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Random Fibonacci sequence

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the random Fibonacci sequence is a <a href="Stochastic" title="Stochastic">stochastic</a> analogue of the <a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> defined by the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}</annotation>

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</math><img src="./fa1e263037c612f118013f1480d0bb42f7c66727.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.213ex; height:2.509ex;" alt="{\displaystyle f_{n}=f_{n-1}\pm f_{n-2}}" loading="lazy">, where the signs + or − are chosen <a href="Bernoulli_distribution" title="Bernoulli distribution">at random</a> with equal probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

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</math><img src="./edef8290613648790a8ac1a95c2fb7c3972aea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" loading="lazy">, <a href="Independence_(probability_theory)" title="Independence (probability theory)">independently</a> for different <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. By a <a href="Theorem" title="Theorem">theorem</a> of <a href="Harry_Kesten" title="Harry Kesten">Harry Kesten</a> and <a href="Hillel_Furstenberg" title="Hillel Furstenberg">Hillel Furstenberg</a>, random recurrent sequences of this kind grow at a certain <a href="Exponential_growth" title="Exponential growth">exponential rate</a>, 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... (sequence <a href="https://oeis.org/A078416" class="extiw external" title="oeis:A078416">A078416</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), a <a href="Mathematical_constant" title="Mathematical constant">mathematical constant</a> that was later named Viswanath's constant.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>

A random Fibonacci sequence is an <a href="Integer" title="Integer">integer</a> <a href="Random_sequence" title="Random sequence">random sequence</a> given by the numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle f_{n}}</annotation>

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</math><img src="./b2702450f0458a5e01a698e248af552a7fab2b50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.358ex; height:2.509ex;" alt="{\displaystyle f_{n}}" loading="lazy"> for <a href="Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}=f_{2}=1}">

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<annotation encoding="application/x-tex">{\displaystyle f_{1}=f_{2}=1}</annotation>

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</math><img src="./2f20ab290720c967d7360eca96383eef26cc64d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.746ex; height:2.509ex;" alt="{\displaystyle f_{1}=f_{2}=1}" loading="lazy"> and the subsequent terms are chosen randomly according to the random recurrence relation

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<mtext> with probability </mtext>

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<mtext> with probability </mtext>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}={\begin{cases}f_{n-1}+f_{n-2},&{\text{ with probability }}{\tfrac {1}{2}};\\f_{n-1}-f_{n-2},&{\text{ with probability }}{\tfrac {1}{2}}.\end{cases}}}</annotation>

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An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a <a href="Fair_coin" title="Fair coin">fair coin</a> 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 <a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> (Fn),

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<annotation encoding="application/x-tex">{\displaystyle 1,1,2,3,5,8,13,21,34,55,\ldots .}</annotation>

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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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<annotation encoding="application/x-tex">{\displaystyle 1,1,0,1,1,0,1,1,0,1,\ldots .}</annotation>

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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:

<mtext> for the signs </mtext>

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<annotation encoding="application/x-tex">{\displaystyle 1,1,2,3,1,-2,-3,-5,-2,-3,\ldots {\text{ for the signs }}+,+,+,-,-,+,-,-,\ldots .}</annotation>

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Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>:

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<annotation encoding="application/x-tex">{\displaystyle {f_{n-1} \choose f_{n}}={\begin{pmatrix}0&1\\\pm 1&1\end{pmatrix}}{f_{n-2} \choose f_{n-1}},}</annotation>

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

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<annotation encoding="application/x-tex">{\displaystyle {f_{n-1} \choose f_{n}}=M_{n}M_{n-1}\ldots M_{3}{f_{1} \choose f_{2}},}</annotation>

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where (Mk) is a sequence of <a href="Independent_and_identically-distributed_random_variables" class="mw-redirect" title="Independent and identically-distributed random variables">independent identically distributed random matrices</a> taking values A or B with probability 1/2:

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}0&1\\1&1\end{pmatrix}},\quad B={\begin{pmatrix}0&1\\-1&1\end{pmatrix}}.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Growth_rate">Growth rate</h2></div>

<a href="Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> discovered that as n increases, the ratio of the successive terms of the Fibonacci sequence (Fn) <a href="Limit_of_a_sequence" title="Limit of a sequence">approaches</a> the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(1+{\sqrt {5}})/2,}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi =(1+{\sqrt {5}})/2,}</annotation>

