# Biased random Fibonacci sequences

I have recently been toying (very superficially) with the random Fibonacci sequence, i.e., defined by $$F_0=1=F_1=1$$ and $$F_{n} = F_{n-1} + \varepsilon_n F_{n-2}$$ where $$(\varepsilon_n)_{n\geq 2}$$ is a sequence of i.i.d. Rademacher random variables.

It is known that this sequence is such that $$|F_n|^{1/n} \xrightarrow[n\to\infty]{\rm a.s.} \nu$$ where $$\nu \approx 1.13$$ is the Viswanath constant (D. Viswanath, 1999).

I am interested in the case where the $$(\varepsilon_n)_{n\geq 2}$$ are no longer uniform in $$\{-1,1\}$$, but instead $$1$$ with probability $$p\in[0,1]$$ (so the standard Fibonacci sequence corresponds to $$p=1$$, and the standard random Fibonacci sequence to $$p=1/2$$). Then my understanding is that the convergence proof goes through, i.e., $$|F_n|^{1/n} \xrightarrow[n\to\infty]{\rm a.s.} \nu(p)$$ I am interested in the behavior of $$\nu(p)$$ as a function of $$p$$. Is there anything known about it?

Through some basic simulations, the behavior seems very regular, but I am not sure what else to infer from it.$\nu(p)$ as a function of p" />

Note that the paper by D. Viswanath does (briefly) discuss a biased version (see, e.g., Figure 5); however, the recurrence considered there is of the form $$F_{n} = \varepsilon_n F_{n-1} + \varepsilon_n' F_{n-2}$$ where $$\varepsilon'_n,\varepsilon_n$$ are i.i.d. The result is the same (because of the absolute values) when $$p=1/2$$, but it is not the case for $$p\neq 1/2$$.

• The paper How do random Fibonacci sequences grow? has some results on this problem and a nonlinear generalization. A preprint is also available at arXiv:math/0611860 in case you don't have access to PTRF. – HMPanzo Nov 8 '20 at 2:40
• Thanks -- Based on the abstract it looks very relevant. – Clement C. Nov 8 '20 at 2:43
• Mmmh. That made me realize I made a mistake in my simulations and attempts, so even better... – Clement C. Nov 8 '20 at 2:57
• @HMPanzo If you want to make this comment an answer, I definitely would upvote it. This would be a very good reference to have archived permanently here. – Clement C. Nov 10 '20 at 18:26

This example is discussed in my 1990 PhD thesis, where is it is shown that $$\nu(p)$$ is a real analytic function of $$p$$. The thesis was written in Hebrew, but this result was generalized in two papers [1], [2]. Note that the existence of the limit you call the Viswanath constant (and more generally, $$\nu(p)$$) is an immediate consequence of the Furstenberg-Kesten Theorem [3] on the existence of the top Lyapunov exponent for random matrix products. This is because the vector $$(F_n,F_{n+1})$$ is obtained by multiplying $$(F_{n-1},F_n)$$ by a random matrix that takes two possible values.

[1] Peres, Yuval Analytic dependence of Lyapunov exponents on transition probabilities. Lyapunov exponents (Oberwolfach, 1990), 64–80, Lecture Notes in Math., 1486, Springer, Berlin, 1991.

[2] Peres, Yuval Domains of analytic continuation for the top Lyapunov exponent. Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), no. 1, 131–148. (Reviewer: Philippe Bougerol) http://www.numdam.org/article/AIHPB_1992__28_1_131_0.pdf

[3] Furstenberg, Harry, and Harry Kesten. "Products of random matrices." The Annals of Mathematical Statistics 31, no. 2 (1960): 457-469.