*Journées de Probabilités 2007*

**Élise Janvresse **

*Exponential growth of random Fibonacci sequences and generalizations *

(Joint work with Thierry de la Rue and Benoît Rittaud)

We study two kinds of generalized random Fibonacci sequences defined by *F*_{1}=*a*, *F*_{2}=*b* and one of the following induction formulas:

linear case: for *n*>0, *F*_{n+2} = *F*_{n+1} *F*_{n}

non-linear case: *n*>0, *F*_{n+2} = | *F*_{n+1} *F*_{n}|

where is of the form for some *k* 3, and each sign is independent and either + with probability *p* or - with probability 1-*p* (0 < *p* 1). For *k*=3, =1 and we recover random Fibonacci sequences.

Our main result is that the exponential growth of *F _{n}* for 0 <

where is an explicit function of

Our method relies on some reduction process and makes use of Rosen continued fraction expansion. In the linear case, it gives an explicit form for Furstenberg's invariant measure. It is easier in the non-linear case to which the standard theory does not apply.