Independent functions and the geometry of Banach spaces

S. V. Astashkin   ; F. A. Sukochev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 65 (2010) no. 6, pp. 1003-1081

Résumé

The main objective of this survey is to present the ‘state of the art’ of those parts of the theory of independent functions which are related to the geometry of function spaces. The ‘size’ of a sum of independent functions is estimated in terms of classical moments and also in terms of general symmetric function norms. The exposition is centred on the Rosenthal inequalities and their various generalizations and sharp conditions under which the latter hold. The crucial tool here is the recently developed construction of the Kruglov operator. The survey also provides a number of applications to the geometry of Banach spaces. In particular, variants of the classical Khintchine–Maurey inequalities, isomorphisms between symmetric spaces on a finite interval and on the semi-axis, and a description of the class of symmetric spaces with any sequence of symmetrically and identically distributed independent random variables spanning a Hilbert subspace are considered. Bibliography: 87 titles.

Détail

Keywords: independent functions, Khintchine inequalities, Kruglov property, Rosenthal inequalities, Kruglov operator, symmetric space, Orlicz space, Lorentz space, Boyd indices, K-functional, real method of interpolation, integral-uniform norm.
Mots-clés : Marcinkiewicz space

Bibliographie

[1] H. P. Rosenthal, “On the subspaces of $L_p$ ($p>2$) spanned by sequences of independent random variables”, Israel J. Math., 8:3 (1970), 273–303 | DOI | MR | Zbl

[2] V. F. Gaposhkin, “Lacunary series and independent functions”, Russian Math. Surveys, 21:6 (1966), 1–82 | DOI | MR | Zbl

[3] G. Peshkir, A. N. Shiryaev, “The Khintchine inequalities and martingale expanding of the sphere of their action”, Russian Math. Surveys, 50:5 (1995), 849–904 | DOI | MR | Zbl

[4] S. V. Astashkin, “Rademacher functions in symmetric spaces”, J. Math. Sci., 169:6 (2010), 725–886 | DOI | MR

[5] S. V. Astashkin, D. V. Zanin, E. M. Semenov, F. A. Sukochev, “Kruglov operator and operators defined by random permutations”, Funct. Anal. Appl., 43:2 (2009), 83–95 | DOI | MR

[6] S. Kwapień, C. Schütt, “Some combinatorial and probabilistic inequalities and their applications to Banach space theory”, Studia Math., 82:1 (1985), 91–106 | MR | Zbl

[7] C. Schütt, “Lorentz spaces that are isomorphic to subspaces of $L_1$”, Trans. Amer. Math. Soc., 314:2 (1989), 583–595 http://www.jstor.org/stable/2001398 | MR | Zbl

[8] M. Junge, Q. Xu, “Noncommutative Burkholder/Rosenthal inequalities. II: Applications”, Israel J. Math., 167:1 (2008), 227–282 | DOI | MR | Zbl

[9] A. Khintchine, “Über dyadische Brüche”, Math. Z., 18:1 (1923), 109–116 | DOI | MR | Zbl

[10] S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen, Subwencji Funduszu Kultury Narodowej, Warszawa, 1935, 298 pp. | MR | Zbl

[11] J. Marcinkiewicz, A. Zygmund, “Remarque sur la loi du logarithme itéré”, Fund. Math., 29 (1937), 215–222 | Zbl

[12] B. S. Kashin, A. A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, 1989, 451 pp. | MR | MR | Zbl | Zbl

[13] A. N. Kolmogoroff, “Über das Gesetz des iterierten Logarithmus”, Math. Ann., 101:1 (1929), 126–135 ; “O zakone povtornogo logarifma”: A. N. Kolmogorov, Teoriya veroyatnostei i matematicheskaya statistika, Nauka, M., 1986, 34–44 ; Рђ. Рќ. Колмогоров, Рђ. Рќ. Колмогоров, Р�збранныРμ труды, 2, Наука, Рњ., 2005, 37–46 | DOI | MR | Zbl | MR | Zbl | MR | Zbl

[14] Yu. V. Prokhorov, “An extremal problem in probability theory”, Theory Probab. Appl., 4 (1959), 201–203 | DOI | MR | MR | Zbl

[15] R. J. Elliott, Stochastic calculus and applications, Appl. Math. (N. Y.), 18, Springer-Verlag, New York, 1982, 302 pp. | MR | MR | Zbl | Zbl

[16] S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolation of linear operators, Transl. Math. Monogr., 54, Amer. Math. Soc., Providence, RI, 1982, 375 pp. | MR | Zbl | Zbl

