Journal of the Mathematical Society of Japan

Random Dirichlet series arising from records

Ron PELED, Yuval PERES, Jim PITMAN, and Ryokichi TANAKA

Full-text: Open access

Abstract

We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by

$S=\sum_{n=1}^{\infty}\frac{I_n}{n^s},$

where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value 1 with probability $1/n^\beta$ and value 0 otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in extreme value theory in statistics. We show that when $s$ > 0 and 0 < $\beta \le 1$ with $s+\beta$ > 1 the distribution of $S$ has a density; otherwise it is purely atomic or not defined because of divergence. In particular, in the case when $s$ > 0 and $\beta=1$, we prove that for every 0 < $s$ < 1 the density is bounded and continuous, whereas for every $s$ > 1 it is unbounded. In the case when $s$ > 0 and 0 < $\beta$ < 1 with $s+\beta$ > 1, the density is smooth. To show the absolute continuity, we obtain estimates of the Fourier transforms, employing van der Corput's method to deal with number-theoretic problems. We also give further regularity results of the densities, and present an example of a non-atomic singular distribution which is induced by the series restricted to the primes.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1705-1723.

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951163

Digital Object Identifier
doi:10.2969/jmsj/06741705

Mathematical Reviews number (MathSciNet)
MR3417510

Zentralblatt MATH identifier
1346.37009

Subjects
Primary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]
Secondary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords
random Dirichlet series records the van der Corput lemma

Citation

PELED, Ron; PERES, Yuval; PITMAN, Jim; TANAKA, Ryokichi. Random Dirichlet series arising from records. J. Math. Soc. Japan 67 (2015), no. 4, 1705--1723. doi:10.2969/jmsj/06741705. https://projecteuclid.org/euclid.jmsj/1445951163


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References

  • B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, Records, Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, 1998. A Wiley-Interscience Publication.
  • L. Breiman, Probability, SIAM the Society for Industrial and Applied Mathematics, Philadelphia, 1992 (originally published by Addison-Wesley, Reading, Massachusetts, 1968).
  • G. Bhowmik and K. Matsumoto, Analytic continuation of random Dirichlet series, Proc. Steklov Inst. Math., 282 (2013), Suppl.,1, S67–S72.
  • R. Durrett, Probability: Theory and Examples, Fourth Edition, Cambridge University Press, Cambridge, 2010.
  • P. D. T. A. Elliott, Probabilistic Number Theory I, Grundlehren der mathematischen Wissenschaften, 239, Springer-Verlag, New York, 1979.
  • P. Erdős, On the smoothness of the asymptotic distribution of additive arithmetical functions, Amer. J. Math., 61 (1939), 722–725.
  • S. W. Graham and G. Kolesnik, Van der Corput's Method of Exponential Sums, London Math. Soc. Lecture Note Series, 126, Cambridge University Press, Cambridge, 1991.
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Sixth Edition, Oxford University Press, Oxford, 2008.
  • M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math., 180 (2014), 773–822.
  • B. Jessen and A. Wintner, Distribution functions and the Riemann zeta functions, Trans. Amer. Math. Soc., 38 (1935), 48–88.
  • Y. Katznelson, An Introduction to Harmonic Analysis, Second Corrected Edition, Dover Publications, Inc., New York, 1976.
  • L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, A Wiley-Interscience Publication, New York, 1974.
  • F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math., 78 (1874), 46–62.
  • V. B. Nevzorov, Records: Mathematical Theory, Translations of Mathematical Monographs, 194, Amer. Math. Soc., Providence, RI, 2001, Translated from the Russian manuscript by D. M. Chibisov.
  • Y. Peres, W. Schlag and B. Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., 46, Birkhäuser, Basel, 2000, 39–65.
  • J. Pitman, Combinatorial Stochastic Processes, Lecture Notes in Math., 1875, Springer-Verlag Berlin Heidelberg, 2006.
  • A. Rényi, On the extreme elements of observations, MTA III, Oszt. Közl, 12, 105–121, 1962, Reprinted in Selected Papers of Alfréd Rényi, 3, Akadémiai Kiadó, Budapest 1976, 50–66.
  • B. Schmuland, Random harmonic series, Amer. Math. Monthly., 110 (2003), 407–416.
  • P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal., 24 (2014), 946–958.