## Monday, June 20, 2022

### The random Fibonacci sequence and the Viswanath's constant

Random Fibonacci recursion

In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation ${\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}$, where the signs + or − are chosen at random with equal probability ${\displaystyle {\tfrac {1}{2}}}$, independently for different ${\displaystyle n}$. By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943...(sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.