[17] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces. II. Function Spaces, Ergeb. Math. Grenzgeb., 97, Springer-Verlag, Berlin–New York, 1979, 243 pp. | MR | Zbl

[18] C. Bennett, R. Sharpley, Interpolation of operators, Pure Appl. Math., 129, Academic Press, Boston, MA, 1988, 469 pp. | MR | Zbl

[19] L. V. Kantorovich, G. P. Akilov, Functional analysis in normed spaces, International Series of Monographs in Pure and Applied Mathematics, 46, Macmillan, New York, 1964, 771 pp. | MR | MR | Zbl

[20] M. A. Krasnosel'skii, Ja. B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen, 1961, 249 pp. | MR | MR | Zbl

[21] J. Bergh, J. Löfstróm, Interpolation spaces. An introduction, Grundlehren Math. Wiss., 223, Springer-Verlag, Berlin–New York, 1976, 207 pp. | MR | MR | Zbl

[22] Ju. A. Brudnyi, N. Ya. Krugljak, Interpolation functors and interpolation spaces, v. I, North-Holland Math. Library, 47, North-Holland, Amsterdam, 1991, 718 pp. | MR | Zbl

[23] A. P. Calderón, “Spaces between $L^1$ and $L^\infty$ and the theorem of Marcinkiewicz”, Studia Math., 26 (1966), 273–299 | MR | Zbl

[24] B. S. Mityagin, “Interpolyatsionnaya teorema dlya modulyarnykh prostranstv”, Matem. sb., 66:4 (1965), 473–482 | MR | Zbl

[25] A. P. Calderon, “Intermediate spaces and interpolation, the complex method”, Studia Math., 24:2 (1964), 113–190 | MR | Zbl

[26] G. Ya. Lozanovskii, “A remark on an interpolational theorem of Calderon”, Funct. Anal. Appl., 6:4 (1972), 333–334 | DOI | MR | Zbl

[27] P. Hitczenko, “Domination inequality for martingale transforms of a Rademacher sequence”, Israel J. Math., 84:1–2 (1993), 161–178 | DOI | MR | Zbl

[28] T. Holmstedt, “Interpolation of quasi-normed spaces”, Math. Scand., 26:1 (1970), 177–199 | MR | Zbl

[29] S. J. Montgomery-Smith, “The distribution of Rademacher sums”, Proc. Amer. Math. Soc., 109:2 (1990), 517–522 | DOI | MR | Zbl

[30] J.-P. Kahane, Some random series of functions, D. C. Heath and Co., Raytheon Education Co., Lexington, MA, 1968, 184 pp. | MR | MR | Zbl | Zbl

[31] R. Latala, “Estimation of moments of sums of independent real random variables”, Ann. Probab., 25:3 (1997), 1502–1513 | DOI | MR | Zbl

[32] E. D. Gluskin, S. Kwapien, “Tail and moment estimates for sums of independent random variables with logarithmically concave tails”, Studia Math., 114:3 (1995), 303–309 | MR | Zbl

[33] P. Hitczenko, S. J. Montgomery-Smith, K. Oleszkiewicz, “Moment inequalities for sums of certain independent symmetric random variables”, Studia Math., 123:1 (1997), 15–42 | MR | Zbl

[34] N. L. Carothers, S. J. Dilworth, “Inequalities for sums of independent random variables”, Proc. Amer. Math. Soc., 104:1 (1988), 221–226 | DOI | MR | Zbl

[35] W. B. Johnson, G. Schechtman, “Sums of independent random variables in rearrangement invariant function spaces”, Ann. Probab., 17:2 (1989), 789–808 | DOI | MR | Zbl

[36] J. Hoffman-Jørgensen, “Sums of independent Banach space valued random variables”, Studia Math., 52 (1974), 159–186 | MR | Zbl

[37] M. B. Marcus, G. Pisier, “Characterization of almost surely continuous $p$-stable random Fourier series and strongly stationary processes”, Acta Math., 152:1 (1984), 245–301 | DOI | MR | Zbl

[38] W. B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc., 19, No 217, 1979, 298 pp. | MR | Zbl

[39] M. Sh. Braverman, Independent random variables and rearrangement invariant spaces, London Math. Soc. Lecture Note Ser., 194, Cambridge Univ. Press, Cambridge, 1994, 116 pp. | MR | Zbl

[40] V. M. Kruglov, “A note on infinitely divisible distributions”, Theory Probab. Appl., 15:2 (1970), 319–324 | DOI | MR | Zbl | Zbl

[41] E. Lukacs, Characteristic functions, 2nd ed., Hafner, New York, 1970, 350 pp. | MR | MR | Zbl | Zbl

