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2 results

  • Probabilistic Proofs of Euler Identities

  • Part of
  • Lars Holst
  • Journal:
    Journal of Applied Probability / Volume 50 / Issue 4 / December 2013
    Published online by Cambridge University Press:
    30 January 2018, pp. 1206-1212
    Print publication:
    December 2013
    • Article
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  • Formulae for ζ(2n) and Lχ4(2n + 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2 / 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

  • On a Class of Distributions Stable Under Random Summation

  • Part of
  • L. B. Klebanov, A. V. Kakosyan, S. T. Rachev, G. Temnov
  • Journal:
    Journal of Applied Probability / Volume 49 / Issue 2 / June 2012
    Published online by Cambridge University Press:
    04 February 2016, pp. 303-318
    Print publication:
    June 2012
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  • We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.