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On a probabilistic analogue of the Fibonacci sequence
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- Journal:
- Journal of Applied Probability / Volume 17 / Issue 4 / December 1980
- Published online by Cambridge University Press:
- 14 July 2016, pp. 1079-1082
- Print publication:
- December 1980
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- Article
- Export citation
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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' 
a.s. as 
