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, · ··, 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'