We study the generalized random Fibonacci sequences defined by their first non-negative terms and for n≥1, Fn+2=λFn+1±Fn (linear case) and ̃Fn+2=|λ̃Fn+1±̃Fn| (non-linear case), where each ± sign is independent and either + with probability p or − with probability 1−p (0k=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0n for 1/k0∞log x dνk, ρ(x),
¶
where ρ is an explicit function of p depending on the case we consider, taking values in [0, 1], and νk, ρ is an explicit probability distribution on ℝ+ defined inductively on generalized Stern–Brocot intervals. We also provide an integral formula for 0
Publié le : 2010-02-15
Classification:
Random Fibonacci sequence,
Rosen continued fraction,
Upper Lyapunov exponent,
Stern–Brocot intervals,
Hecke group,
37H15,
60J05,
11J70
@article{1267454112,
author = {Janvresse, \'Elise and Rittaud, Beno\^\i t and de la Rue, Thierry},
title = {Almost-sure growth rate of generalized random Fibonacci sequences},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {46},
number = {1},
year = {2010},
pages = { 135-158},
language = {en},
url = {http://dml.mathdoc.fr/item/1267454112}
}
Janvresse, Élise; Rittaud, Benoît; de la Rue, Thierry. Almost-sure growth rate of generalized random Fibonacci sequences. Ann. Inst. H. Poincaré Probab. Statist., Tome 46 (2010) no. 1, pp. 135-158. http://gdmltest.u-ga.fr/item/1267454112/