Random Fibonacci sequence: Difference between revisions

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

'''Viswanath's constant''' is a mathematical [[constant]], occuring in [[number theory]] - more specifically in the study of randomized [[Fibonacci sequence]]s. The value of the number is 1.13198824 ''(approximated to 8 decimal digits)''.

In [[mathematics]], the '''random Fibonacci sequence''' is a [[stochastic]] analogue of the [[Fibonacci sequence]] defined by the [[recurrence relation]] <math>f_n=f_{n-1}\pm f_{n-2}</math>, where the signs + or − are chosen [[Bernoulli distribution|at random]] with equal probability <math>\tfrac12</math>, [[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... {{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>

==DefinitionDescription==

A random Fibonacci sequence is an [[integer]] [[random sequence]] given by the numbers <math>f_n</math> for [[natural number]]s <math>n</math>, where <math>f_1=f_2=1</math> and the subsequent terms are chosen randomly according to the random recurrence relation

<math display=block>

f_n = \begin{cases}

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

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

\end{cases}

</math>

An instance 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 instance is the [[Fibonacci sequence]] (''F''_''n''),

<math display=block> 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>

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

The constant is defined as the [[exponential]] rate at which the average absolute value of a random [[Fibonacci sequence]] increases. A "random Fibonacci sequence" is a sequence of Fibonacci numbers that have the following recursive definition.

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

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

'''Initial conditions:'''

:<math>f(0)=1</math>

:<math>f(1)=1</math>

'''Recursive step:'''

:<math>

f(n)=\left\{\begin{matrix} f_+(n), & \mbox{with probability 0.5}\\ f_-(n), & \mbox{with probability 0.5}\end{matrix}\right.

</math>

where,

:<math>f_+(n)=f(n-1) + f(n-2)</math>

:<math>f_-(n)=f(n-1) - f(n-2)</math>

or in other words, the decision whether to add or subtract the previous two elements of the sequence to get the next element, is taken at random with a probability of 0.5 favouring each decision (say with a toss of a fair coin.)

Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via [[matrix (mathematics)|matrices]]:

In a sequence, thus constructed, with a probability of 1 (i.e. with extremely rare exceptions, [[almost surely]]) the ratio of the absolute values of successive terms converges to the value of the constant, for large values of ''n''.

<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>

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

==Explication==

<math display=block>{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

B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}. </math>

==Growth rate==

The constant was discovered by [[Divakar Viswanath]] in 1999.

[[Johannes Kepler]] discovered that as ''n'' increases, the ratio of the successive terms of the Fibonacci sequence (''F''_''n'') [[limit of a sequence|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>

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

[[Johannes Kepler]] had shown that for normal Fibonacci sequences, (where the randomness of the sign does not occur), the ratio of the successive numbers converged to the [[golden mean]]. Thus, for any large ''n'', the golden mean constant raised to the power of ''n'' yields the nth term of the sequence, with astonishing accuracy.

In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] showed that for a general class of [[random 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 [[nth root|''n''th root]] of |''f''_''n''| converges to a constant value ''[[almost surely]]'', or with probability one:

Though it seems surprising that a similar ratio could be obtained for a series of elements obtained by randomly chosen signs, a little thought shows that there are extremely rare cases where this ratio does not hold. For example, consider the following series

<math display=block> \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]] arithmetic validated by an analysis of the [[rounding error]].

1,1,0,1,-1,0,...

==Generalization==

This series is not allowed to "grow beyond" 1 or -1, only because the oscillating signs of + and - appear in a systematic pattern, but in a random experiment, the chances that such patterns of + and - are obtained are negligible.

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

<math display=block> f_n=\pm 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>

==Significance==

==References==

In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] had shown that for a a general class of random [[matrix]] products, the absolute value of the ''n''th term converges to a power of a fixed constant. This is a broad class of random sequence-generating processes, which includes the random Fibonacci sequence. This proof was significant in advances in [[laser]] technology and the study of [[glass]]es. The [[Nobel Prize for Physics]] in [[1977]] went to [[Philip Warren Anderson]] of [[Bell Laboratories]], [[Sir Nevill Francis Mott]] of [[Cambridge University]] in [[England]], and [[John Hasbrouck van Vleck ]] of [[Harvard]] "for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems".

{{reflist}}

==External Linkslinks==

Viswanath's proof, by specifying the value of the constant number in this case, has helped make this area more accessible for direct study. Viswanath's constant may explain the case of rabbits randomly allowed to prey on each other. (See [[Fibonacci sequence]] for the original statement of the rabbit problem.) This step would allow closer simulation of real world scenarios in various applications.

* {{MathWorld|urlname=RandomFibonacciSequence|title=Random Fibonacci Sequence}}

* {{OEIS el|sequencenumber=A078416|name=Decimal expansion of Viswanath's constant}}

*[https://www.youtube.com/watch?v=ELA8gNNMHoU Random Fibonacci Numbers]. [[Numberphile]]'s video about the random Fibonnaci sequence.

[[Category:Fibonacci numbers]]

==External Links==

[[Category:Mathematical constants]]

*[http://sciencenews.org/sn_arc99/6_12_99/bob1.htm A brief explanation]

[[Category:Number theory]]

[[Category:Stochastic processes]]