On a probabilistic analogue of the Fibonacci sequence

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, · ··, X 0) = 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'