Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T15:38:56.382Z Has data issue: false hasContentIssue false
  • English
  • Français

On a probabilistic analogue of the Fibonacci sequence

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
CSIRO Division of Mathematics and Statistics, Canberra
*
∗Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 1965, Canberra City, A.C.T. 2601, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

One of the earliest population models to be studied gives rise to the Fibonacci sequence and has a history dating back more than 750 years. A stochastic version of the model is discussed in this paper, its basic defining property being E(Xn | Xn−1, · ··, X0) = Xn−1 + Xn−2 a.s. The process {Xn} mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions, a.s. as n → where α is the ‘golden ratio'

Keywords

POPULATION PROCESSESGROWTH RATESFIBONACCI NUMBERSGOLDEN RATIOGENERALIZED MARTINGALES
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 1079 - 1082
Copyright
Copyright © Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Coxeter, H. S. M. (1953) The golden section, phyllotaxis, and Wythoff's game. Scripta Math. 19, 135–143.Google Scholar
Hedían, H. (1976) The golden section and the artist. The Fibonacci Quarterly 14, 406–418.Google Scholar
Mcqueen, J. B. (1973) A linear extension of the martingale convergence theorem. Ann. Prob. 1, 263–271.Google Scholar
De Vita, J. (1978) Fibonacci, insects, and flowers. The Fibonacci Quarterly 16, 315–317.Google Scholar
Vorobev, N. N. (1961) Fibonacci Numbers. Blaisdell. New York.Google Scholar