Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Fibonacci numbers, Lucas numbers
and integrals of certain Gaussian processes


Author: Ludwig Baringhaus
Journal: Proc. Amer. Math. Soc. 124 (1996), 3875-3884
MSC (1991): Primary 60E05; Secondary 11B35
DOI: https://doi.org/10.1090/S0002-9939-96-03691-X
MathSciNet review: 1363410
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the distributions of integrals of Gaussian processes arising as limiting distributions of test statistics proposed for treating a goodness of fit or symmetry problem. We show that the cumulants of the distributions can be expressed in terms of Fibonacci numbers and Lucas numbers.


References [Enhancements On Off] (What's this?)

  • [1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
  • [2] Aloisio Araujo and Evarist Giné, The central limit theorem for real and Banach valued random variables, John Wiley & Sons, New York-Chichester-Brisbane, 1980. Wiley Series in Probability and Mathematical Statistics. MR 576407
  • [3] L. Baringhaus and N. Henze, A consistent test for multivariate normality based on the empirical characteristic function, Metrika 35 (1988), no. 6, 339–348. MR 980849, https://doi.org/10.1007/BF02613322
  • [4] Sándor Csörgő, Multivariate empirical characteristic functions, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 2, 203–229. MR 608017, https://doi.org/10.1007/BF00535160
  • [5] D. A. Darling, The Cramér-Smirnov test in the parametric case, Ann. Statist. 26 (1955), 1-20. MR 16:729g
  • [6] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [7] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vol. 2, McGraw-Hill, New York, 1953. MR 15:419i
  • [8] -, Tables of Integral Transforms. Vols. 1, 2, McGraw-Hill, New York, 1954. MR 15:868a; MR 16:468c
  • [9] Andrey Feuerverger and Roman A. Mureika, The empirical characteristic function and its applications, Ann. Statist. 5 (1977), no. 1, 88–97. MR 0428584
  • [10] H.-D. Keller, Einige Untersuchungen zur empirischen charakteristischen Funtion und deren Anwendungen, Dissertation, Dortmund, 1979.
  • [11] S. Vajda, Fibonacci & Lucas numbers, and the golden section, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1989. Theory and applications; With chapter XII by B. W. Conolly. MR 1015938
  • [12] Ward Whitt, Weak convergence of probability measures on the function space 𝐶[0,∞), Ann. Math. Statist. 41 (1970), 939–944. MR 0261646, https://doi.org/10.1214/aoms/1177696970

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 60E05, 11B35

Retrieve articles in all journals with MSC (1991): 60E05, 11B35


Additional Information

Ludwig Baringhaus
Affiliation: Institut für Mathematische Stochastik, Universität Hannover, D-30167 Hannover, Germany
Email: baringhaus@mbox.stochastik.uni-hannover.de

DOI: https://doi.org/10.1090/S0002-9939-96-03691-X
Keywords: Gaussian processes, Fibonacci numbers, Lucas numbers, integral equations, empirical Fourier transform, testing for normality, testing for symmetry
Received by editor(s): May 15, 1995
Communicated by: Wei-Yin Loh
Article copyright: © Copyright 1996 American Mathematical Society