Almost-sure growth rate of generalized random Fibonacci sequences

作者：Janvresse Elise^*; Rittaud Benoit; de la Rue Thierry

来源：Annales de l Institut Henri Poincare-Probabilites et Statistiques, 2010, 46(1): 135-158.

DOI：10.1214/09-AIHP312

• Random Fibonacci sequence
• Rosen continued fraction
• Upper Lyapunov exponent
• Stern-Brocot intervals
• Hecke group

We study the generalized random Fibonacci sequences defined by their first non-negative terms and for n >= 1, Fn+2 = lambda Fn+1 +/- F-n (linear case) and (F) over tilde (n+2) = vertical bar lambda(F) over tilde (n+1) +/- (F) over tilde (n)vertical bar (non-linear case), where each sign is independent and either + with probability p or - with probability 1 - p (0 < p <= 1). Our main result is that, when lambda is of the form lambda(k) = 2 cos(pi/k) for some integer k >= 3, the exponential growth of F-n for 0 < p <= 1, and of (F) over tilde (n) for 1/k < p <= 1, is almost surely positive and given by
integral(infinity)(0) log x d upsilon(k,rho)(x),
where rho is an explicit function of p depending on the case we consider, taking values in [0, 1], and upsilon(k,rho) is an explicit probability distribution on R+ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for 0 < p <= 1 in the easier case lambda >= 2. Finally, we study the variations of the exponent as a function of p.

• 出版日期2010-2