An elementary proof that random Fibonacci sequences grow exponentially

Abstract

We consider random Fibonacci sequences given by x n + 1 = ± β x n + x n − 1 . Viswanath [Divakar Viswanath, Random Fibonacci sequences and the number 1.13198824…, Math. Comp. 69 (231) (2000) 1131–1155, MR MR1654010 (2000j:15040)] following Furstenberg [Harry Furstenberg, Noncommuting random products, Trans. Amer. Math. Soc. 108 (1963) 377–428, MR MR0163345 (29 #648)] showed that when β = 1 , lim n → ∞ | x n | 1 / n = 1.13 … , but his proof involves the use of floating point computer calculations. We give a completely elementary proof that 1.23375 ⩾ ( E ( | x n | ) ) 1 / n ⩾ 1.12095 where E ( | x n | ) is the expected value for the absolute value of the nth term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible nth terms for such sequences. In addition, we give upper and lower bounds for the second moment of the | x n | . Finally, we consider the conjecture of Embree and Trefethen [Mark Embree, Lloyd N. Trefethen, Growth and decay of random Fibonacci sequences, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1987) (1999) 2471–2485, MR MR1807827 (2001i:11098)], derived using computational calculations, that for values of β < 0.702585 such sequences decay. We show that as β decreases, the critical value where growth can change to decay is in fact 1 / 2 .