JFM 52.0255.02
[8] Chen, Y, Karagiannidis, GK, Lu, H, Cao, N: Novel approximations to the statistics of products of independent random variables and their applications in wireless communications. IEEE Trans. Veh. Tech. 61, 443-454 (2012). · doi:10.1109/TVT.2011.2178441
[9] Chihara, TS: On indeterminate Hamburger moment problems. Pacific J. Math. 27, 475-484 (1968). · Zbl 0167.41901 · doi:10.2140/pjm.1968.27.475
[10] Chihara, TS: A characterization and a class of distribution functions for the Stieltjes-Wigert polynomials. Canad. Math. Bull. 13, 529-532 (1970). · Zbl 0205.07604 · doi:10.4153/CMB-1970-098-7
[11] Chow, YS, Teicher, H: Probability Theory: Independence, Interchangeability, Martingales. 3rd edn. Springer, New York (1997). · Zbl 0891.60002 · doi:10.1007/978-1-4612-1950-7
[12] Christiansen, JS: The moment problem associated with the Stieltjes-Wigert polynomials. J. Math. Anal. Appl. 277, 218-245 (2003). · Zbl 1019.44005 · doi:10.1016/S0022-247X(02)00534-6
[13] Fischer, H: A History of the Central Limit Theorems: From Classical to Modern Probability Theory. Springer, New York (2011). · Zbl 1226.60004 · doi:10.1007/978-0-387-87857-7
[14] Fréchet, M, Shohat, J: A proof of the generalized second limit theorem in the theory of probability. Trans. Amer. Math. Soc. 33, 533-543 (1931). · Zbl 0002.28003 · doi:10.2307/1989421
[15] Galambos, J, Simonelli, I: Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions. Marcel Dekker, New York (2004). · Zbl 1172.60300
[16] Garnett, JB: Bounded Analytic Functions. Springer, New York (1981). · Zbl 0469.30024
[17] Goffman, C, Pedrick, G: First Course in Functional Analysis. Prentice Hall of India, New Delhi (2002). · Zbl 0122.11206
[18] Gómez, R, López-García, M: A family of heat functions as solutions of indeterminate moment problems. Int. J. Math. Math. Sci. Article ID 41526, 1-11 (2007). · Zbl 1156.44004
[19] Graffi, S, Grecchi, V: Borel summability and indeterminacy of the Stieltjes moment problem: Application to the anharmonic oscillators. J. Math. Phys. 19, 1002-1006 (1978). · Zbl 0432.40007 · doi:10.1063/1.523760
[20] Gut, A: On the moment problem. Bernoulli. 8, 407-421 (2002). · Zbl 1006.60016
[21] Gut, A: On the moment problem for random sums. J. Appl. Probab. 40, 797-802 (2003). · Zbl 1042.60006 · doi:10.1017/S0021900200019732
[22] Hamburger, H: Über eine Erweiterung des Stieltjesschen Momentenproblems (Teil I). Math. Ann. 81, 235-319 (1920). · JFM 47.0427.04 · doi:10.1007/BF01564869
[23] Hamburger, H: Über eine Erweiterung des Stieltjesschen Momentenproblems (Teil II). Math. Ann. 82, 120-164 (1921), 168-187 (1921). · JFM 48.0535.01
[24] Hardy, GH: On Stieltjes’ ‘problème des moments’. Messenger of Math. 46, 175-182 (1917). · JFM 46.0406.02
[25] Hardy, GH: On Stieltjes’ ‘problème des moments’. Messenger of Math. 47, 81-88 (1918). [Collected Papers of G.H. Hardy, Vol. VII, pp. 75-83, 84-91 (1979) Oxford University Press, Oxford].
