Random Fibonacci Sequence - Growth Rate

Growth Rate

Johannes Kepler discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence {F_n} approaches the golden ratio which is approximately 1.61803. In 1765, Leonhard Euler published an explicit formula, known today as the Binet formula,

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 norm grows as λn, 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 of |f_n| converges to a constant value almost surely, or with probability one:

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 measure on the Stern–Brocot tree. Moreover, Viswanath computed the numerical value above using floating point arithmetics validated by an analysis of the rounding error.