Editing Random Fibonacci sequence (section)

==Growth rate==
[[Johannes Kepler]] discovered that as ''n'' increases, the ratio of the successive terms of the Fibonacci sequence (''F''<sub>''n''</sub>) [[limit of a sequence|approaches]] the [[golden ratio]] <math>\varphi=(1+\sqrt{5})/2,</math> which is approximately 1.61803. In 1765, [[Leonhard Euler]] published an explicit formula, known today as the [[Binet formula]],
<math display=block>F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}. </math>

It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio ''φ''.

In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] showed that for a general class of random matrix products, the [[matrix norm|norm]] grows as ''λ''<sup>''n''</sup>, where ''n'' is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the [[nth root|''n''th root]] of |''f''<sub>''n''</sub>| converges to a constant value ''[[almost surely]]'', or with probability one:
<math display=block> \sqrt[n]{|f_n|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>

An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the [[Lyapunov exponent]] of a random matrix product and integration over a certain [[fractal|fractal measure]] on the [[Stern–Brocot tree]]. Moreover, Viswanath computed the numerical value above using [[floating point]] arithmetic validated by an analysis of the [[rounding error]].