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Growth and Decay of Random Fibonacci Sequences (1999)
| Venue: | London Proceedings, Series A, Mathematical, Physical and Engineering Sciences |
| Citations: | 22 - 2 self |
BibTeX
@ARTICLE{Embree99growthand,
author = {Mark Embree and Lloyd N. Trefethen},
title = {Growth and Decay of Random Fibonacci Sequences},
journal = {London Proceedings, Series A, Mathematical, Physical and Engineering Sciences},
year = {1999},
volume = {455},
pages = {455}
}
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Abstract
Introduction In a remarkable recent paper, Viswanath (1998) has considered the large n behaviour of solutions to the `random Fibonacci recurrence' x n+1 = \Sigma x n \Sigma x n\Gamma1 ; (1.1) where the signs are chosen independently and with equal probabilities, and x 0 = x 1 = 1. Computer experiments, as in figure 1, show exponential growth with n. The problem of large n behaviour of (1.1) has been mentioned at least since 1963, when Furstenberg (1963) established exponential growth with probability 1, but Viswanath's contribution represents an intriguing new development. By an ingenious application of a Stern--Brocot tree (Graham et al. 1994), he proved that solutions to (1.1) satisfy lim n!1 jx n j<F8.064