[42] S. V. Astashkin, F. A. Sukochev, “Series of independent random variables in rearrangement invariant spaces: An operator approach”, Israel J. Math., 145:1 (2005), 125–156 | DOI | MR | Zbl

[43] S. V. Astashkin, F. A. Sukochev, “Comparison of sums of independent and disjoint functions in symmetric spaces”, Math. Notes, 76:3–4 (1970), 449–454 | DOI | MR | Zbl

[44] S. V. Astashkin, F. A. Sukochev, “Best constants in Rosenthal-type inequalities and the Kruglov operator”, Ann. Probab., 38:5 (2010), 1986–2008 ; arXiv: 1011.1381 | DOI | MR | Zbl

[45] S. Kwapień, W. A. Woyczyński, Random series and stochastic integrals: single and multiple, Probab. Appl., Birkhäuser, Boston, MA, 1992, 360 pp. | MR | Zbl

[46] V. I. Ovchinnikov, “The method of orbits in interpolation theory”, Math. Rep., 1:2 (1984), 349–515 | MR | Zbl

[47] S. Montgomery-Smith, E. M. Semenov, “Random rearrangements and operators”, Amer. Math. Soc. Transl. Ser. 2, 184, Amer. Math. Soc., Providence, RI, 1998, 157–183 | MR | Zbl

[48] J. V. Ryff, “Orbits of $L_1$ functions under doubly stochastic transformations”, Trans. Amer. Math. Soc., 117 (1965), 92–100 http://www.jstor.org/stable/1994198 | MR | Zbl

[49] V. I. Chilin, A. V. Krygin, F. A. Sukochev, “Extreme points of convex fully symmetric sets of measurable operators”, Integral Equations Operator Theory, 15:2 (1992), 186–226 | DOI | MR | Zbl

[50] A. A. Borovkov, Probability theory, Gordon and Breach, Abingdon, Oxon, 1998, 474 pp. | MR | MR | Zbl | Zbl

[51] Yu. V. Prokhorov, “Strong stability of sums and infinitely divisible distributions”, Theory Probab. Appl., 3:2 (1958), 141–153 | DOI | Zbl

[52] S. V. Astashkin, F. A. Sukochev, “Series of independent, mean zero random variables in rearrangement-invariant spaces having the Kruglov property”, J. Math. Sci. (N. Y.), 148:6 (2008), 795–809 | DOI | MR

[53] F. A. Sukochev, D. V. Zanin, Khintchine inequalities in quasi-normed spaces, manuscript

[54] Y. Gordon, A. Litvak, C. Schütt, E. Werner, “Orlicz norms of sequences of random variables”, Ann. Probab., 30:4 (2002), 1833–1853 | DOI | MR | Zbl

[55] Y. Gordon, A. Litvak, C. Schütt, E. Werner, “Geometry of spaces between polytopes and related zonotopes”, Bull. Sci. Math., 126:9 (2002), 733–762 | DOI | MR | Zbl

[56] P. Hitczenko, S. Montgomery-Smith, “Measuring the magnitude of sums of independent random variables”, Ann. Probab., 29:1 (2001), 447–466 | DOI | MR | Zbl

[57] S. Montgomery-Smith, “Rearrangement invariant norms of symmetric sequence norms of independent sequences of random variables”, Israel J. Math., 131:1 (2002), 51–60 | DOI | MR | Zbl

[58] M. Junge, “The optimal order for the $p$-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates”, Positivity, 10:2 (2006), 201–230 | DOI | MR | Zbl

[59] S. V. Astashkin, E. M. Semenov, F. A. Sukochev, “Banach–Saks type properties in rearrangement-invariant spaces with Kruglov property”, Houston J. Math., 35:3 (2009), 959–973 | MR | Zbl

[60] S. V. Astashkin, K. E. Tikhomirov, “O nekotorykh veroyatnostnykh analogakh neravenstva Rozentalya”, Matem. zametki (to appear)

[61] S. V. Astashkin, F. A. Sukochev, “Symmetric quasi-norms of sums of independent random variables in symmetric function spaces with the Kruglov property”, Israel J. Math. (to appear)

[62] A. V. Bukhvalov, “Interpolation of linear operators in spaces of vector-valued functions and with mixed norm”, Siberian Math. J., 28:1 (1987), 24–36 | MR | Zbl

[63] D. L. Burkholder, “A sharp inequality for martingale transforms”, Ann. Probab., 7:5 (1979), 858–863 | DOI | MR | Zbl

[64] W. Johnson, G. Schechtman, J. Zinn, “Best constants in moment inequalities for linear combinations of independent and exchangeable random variables”, Ann. Probab., 13:1 (1985), 234–253 | DOI | MR | Zbl

