How do random Fibonacci sequences grow?
成果类型：
Article
署名作者：
Janvresse, Elise; Rittaud, Benoit; de la Rue, Thierry
署名单位：
Universite de Rouen Normandie; Centre National de la Recherche Scientifique (CNRS); Universite Paris 13
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-007-0117-7
发表日期：
2008
页码：
619-648
关键词：
摘要：
We study the random Fibonacci sequences defined by F-1 = F-2 = (F) over tilde (1) = (F) over tilde (2) = 1 and for n >= 1, Fn+2 = Fn+1 +/- F-n (linear case) and (F) over tilde (n+2) = vertical bar(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 the exponential growth of F-n for 0 < p <= 1, and of F-n for 0 < p < <= 1, and of (F) over tilde (n) for 1/3 <= p <= 1 is almost surely given by integral(infinity)(0) log x dv(alpha)(x), where alpha is an explicit function of p depending on the case we consider, and nu(alpha) is an explicit probability distribution on R+ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of p, since we prove that it is equal to zero for 0 < p <= 1/3. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.
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