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</math><img src="./9caa1c8e103e488810fe00214434f5fa864b4d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.501ex; height:3.009ex;" alt="{\displaystyle \varphi =(1+{\sqrt {5}})/2,}" loading="lazy"> which is approximately 1.61803. In 1765, <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> published an explicit formula, known today as the <a href="Binet_formula" class="mw-redirect" title="Binet formula">Binet formula</a>,

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<annotation encoding="application/x-tex">{\displaystyle F_{n}={{\varphi ^{n}-(-1/\varphi )^{n}} \over {\sqrt {5}}}.}</annotation>

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

In 1960, <a href="Hillel_Furstenberg" title="Hillel Furstenberg">Hillel Furstenberg</a> and <a href="Harry_Kesten" title="Harry Kesten">Harry Kesten</a> showed that for a general class of <a href="Random_matrix" title="Random matrix">random matrix</a> products, the <a href="Matrix_norm" title="Matrix norm">norm</a> grows as λn, 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 <a href="Nth_root" title="Nth root">nth root</a> of |fn| converges to a constant value <a href="Almost_surely" title="Almost surely">almost surely</a>, or with probability one:

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{|f_{n}|}}\to 1.1319882487943\dots {\text{ as }}n\to \infty .}</annotation>

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An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the <a href="Lyapunov_exponent" title="Lyapunov exponent">Lyapunov exponent</a> of a random matrix product and integration over a certain <a href="Fractal" title="Fractal">fractal measure</a> on the <a href="Stern%E2%80%93Brocot_tree" title="Stern–Brocot tree">Stern–Brocot tree</a>. Moreover, Viswanath computed the numerical value above using <a href="Floating_point" class="mw-redirect" title="Floating point">floating point</a> arithmetic validated by an analysis of the <a href="Rounding_error" class="mw-redirect" title="Rounding error">rounding error</a>.

<div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2></div>

<a href="Mark_Embree" title="Mark Embree">Mark Embree</a> and <a href="Nick_Trefethen" title="Nick Trefethen">Nick Trefethen</a> showed in 1999 that the sequence

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<annotation encoding="application/x-tex">{\displaystyle f_{n}=\pm f_{n-1}\pm \beta f_{n-2}}</annotation>

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decays almost surely if β is less than a critical value 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 <a href="Fractal" title="Fractal">fractal</a> structure, with a global minimum near βmin ≈ 0.36747 approximately equal to σ(βmin) ≈ 0.89517.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFViswanath1999" class="citation journal cs1">Viswanath, D. (1999). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-99-01145-X">"Random Fibonacci sequences and the number 1.13198824..."</a> Mathematics of Computation. 69 (231): 1131–1155. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-99-01145-X">10.1090/S0025-5718-99-01145-X</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFOliveiraDe_Figueiredo2002" class="citation journal cs1">Oliveira, J. O. B.; De Figueiredo, L. H. (2002). "Interval Computation of Viswanath's Constant". Reliable Computing. 8 (2): 131. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1014702122205">10.1023/A:1014702122205</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29600050">29600050</a>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMakoverMcGowan2006" class="citation journal cs1">Makover, E.; McGowan, J. (2006). "An elementary proof that random Fibonacci sequences grow exponentially". Journal of Number Theory. 121: 40–44. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.NT/0510159">math.NT/0510159</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jnt.2006.01.002">10.1016/j.jnt.2006.01.002</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119169165">119169165</a>.</cite>

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFEmbreeTrefethen1999" class="citation journal cs1"><a href="Mark_Embree" title="Mark Embree">Embree, M.</a>; <a href="Lloyd_N._Trefethen" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, L. N.</a> (1999). <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/~trefethen/publication/PDF/1999_86.pdf">"Growth and decay of random Fibonacci sequences"</a> (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 455 (1987): 2471. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999RSPSA.455.2471T">1999RSPSA.455.2471T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1999.0412">10.1098/rspa.1999.0412</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16404862">16404862</a>.</cite>

</li>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RandomFibonacciSequence.html">"Random Fibonacci Sequence"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A078416">sequence A078416 (Decimal expansion of Viswanath's constant)</a></li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=ELA8gNNMHoU">Random Fibonacci Numbers</a>. <a href="Numberphile" title="Numberphile">Numberphile</a>'s video about the random Fibonnaci sequence.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-23" href="https://en.wikipedia.org/wiki/?title=Random_Fibonacci_sequence&oldid=1296948934">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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