[65] S. V. Astashkin, “Extrapolation functors on a family of scales generated by the real interpolation method”, Siberian Math. J., 46:2 (2005), 205–225 | DOI | MR | Zbl

[66] V. A. Rodin, E. M. Semyonov, “Rademacher series in symmetric spaces”, Anal. Math., 1:3 (1975), 207–222 | DOI | MR | Zbl

[67] S. V. Astashkin, “Independent functions in rearrangement invariant spaces and the Kruglov property”, Sb. Math., 199:7 (2008), 945–963 | DOI | MR

[68] S. V. Astashkin, “A generalized Khintchine inequality in rearrangement invariant spaces”, Funct. Anal. Appl., 42:2 (2008), 144–147 | DOI | MR | Zbl

[69] V. I. Dmitriev, S. G. Krein, V. I. Ovchinnikov, “Osnovy teorii interpolyatsii lineinykh operatorov”, Geometriya lineinykh prostranstv i teoriya operatorov, Yaroslavl, 1977, 31–74 | MR | Zbl

[70] S. V. Astashkin, “About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system”, Int. J. Math. Math. Sci., 25:7 (2001), 451–465 | DOI | MR | Zbl

[71] S. J. Szarek, “On the best constant in the Khintchine inequality”, Studia Math., 58:2 (1976), 197–208 | MR | Zbl

[72] S. V. Astashkin, M. Sh. Braverman, “Podprostranstvo simmetrichnogo prostranstva, porozhdennoe sistemoi Rademakhera s vektornymi koeffitsientami”, Operatornye uravneniya v funktsionalnykh prostranstvakh, Izd-vo VGU, Voronezh, 1986, 3–10 | MR | Zbl

[73] R. Salem, A. Zygmund, “Some properties of trigonometric series whose terms have random signs”, Acta Math., 91:1 (1954), 245–301 | DOI | MR | Zbl

[74] M. B. Marcus, G. Pisier, Random Fourier series with applications to harmonic analysis, Ann. of Math. Stud., 101, Princeton Univ. Press, Princeton, NJ; Univ. Tokyo Press, Tokyo, 1981, 151 pp. | MR | Zbl

[75] M. Ledoux, M. Talagrand, Probability in Banach spaces, Ergeb. Math. Grenzgeb. (3), 23, Springer-Verlag, Berlin, 1991, 480 pp. | MR | Zbl

[76] B. Kashin, L. Tzafriri, “Lower estimates for the supremum of some random processes”, East J. Approx., 1:1 (1995), 125–139 | MR | Zbl

[77] B. Kashin, L. Tzafriri, “Lower estimates for the supremum of some random processes. II”, East J. Approx., 1:3 (1995), 373–377 | MR | Zbl

[78] S. V. Astashkin, “Extraction of subsystems ‘majorized’ by the Rademacher system”, Math. Notes, 65:4 (1999), 407–417 | DOI | MR | Zbl

[79] P. G. Grigor'ev, “Random linear combinations of functions from $L_1$”, Math. Notes, 74:1–2 (2003), 185–211 | DOI | MR | Zbl

[80] P. G. Grigoriev, “Estimates for norms of random polynomials”, East J. Approx., 7:4 (2001), 445–469 | MR | Zbl

[81] P. G. Grigor'ev, “Estimates for norms of random polynomials and their application”, Math. Notes, 69:5-6 (2003), 868–872 | DOI | MR | Zbl

[82] B. S. Mityagin, “The homotopy structure of the linear group of a Banach space”, Russian Math. Surveys, 25:5 (1970), 59–103 | DOI | MR | Zbl

[83] S. V. Astashkin, “Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis”, J. Funct. Anal., 260:1 (2010), 195–207 | DOI

[84] L. E. Dor, J. T. Starbird, “Projections of $L_p$ onto subspaces spanned by independent random variables”, Compositio Math., 39:2 (1979), 141–175 | MR | Zbl

[85] S. V. Astashkin, F. A. Sukochev, “Sequences of independent identically distributed functions in rearrangement invariant spaces”, Function spaces VIII, Banach Center Publ., 79, Polish Acad. Sci. Inst. Math., Warsaw, 2008, 27–37 | MR | Zbl

[86] S. V. Astashkin, “Interpolation of intersections by the real method”, St. Petersburg Math. J., 17:2 (2006), 239–265 | DOI | MR | Zbl

[87] G. G. Lorentz, T. Shimogaki, “Interpolation theorems for the pairs of spaces $(L^p,L^\infty)$ and $(L^1,L^q)$”, Trans. Amer. Math. Soc., 159 (1971), 207–221 | MR | Zbl