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The fractional Fisher information and the central limit theorem for stable laws. (English) Zbl 1366.60064
Summary: A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of $$n$$ independent random variables to the information contained in sums over subsets containing $$n-1$$ of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate.

##### MSC:
 60F05 Central limit and other weak theorems 26A33 Fractional derivatives and integrals 94A17 Measures of information, entropy
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##### References:
 [1] Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379-423, 623-656 (1948) · Zbl 1154.94303 [2] Stam, AJ, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inf. Control, 2, 101-112, (1959) · Zbl 0085.34701 [3] Blachman, NM, The convolution inequality for entropy powers, IEEE Trans. Inf. Theory, 2, 267-271, (1965) · Zbl 0134.37401 [4] Lieb, EH, Proof of an entropy conjecture of wehrl, Commun. Math. Phys., 62, 35-41, (1978) · Zbl 0385.60089 [5] Barron, AR, Entropy and the central limit theorem, Ann. Probab., 14, 336-342, (1986) · Zbl 0599.60024 [6] Johnson, O, Entropy inequalities and the central limit theorem, Stoch. Process. Appl., 88, 291-304, (2000) · Zbl 1045.60023 [7] Johnson, O; Barron, AR, Fisher information inequalities and the central limit theorem, Probab. Theory Relat. Fields, 129, 391-409, (2004) · Zbl 1047.62005 [8] Artstein, S; Ball, KM; Barthe, F; Naor, A, Solution of shannon’s problem on the monotonicity of entropy, J. Am. Math. Soc., 17, 975-982, (2004) · Zbl 1062.94006 [9] Artstein, S; Ball, KM; Barthe, F; Naor, A, On the rate of convergence in the entropic central limit theorem, Probab. Theory Relat. Fields, 129, 381-390, (2004) · Zbl 1055.94004 [10] Madiman, M., Barron, A.R.: The monotonicity of information in the central limit theorem and entropy power inequalities. In: Proceedings of IEEE International Symposium Information Theory, pp. 1021-1025. Seattle, WA (2006) · Zbl 0848.60017 [11] Madiman, M; Barron, A, Generalized entropy power inequalities and monotonicity properties of information, IEEE Trans. Inf. Theory, 53, 2317-2329, (2007) · Zbl 1326.94034 [12] Tulino, AM; Verdú, S, Monotonic decrease of the non-gaussianness of the sum of independent random variables: A simple proof, IEEE Trans. Inform. Theory, 52, 4295-4297, (2006) · Zbl 1320.60111 [13] Carlen, EA; Soffer, A, Entropy production by block variable summation and central limit theorems, Commun. Math. Phys., 140, 339-371, (1991) · Zbl 0734.60024 [14] Brown, L.D.: A proof of the central limit theorem motivated by the Cramér-Rao inequality, in Statistics and Probability: Essays in Honor of C.R. Rao, pp. 141-148. Amsterdam, The Netherlands: North-Holland (1982) · Zbl 1047.62005 [15] Linnik, YuV, An information-theoretic proof of the central limit theorem with the lindeberg condition, Theory Probab. Appl., 4, 288-299, (1959) · Zbl 0097.13103 [16] Bobkov, SG; Chistyakov, GP; Götze, F, Fisher information and convergence to stable laws, Bernoulli, 20, 1620-1646, (2014) · Zbl 1315.60031 [17] Bobkov, SG; Chistyakov, GP; Götze, F, Bounds for characteristic functions in terms of quantiles and entropy, Electron. Commun. Probab., 17, 1-9, (2012) · Zbl 1258.60022 [18] Bobkov, SG; Chistyakov, GP; Götze, F, Fisher information and the central limit theorem, Probab. Theory Relat. Fields, 159, 1-59, (2014) · Zbl 1372.60018 [19] Feller, W.: An introduction to probability theory and its applications, vol II., Second edn. John Wiley & Sons Inc., New York (1971) · Zbl 0219.60003 [20] Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge (1954) · Zbl 0056.36001 [21] Laha, R.G., Rohatgi, v.K.: Probability theory, John Wiley & Sons, New York-Chichester-Brisbane, Wiley Series in Probability and Mathematical Statistics (1979) · Zbl 0409.60001 [22] Furioli, G; Pulvirenti, A; Terraneo, E; Toscani, G, On