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| In mathematics, the '''random Fibonacci sequence''' is a stochastic analogue of the [[Fibonacci sequence]] defined by the [[recurrence relation]] ''f''<sub>''n''</sub> = ''f''<sub>''n''−1</sub> ± ''f''<sub>''n''−2</sub>, where the signs + or − are chosen [[Bernoulli distribution|at random]] with equal probability 1/2, [[Independence (probability theory)|independently]] for different ''n''. By a theorem of [[Harry Kesten]] and [[Hillel Furstenberg]], random recurrent sequences of this kind grow at a certain [[exponential growth|exponential rate]], but it is difficult to compute the rate explicitly. In 1999, [[Divakar Viswanath]] showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…, a [[mathematical constant]] that was later named Viswanath's constant.<ref>{{cite doi|10.1090/S0025-5718-99-01145-X}}</ref><ref>{{cite doi|10.1023/A:1014702122205}}</ref><ref>{{cite doi|10.1016/j.jnt.2006.01.002}} {{arxiv|math.NT/0510159}}</ref>
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| == Description ==
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| The random Fibonacci sequence is an integer random sequence {''f''<sub>''n''</sub>}, where ''f''<sub>1</sub> = ''f''<sub>2</sub> = 1 and the subsequent terms are determined from the random recurrence relation
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| :<math>
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| f_n = \begin{cases}
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| f_{n-1}+f_{n-2}, & \text{ with probability 1/2}; \\
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| f_{n-1}-f_{n-2}, & \text{ with probability 1/2}.
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| \end{cases}
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| </math>
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| A run of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a [[fair coin]] toss: given two consecutive elements of the sequence, the next element is either their sum or their difference with probability 1/2, independently of all the choices made previously. If in the random Fibonacci sequence the plus sign is chosen at each step, the corresponding run is the [[Fibonacci sequence]] {''F''<sub>''n''</sub>},
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| : <math> 1,1,2,3,5,8,13,21,34,55,\ldots. </math> | |
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| If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence
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| : <math> 1,1,0,1,1,0,1,1,0,1,\ldots.</math>
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| However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:
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| : <math> 1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \ldots
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| \text{ for the signs } +, +, -, -, -, +, -, -, \ldots.</math>
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| Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:
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| :<math>{f_{n-1} \choose f_{n}} = \begin{pmatrix} 0 & 1 \\ \pm 1 & 1 \end{pmatrix} {f_{n-2} \choose f_{n-1}},</math>
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| where the signs are chosen independently for different ''n'' with equal probabilities for + or −. Thus
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| :<math>{f_{n-1} \choose f_{n}} = M_{n}M_{n-1}\ldots M_3{f_{1} \choose f_{2}},</math>
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| where {''M''<sub>''k''</sub>} is a sequence of [[Independent and identically-distributed random variables|independent identically distributed random matrices]] taking values ''A'' or ''B'' with probability 1/2:
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| : <math> A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad | |
| B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}. </math>
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| == Growth rate ==
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| [[Johannes Kepler]] discovered that as ''n'' increases, the ratio of the successive terms of the Fibonacci sequence {''F''<sub>''n''</sub>} approaches the [[golden ratio]] <math>\varphi=(1+\sqrt{5})/2,</math> which is approximately 1.61803. In 1765, [[Leonhard Euler]] published an explicit formula, known today as the [[Binet formula]],
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| :<math> F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}. </math>
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| It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio ''φ''.
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| In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] showed that for a general class of random [[matrix (math)|matrix]] products, the [[matrix norm|norm]] grows as ''λ''<sup>''n''</sup>, where ''n'' is the number of factors. Their results apply to a broad class of random sequence generating processes that includes the random Fibonacci sequence. As a consequence, the ''n''th root of |''f''<sub>''n''</sub>| converges to a constant value ''[[almost surely]]'', or with probability one:
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| :<math> \sqrt[n]{|f_n|} \to 1.13198824\dots \text{ as } n \to \infty. </math> | |
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| An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the [[Lyapunov exponent]] of a random matrix product and integration over a certain [[fractal|fractal measure]] on the [[Stern–Brocot tree]]. Moreover, Viswanath computed the numerical value above using [[floating point]] arithmetics validated by an analysis of the [[rounding error]].
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| ==Related work==
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| The [[Embree–Trefethen constant]] describes the qualitative behavior of the random sequence with the recurrence relation
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| : <math> f_n=f_{n-1}\pm \beta f_{n-2}</math> | |
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| for different values of β.<ref>{{cite doi|10.1098/rspa.1999.0412|noedit}}</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://sciencenews.org/sn_arc99/6_12_99/bob1.htm A brief explanation]
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| * {{MathWorld|urlname=RandomFibonacciSequence|title=Random Fibonacci Sequence}}
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| * {{SloanesRef|sequencenumber=A078416}}
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| [[Category:Fibonacci numbers]]
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| [[Category:Mathematical constants]]
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| [[Category:Number theory]]
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