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 (\cite{viswanath}), following Furstenberg (\cite{furst}) 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
1.25577≥(E(|
x
n
|)
)
1/n
≥1.12095
where
E(|
x
n
|)
is the expected value for the absolute value of the
n
th term in a random Fibonacci sequence. We compute this expected value using recurrence relations which bound the sum of all possible
n
th terms for such sequences. In addition, we give upper an lower