Geodesic
Log-integrability of Rademacher Fourier series, with applications to random analytic functions
Algebra i analiz, Tome 25 (2013) no. 3, pp. 147-184.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that any power of the logarithm of a Fourier series with random signs is integrable. This result has applications to the distribution of values of random Taylor series, one of which answers a long-standing question by J.-P. Kahane.
Keywords: random Taylor series, reduction principle. 
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F. Nazarov; A. Nishry; M. Sodin. Log-integrability of Rademacher Fourier series, with applications to random analytic functions. Algebra i analiz, Tome 25 (2013) no. 3, pp. 147-184. http://geodesic.mathdoc.fr/item/AA_2013_25_3_a5/

[1] Erdős L., Knowles A., Yau H.-T., Yin J., Delocalization and diffusion profile for random band matrices, arXiv: 1205.5669v1 | MR

[2] Favorov S. Yu., “Rost i raspredelenie znachenii golomorfnykh otobrazhenii konechnomernogo prostranstva v banakhovo”, Sib. mat. zh., 31:1 (1990), 161–171 | MR | Zbl

[3] Favorov S. Yu., “On the growth of holomorphic mappings from a finite-dimensional space into a Banach space”, Mat. Fiz. Anal. Geom., 1 (1994), 240–251 | MR | Zbl

[4] Kahane J. P., Some Random Series of Functions, 2-nd ed., Cambirdge Univ. Press, Cambridge, 1985 | MR | Zbl

[5] M. Jacob, A. C. Offord, “The distribution of the values of a random power series in the unit disk”, Proc. Roy. Soc. Edinburgh Sect. A, 94 (1983), 251–263 | DOI | MR

[6] Levin B. Ya., Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956

[7] Littlewood J. E., Offord A. C., “On the number of real roots of random algebraic equation. I”, J. London Math. Soc., 13 (1938), 288–295 ; “II”, Proc. Cambridge Phil. Soc., 35 (1939), 133–148 | DOI | MR | DOI | Zbl

[8] Littlewood J. E., Offord A. C., “On the distribution of zeros and $a$-values of a random integral function. II”, Ann. of Math. (2), 49 (1948), 885–952 ; “errata”, Ann. of Math. (2), 50 (1949), 990–991 | DOI | MR | DOI | MR | Zbl

[9] Murai T., “The value-distribution of lacunary series and a conjecture of Paley”, Ann. Inst. Fourier (Grenoble), 31 (1981), 135–156 | DOI | MR | Zbl

[10] Murai T., “The value-distribution of random Taylor series in the unit disk”, J. London Math. Soc. (2), 24 (1981), 480–494 | DOI | MR | Zbl

[11] Nazarov F. L., “Lokalnye otsenki eksponentsialnykh polinomov i ikh prilozheniya k neravenstvam tipa printsipa neopredelennosti”, Algebra i analiz, 5:4 (1993), 3–66 | MR | Zbl

[12] Nazarov F., Summability of large powers of logarithm of classic lacunary series and its simplest consequences, Preprint, 1995

[13] Offord A. C., “The distribution of the values of an entire function whose coefficients are independent random variables. I”, Proc. London Math. Soc. (3), 14a (1965), 199–238 | DOI | MR | Zbl

[14] Offord A. C., “The distribution of zeros of power series whose coefficients are independent random variables”, Indian J. Math., 9 (1967), 175–196 | MR | Zbl

[15] Offord A. C., “The distribution of values of a random function in the unit disk”, Studia Math., 41 (1972), 71–106 | MR | Zbl

[16] Offord A. C., “The distribution of the values of an entire function whose coefficients are independent random variables. II”, Math. Proc. Cambridge Phil. Soc., 118 (1995), 527–542 | DOI | MR | Zbl

[17] Ullrich D. C., “An extension of the Kahane–Khinchine inequality in a Banach space”, Israel J. Math., 62 (1988), 56–62 | DOI | MR | Zbl

[18] Ullrich D. C., “Khinchin's inequality and the zeroes of Bloch functions”, Duke Math. J., 57 (1988), 519–535 | DOI | MR | Zbl