[26] Heyde, CC: On a property of the lognormal distribution. J. Roy. Statist. Soc. Ser. B. 25, 392-393 (1963a). · Zbl 0114.33802
[27] Heyde, CC: Some remarks on the moment problem (I). Quart. J. Math. Oxford. 14, 91-96 (1963b). · Zbl 0112.10005
[28] Hörfelt, P: The moment problem for some Wiener functionals: corrections to previous proofs (with an appendix by H. L. Pedersen). J. Appl. Probab. 42, 851-860 (2005). · Zbl 1138.60335 · doi:10.1017/S0021900200000826
[29] Kjeldsen, TH: The early history of the moment problem. Historia Math. 20, 19-44 (1993). · Zbl 0769.01008 · doi:10.1006/hmat.1993.1004
[30] Klebanov, LB, Mkrtchyan, ST: Estimation of the closeness of distributions in terms of identical moments. In: Stability Problems for Stochastic Models, (Proc. Fourth All-Union Sem., Palanga, 1979) (Russian), Zobotarev, VM, Kalashnikov, VV (eds.), pp. 64-72, Moscow (1980). Translations: J. Soviet Math. 32, 54-60 (1986); Selected Translations in Mathematical Statistics and Probability 16, 1-10 (Estimating the proximity of distributions in terms of coinciding moments) (1985). · Zbl 1120.60007
[31] Kleiber, C: On moment indeterminacy of the Benini income distribution. Statist. Papers. 54, 1121-1130 (2013). · Zbl 1277.60030 · doi:10.1007/s00362-013-0535-9
[32] Kleiber, C: The generalized lognormal distribution and the Stieltjes moment problem. J. Theor. Probab. 27, 1167-1177 (2014). · Zbl 1310.60009 · doi:10.1007/s10959-013-0477-0
[33] Koosis, P: The Logarithmic Integral I. Cambridge Univ. Press, Cambridge (1988). · Zbl 0665.30038 · doi:10.1017/CBO9780511566196
[34] Kopanov, P, Stoyanov, J: Lin’s condition for functions of random variables and moment determinacy of probability distributions. C. R. Bulg. Acad. Sci. 70, 611-618 (2017). · Zbl 1389.60006
[35] Krein, M: On a problem of extrapolation of A.N. Kolmogoroff. Comptes Rendus (Doklady) l’Academie Sci l’URSS XLVI, 306-309 (1945). [Dokl. Akad. Nauk SSSR 46, 339-342, (1944)]. · Zbl 1358.60030
[36] Lariviere, MA: A note on probability distributions with increasing generalized failure rates. Oper. Res. 54, 602-604 (2006). · Zbl 1167.60309 · doi:10.1287/opre.1060.0282
[37] Leipnik, R: The lognormal distribution and strong non-uniqueness of the moment problem. Theory Probab. Appl. 26 863-865 (1981, Russian edition). SIAM version, 850-852 (1982). · Zbl 0488.60024
[38] Lin, GD: On the moment problems. Statist. Probab. Lett. 35, 85-90 (1997). Erratum: ibid 50, 205 (2000). · Zbl 0904.62021 · doi:10.1016/S0167-7152(96)00220-9
[39] Lin, GD, Huang, JS: The cube of a logistic distribution is indeterminate. Austral. J. Statist. 39, 247-252 (1997). · Zbl 0898.62015 · doi:10.1111/j.1467-842X.1997.tb00689.x
[40] Lin, GD, Stoyanov, J: On the moment determinacy of the distribution of compound geometric sums. J. Appl. Probab. 39, 545-554 (2002). · Zbl 1026.60015 · doi:10.1017/S0021900200021781
[41] Lin, GD, Stoyanov, J: The logarithmic skew-normal distributions are moment-indeterminate. J. Appl. Probab. 46, 909-916 (2009). · Zbl 1175.60013 · doi:10.1017/S0021900200005945
[42] Lin, GD, Stoyanov, J: Moment determinacy of powers and products of nonnegative random variables. J. Theoret. Probab. 28, 1337-1353 (2015). · Zbl 1358.60030 · doi:10.1007/s10959-014-0546-z
[43] Lin, GD, Stoyanov, J: On the moment determinacy of products of non-identically distributed random variables. Probab. Math. Statist. 36, 21-33 (2016). · Zbl 1342.60014
[44] López-García, M: Characterization of solutions to the log-normal moment problem. Theory Probab. Appl. 55, 303-307 (2011). · Zbl 1225.60024 · doi:10.1137/S0040585X97984875
[45] Ostrovska, S: Constructing Stieltjes classes for M-indeterminate absolutely continuous probability distributions. ALEA. Lat. Am. J. Probab. Math. Stat. 11, 253-258 (2014). · Zbl 1290.60015
[46] Ostrovska, S: On the powers of polynomial logistic distributions. Braz. J. Probab. Stat. 30, 676-690 (2016). · Zbl 1376.60032 · doi:10.1214/15-BJPS298
[47] Ostrovska, S, Stoyanov, J: Stieltjes classes for M-indeterminate powers of inverse Gaussian distributions. Statist. Probab. Lett. 71, 165-171 (2005). · Zbl 1081.62006 · doi:10.1016/j.spl.2004.10.024
[48] Pakes, AG: Length biasing and laws equivalent to the log-normal. J. Math. Anal. Appl. 197, 825-854 (1996). · Zbl 0852.60013 · doi:10.1006/jmaa.1996.0056
