Abstract: We are going to watch a video lecture.
Random Fibonacci sequences are given by their first two terms F_0 and F_1, and one of the following stochastic inductions:
F_(n+1) = F_n +/- F_(n-1) (linear random Fibonacci)
F_(n+1) = | F_n +/- F_(n-1) | (non-linear random Fibonacci),
where the +/- signs are given by an i.i.d. Bernoulli process of parameter p in [0,1] (p is the probability of a +). We are mainly interested in the exponential growth (largest Lyapunov exponent) of these sequences. We show that this exponents can in both cases be computed as an integral with respect to some explicit probability measure inductively defined on Stern-Brocot intervals. This allows us to study the variations of the exponent with respect to the parameter p. In particular, we can show that this exponent is positive for any p>0 in the linear case, whereas in the non-linear case the exponential growth holds only when p>1/3. To get the integral formula, we develop an original method whose main ingredients are - a reduction process of the random Fibonacci sequence, allowed by some nontrivial identities satisfied by the 2X2 matrices which are involved in the induction; - elementary properties of continued fraction expansion and Stern-Brocot intervals. This method can be extended to random Fibonacci sequences with multiplicative coefficient:
F_(n+1) = lambda F_n +/- F_(n-1) (linear case)
F_(n+1) = | lambda F_n +/- F_(n-1) | (non-linear case),
where lambda is of the form lambda = lambda_k = 2 cos pi/k (k=2,3,4,...) In these cases, we have to introduce generalizations of Stern-Brocot intervals and work with so-called Rosen continued fraction expansions.