rosenau-type approximations to fractional diffusion equations, Commun. Math. Sci., 13, 1163-1191, (2015) · Zbl 1332.35379 [23] Klafter, J; Zumofen, G; Shlesinger, MF; Mallamace, F (ed.); Stanley, HE (ed.), Long-tailed distributions and non-Brownian transport in complex systems, 85-94, (1997), Amsterdam [24] Benson, DA; Wheatcraft, SW; Meerschaert, MM, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36, 1413-1423, (2000) · Zbl 1064.93035 [25] Chaves, AS, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239, 13-16, (1998) · Zbl 1026.82524 [26] Gorenko, R; Mainardi, F, Fractional calculus and stable probability distributions, Arch. Mech., 50, 377-388, (1998) · Zbl 0934.35008 [27] Molz, FJ; Fix, GJ; Lu, S, A physical interpretation for the fractional derivative in levy diffusion, Appl. Math. Lett., 15, 907-911, (2002) · Zbl 1043.76056 [28] Schumer, R; Benson, DA; Meerschaert, MM; Wheatcraft, SW, Eulerian derivation of the fractional advection-dispersion equation, J. Contam. Hydrol., 48, 69-88, (2001) [29] Caffarelli, L; Vazquez, JL, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202, 537-565, (2011) · Zbl 1264.76105 [30] Carrillo, JA; Huang, Y; Santos, MC; Vázquez, JL, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure, J. Differ. Equ., 258, 736-763, (2015) · Zbl 1307.35311 [31] Vázquez, JL, Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc. (JEMS), 16, 769-803, (2014) · Zbl 1297.35279 [32] Vázquez, JL, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7, 857-885, (2014) · Zbl 1290.26010 [33] Riesz, M, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81, 1-223, (1949) · Zbl 0033.27601 [34] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970) · Zbl 0207.13501 [35] Lieb, EH, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118, 349-374, (1983) · Zbl 0527.42011 [36] Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman & Hall, London (1974) · Zbl 0334.62003 [37] Guo, D.: Relative Entropy and Score Function: New Information-Estimation Relationships through Arbitrary Additive Perturbation. In: Proceedings IEEE International Symposium Information Theory, 2009, pp. 814-818. Seoul, Korea (2009) · Zbl 1047.62005 [38] Hoeffding, W, A class of statistics with asymptotically normal distribution, Ann. Math. Stat., 19, 293-325, (1948) · Zbl 0032.04101 [39] Csiszar, I, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hung., 2, 299-318, (1967) · Zbl 0157.25802 [40] Kullback, S, A lower bound for discrimination information in terms of variation, IEEE Trans. Inf. Theory, 4, 126-127, (1967) [41] Johnson, O.: A de Bruijn identity for symmetric stable laws. arXiv:1310.2045v1 (2013) · Zbl 0527.42011 [42] Ibragimov, I.A., Linnik, Yu.V.: Independent and stationary sequences of random variables, With a supplementary chapter by I. A. Ibragimov and v. V. Petrov, Translation from the Russian edited by J. F. C. Kingman, pp. 443. Wolters-Noordhoff Publishing, Groningen (1971) · Zbl 0599.60024 [43] Bassetti, F; Ladelli, L; Matthes, D, Central limit theorem for a class of one-dimensional kinetic equations, Probab. Theory Relat. Fields, 150, 77-109, (2011) · Zbl 1225.82055 [44] Bassetti, F; Ladelli, L; Regazzini, E, Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model, J. Stat. Phys., 133, 683-710, (2008) · Zbl 1161.82337 [45] Wong, R, Distributional derivation of an asymptotic expansion, Proc. AMS, 80, 266-270, (1980) · Zbl 0441.41024 [46] Linnik, YuV, Linear forms and statistical criteria. II, Ukrainskii Mat. Zhournal, 5, 247-290, (1953) · Zbl 0052.36701 [47] Linnik, Y.V.: Linear forms and statistical criteria. I,II, Selected Transl. Math. Statist. and Prob., vol. 3, pp. 1-90. Amer. Math. Soc., Providence, RI (1962) · Zbl 1225.82055 [48] Kotz, S; Ostrovskii, IV, A mixture representation of the linnik distribution, Stat. Probab. Lett., 26, 61-64, (1996) · Zbl 0848.60017
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