[49] Pakes, AG: Remarks on converse Carleman and Krein criteria for the classical moment problem. J. Aust. Math. Soc. 71, 81-104 (2001). · Zbl 0992.44003 · doi:10.1017/S1446788700002731
[50] Pakes, AG: Structure of Stieltjes classes of moment-equivalent probability laws. J. Math. Anal. Appl. 326, 1268-1290 (2007). · Zbl 1120.60007 · doi:10.1016/j.jmaa.2006.03.035
[51] Pakes, AG: On generalized stable and related laws. J. Math. Anal. Appl. 411, 201-222 (2014). · Zbl 1349.60020 · doi:10.1016/j.jmaa.2013.09.041
[52] Pakes, AG, Hung, W-L, Wu, J-W: Criteria for the unique determination of probability distributions by moments. Aust. N.Z. J. Statist. 43, 101-111 (2001). · Zbl 0997.60001 · doi:10.1111/1467-842X.00158
[53] Pakes, AG, Khattree, R: Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist. 34, 307-322 (1992). · Zbl 0759.62005 · doi:10.1111/j.1467-842X.1992.tb01363.x
[54] Pedersen, HL: On Krein’s theorem for indeterminacy of the classical moment problem. J. Approx. Theory. 95, 90-100 (1998). · Zbl 0924.44009 · doi:10.1006/jath.1998.3188
[55] Penson, KA, Blasiak, P, Duchamp, GHE, Horzela, A, Solomon, AI: On certain non-unique solutions of the Stieltjes moment problem. Discrete Math. Theor. Comput. Sci. 12, 295-306 (2010). · Zbl 1213.60043
[56] Prohorov, YuV, Rozanov, YuA: Probability Theory. Translated by K. Krickeberg and H. Urmitzer. Springer, New York (1969). · Zbl 0186.49701
[57] Rao, CR, Shanbhag, DN, Sapatinas, T, Rao, MB: Some properties of extreme stable laws and related infinitely divisible random variables. J. Statist. Plann. Inference. 139, 802-813 (2009). · Zbl 1158.60313 · doi:10.1016/j.jspi.2008.05.050
[58] Serfling, RJ: Approximation Theorems of Mathematical Statistics. Wiley, New York (1980). · Zbl 0538.62002 · doi:10.1002/9780470316481
[59] Shohat, JA, Tamarkin, JD: The Problem of Moments. Amer. Math. Soc.New York (1943). · Zbl 0063.06973
[60] Slud, EV: The moment problem for polynomial forms in normal random variables. Ann. Probab. 21, 2200-2214 (1993). · Zbl 0788.60028 · doi:10.1214/aop/1176989017
[61] Springer, MD: The Algebra of Random Variables. Wiley, New York (1979). · Zbl 0399.60002
[62] Stieltjes, TJ: Recherches sur les fractions continues. Ann. Fac. Sci. Univ. Toulouse Math.8(J), 1-122 (1894). 9(A), 1-47 (1895). Also in: Stieltjes, T.J. Oeuvres Completes. Noordhoff, Gröningen 2, 402-566 (1918). · JFM 25.0326.01
[63] Stirzaker, D: The Cambridge Dictionary of Probability and Its Applications. Cambr. Univ. Press, Cambridge (2015). · Zbl 1359.60001
[64] Stoyanov, J: Inverse Gaussian distribution and the moment problem. J. Appl. Statist. Sci. 9, 61-71 (1999). · Zbl 0947.60012
[65] Stoyanov, J: Krein condition in probabilistic moment problems. Bernoulli. 6, 939-949 (2000). · Zbl 0971.60017 · doi:10.2307/3318763
[66] Stoyanov, J: Stieltjes classes for moment-indeterminate probability distributions. J. Appl. Probab. 41A, 281-294 (2004). · Zbl 1070.60012 · doi:10.1239/jap/1082552205
[67] Stoyanov, JM: Counterexamples in Probability. 3rd edn. Dover Publications, New York (2013). [First and second edns: Chichester: Wiley, 1987 and 1997]. · Zbl 1287.60004
[68] Stoyanov, J, Kopanov, P: Lin’s condition and moment determinacy of functions of random variables. Submitted (2017). · Zbl 1389.60006
[69] Stoyanov, J, Lin, GD: Hardy’s condition in the moment problem for probability distributions. Theory Probab. Appl. 57, 811-820 (2012) (Russian edition). SIAM version, 699-708 (2013). · Zbl 1295.60020
[70] Stoyanov, J, Lin, GD, DasGupta, A: Hamburger moment problem for powers and products of random variables. J. Statist. Plann. Inference. 154, 166-177 (2014). · Zbl 1300.60031 · doi:10.1016/j.jspi.2013.11.002
[71] Stoyanov, J, Tolmatz, L: New Stieltjes classes involving generalized gamma distributions. Statist. Probab. Lett. 69, 213-219 (2004). · Zbl 1066.60016 · doi:10.1016/j.spl.2004.06.032
[72] Stoyanov, J, Tolmatz, L: Method for constructing Stieltjes classes for M-indeterminate probability distributions. Appl. Math. Comput. 165, 669-685 (2005). · Zbl 1069.60016
[73] Targhetta, ML: On a family of indeterminate distributions. J. Math. Anal. Appl. 147, 477-479 (1990). · Zbl 0701.62023 · doi:10.1016/0022-247X(90)90362-J
[74] Wang, J: Constructing Stieltjes classes for power-order M-indeterminate distributions. J. Appl. Probab. Statist. 7, 41-52 (2012). · Zbl 1275.60022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.