On a probabilistic analogue of the Fibonacci sequence. (English) Zbl 0443.60043
Summary: 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
\[ E(X_n \mid X_{n -1}, \ldots, X_0) = X_{n -1} + X_{n -2} \quad\text{a.s.} \]
The process \(\{X_n\}\) mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions \(X_{n +1}X_n^{-1}\to\alpha\), a.s. as \(n \to\infty\) where \(\alpha\) is the ‘golden ratio’ \(\tfrac12 (1+\sqrt 5)\).
\[ E(X_n \mid X_{n -1}, \ldots, X_0) = X_{n -1} + X_{n -2} \quad\text{a.s.} \]
The process \(\{X_n\}\) mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions \(X_{n +1}X_n^{-1}\to\alpha\), a.s. as \(n \to\infty\) where \(\alpha\) is the ‘golden ratio’ \(\tfrac12 (1+\sqrt 5)\).
MSC:
| 60G48 | Generalizations of martingales |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
