**Problem**

I am interested in the random recurrence relation of the form $x_{n+1}=\alpha x_n \pm \beta x_{n-1}$ where $\alpha$, $\beta$ are known constants and the $\pm$ sign is chosen with equal probability.

In particular I would like to investigate the limit of $|x_n|^{1/n}$ as $n\rightarrow\infty$. (i.e the exponential growth rate)

**Motivation**

I was considering the problem of finding $\lim_{n\rightarrow\infty}|x_n|^{\frac{1}{n}}$ where $x_n$ is the solution to $x_{n+1}=2x_n \pm x_{n-1}$, $x_0=x_1=1$. All numerical evidence suggests that $\lim_{n\rightarrow\infty}|x_n|^{\frac{1}{n}}\approx 1.91$ almost surely but I am having difficulty even proving that the sequence almost surely convergences.

**Survey**

The solution to the recurrence relation of the from $x_{n+1}=x_n \pm \beta x_{n-1}$ has been show to have expontential growth almost surely. See http://en.wikipedia.org/wiki/Embree%E2%80%93Trefethen_constant

Also there has been work on the random fibonacci sequence $x_{n+1}=x_n \pm x_{n-1}$ for example Viswanath(2000), "Random Fibonacci sequences and the number 1.13198824"