Almost-sure growth rate of generalized random Fibonacci sequences

Élise Janvresse; Benoît Rittaud; Thierry de la Rue

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 135-158
  • ISSN: 0246-0203

Abstract

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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 (0<p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0<p≤1, and of ̃Fn for 1/k<p≤1, is almost surely positive and given by ∫0∞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<p≤1 in the easier case λ≥2. Finally, we study the variations of the exponent as a function of p.

How to cite

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Janvresse, Élise, Rittaud, Benoît, and de la Rue, Thierry. "Almost-sure growth rate of generalized random Fibonacci sequences." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 135-158. <http://eudml.org/doc/243041>.

@article{Janvresse2010,
abstract = {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 (0&lt;p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0&lt;p≤1, and of ̃Fn for 1/k&lt;p≤1, is almost surely positive and given by ∫0∞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&lt;p≤1 in the easier case λ≥2. Finally, we study the variations of the exponent as a function of p.},
author = {Janvresse, Élise, Rittaud, Benoît, de la Rue, Thierry},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random Fibonacci sequence; Rosen continued fraction; upper Lyapunov exponent; Stern–Brocot intervals; Hecke group; Lyapunov exponent},
language = {eng},
number = {1},
pages = {135-158},
publisher = {Gauthier-Villars},
title = {Almost-sure growth rate of generalized random Fibonacci sequences},
url = {http://eudml.org/doc/243041},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Janvresse, Élise
AU - Rittaud, Benoît
AU - de la Rue, Thierry
TI - Almost-sure growth rate of generalized random Fibonacci sequences
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 135
EP - 158
AB - 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 (0&lt;p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0&lt;p≤1, and of ̃Fn for 1/k&lt;p≤1, is almost surely positive and given by ∫0∞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&lt;p≤1 in the easier case λ≥2. Finally, we study the variations of the exponent as a function of p.
LA - eng
KW - random Fibonacci sequence; Rosen continued fraction; upper Lyapunov exponent; Stern–Brocot intervals; Hecke group; Lyapunov exponent
UR - http://eudml.org/doc/243041
ER -

References

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  9. [9] Y. Peres. Analytic dependence of Lyapunov exponents on transition probabilities. In Lyapunov Exponents (Oberwolfach, 1990) 64–80. Lecture Notes in Math. 1486. Springer, Berlin, 1991. Zbl0762.60005MR1178947
  10. [10] B. Rittaud. On the average growth of random Fibonacci sequences. J. Int. Seq. 10 (2007) 1–32 (electronic). Zbl1127.11013MR2276788
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