Random Fibonacci sequence: Difference between revisions

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{{short description|Randomized mathematical sequence based upon the Fibonacci sequence}}

In mathematics, the '''random Fibonacci sequence''' is a stochastic analogue of the [[Fibonacci sequence]] defined by the [[recurrence relation]] ''f''<submath>''n''</sub> f_n= ''f''_{''f_{n''−1}-1}\pm ± ''f''<sub>''f_{n''−2-2}</submath>, where the signs + or − are chosen [[Bernoulli distribution|at random]] with equal probability 1<math>\tfrac12</2math>, [[Independence (probability theory)|independently]] for different ''<math>n''</math>. 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…1319882487943...{{OEIS|A078416}}, a [[mathematical constant]] that was later named Viswanath's constant.<ref>{{Cite journal | last1 = Viswanath | first1 = D. | title = Random Fibonacci sequences and the number 1.13198824... | doi = 10.1090/S0025-5718-99-01145-X | journal = Mathematics of Computation | volume = 69 | issue = 231 | pages = 1131–1155 | year = 1999 | doi-access = free }}</ref><ref>{{Cite journal | last1 = Oliveira | first1 = J. O. B. | last2 = De Figueiredo | first2 = L. H. | journal = Reliable Computing | volume = 8 | issue = 2 | pages = 131 |title=Interval Computation of Viswanath's Constant| year = 2002 | doi = 10.1023/A:1014702122205 | s2cid = 29600050 }}</ref><ref>{{Cite journal | last1 = Makover | first1 = E. | last2 = McGowan | first2 = J. | doi = 10.1016/j.jnt.2006.01.002 | title = An elementary proof that random Fibonacci sequences grow exponentially | journal = Journal of Number Theory | volume = 121 | pages = 40–44 | year = 2006 |arxiv=math.NT/0510159| s2cid = 119169165 }}</ref>

== Description ==

TheA random Fibonacci sequence is an integer random sequence given by the numbers {''f''<submath>''n''f_n</submath>}, wherefor [[natural number]]s ''f''<submath>1n</submath>&nbsp;=&nbsp;''f'', where <submath>2f_1=f_2=1</submath>&nbsp;=&nbsp;1 and the subsequent terms are determinedchosen randomly according fromto the random recurrence relation

<math display=block>

:<math>

f_n = \begin{cases}

f_{n-1}+f_{n-2}, & \text{ with probability 1/2} \tfrac12; \\

f_{n-1}-f_{n-2}, & \text{ with probability 1/2} \tfrac12.

\end{cases}

</math>

AAn runinstance 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 runinstance is the [[Fibonacci sequence]] {''F''_''n''},

: <math display=block> 1,1,2,3,5,8,13,21,34,55,\ldots. </math>

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''_''n''},

: <math> 1,1,2,3,5,8,13,21,34,55,\ldots. </math>

If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence

: <math display=block> 1,1,0,1,1,0,1,1,0,1,\ldots.</math>

: <math> 1,1,0,1,1,0,1,1,0,1,\ldots.</math>

However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern:

: <math display=block> 1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \ldots

: <math> 1, 1, 2, 3, 1, -2, -3, -5, -2, -3, \ldots

\text{ for the signs } +, +, +, -, -, +, -, -, \ldots.</math>

Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:

:<math display=block>{f_{n-1} \choose f_{n}} = \begin{pmatrix} 0 & 1 \\ \pm 1 & 1 \end{pmatrix} {f_{n-2} \choose f_{n-1}},</math>

:<math>{f_{n-1} \choose f_{n}} = \begin{pmatrix} 0 & 1 \\ \pm 1 & 1 \end{pmatrix} {f_{n-2} \choose f_{n-1}},</math>

where the signs are chosen independently for different ''n'' with equal probabilities for + or −. Thus

:<math display=block>{f_{n-1} \choose f_{n}} = M_{n}M_{n-1}\ldots M_3{f_{1} \choose f_{2}},</math>

:<math>{f_{n-1} \choose f_{n}} = M_{n}M_{n-1}\ldots M_3{f_{1} \choose f_{2}},</math>

where {''M''_''k''} is a sequence of [[Independent and identically-distributed random variables|independent identically distributed random matrices]] taking values ''A'' or ''B'' with probability 1/2:

: <math display=block> A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad

: <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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[[Johannes Kepler]] discovered that as ''n'' increases, the ratio of the successive terms of the Fibonacci sequence {''F''_''n''} 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]],

:<math> display=block>F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}. </math>

:<math> F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}. </math>

It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio ''φ''.

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''^''n'', 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''_''n''| converges to a constant value ''[[almost surely]]'', or with probability one:

:<math display=block> \sqrt[n]{|f_n|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>

:<math> \sqrt[n]{|f_n|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>

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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==Generalization==

[[Mark Embree]] and [[Nick Trefethen]] showed in 1999 that the sequence

: <math display=block> f_n=f_{n-1}\pm \beta f_{n-2}</math>

: <math> f_n=f_{n-1}\pm \beta f_{n-2}</math>

decays almost surely if ''β'' is less than a critical value {{math|''β''* ≈ 0.70258}}, known as the Embree–Trefethen constant, and otherwise grows almost surely. They also showed that the asymptotic ratio ''σ''(''β'') between consecutive terms converges almost surely for every value of ''β''. The graph of ''σ''(''β'') appears to have a [[fractal]] structure, with a global minimum near {{math|''β''_min ≈ 0.36747}} approximately equal to {{math|''σ''(''β''_min) ≈ 0.89517}}.<ref>{{Cite journal | last1 = Embree | first1 = M. | author-link1 = Mark Embree| last2 = Trefethen | first2 = L. N. | author-link2 = Lloyd N. Trefethen| doi = 10.1098/rspa.1999.0412 | title = Growth and decay of random Fibonacci sequences | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 455 | issue = 1987 | pages = 2471 | year = 1999 | url = http://people.maths.ox.ac.uk/~trefethen/publication/PDF/1999_86.pdf|bibcode = 1999RSPSA.455.2471T | s2cid = 16404862 }}</ref>