Random Fibonacci sequence - Revision history

Blansheflur: /* Generalization */

2023-04-18T23:15:03Z

Generalization

← Previous revision		Revision as of 23:15, 18 April 2023
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	==Generalization==		==Generalization==
	[[Mark Embree]] and [[Nick Trefethen]] showed in 1999 that the sequence		[[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 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''<sub>min</sub> ≈ 0.36747}} approximately equal to {{math\|''σ''(''β''<sub>min</sub>) ≈ 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>		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''<sub>min</sub> ≈ 0.36747}} approximately equal to {{math\|''σ''(''β''<sub>min</sub>) ≈ 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>

Joel Brennan: added wikilinks

2023-01-26T22:04:41Z

added wikilinks

77551enpassant: /* Growth rate */ arithmetics -> arithmetic

2022-11-09T23:35:36Z

Growth rate: arithmetics -> arithmetic

← Previous revision		Revision as of 23:35, 9 November 2022
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	<math display=block> \sqrt[n]{\|f_n\|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>		<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]] ~~arithmetics~~ validated by an analysis of the [[rounding error]].		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]].

	==Generalization==		==Generalization==

Comp.arch at 21:36, 15 May 2022

2022-05-15T21:36:48Z

← Previous revision		Revision as of 21:36, 15 May 2022
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	{{~~short~~ description\|Randomized mathematical sequence based upon the Fibonacci sequence}}		{{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]] <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>		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>

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

	== Growth rate ==		==Growth rate==

	[[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]],		[[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]],
	<math display=block>F_n = {{\varphi^n-(-1/\varphi)^{n}} \over {\sqrt 5}}. </math>		<math display=block>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''.		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''<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:		In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] showed that for a general class of random [[matrix (mathematics)\|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:
	<math display=block> \sqrt[n]{\|f_n\|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>		<math display=block> \sqrt[n]{\|f_n\|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>

David Eppstein: $; some copyediting$

2022-05-15T00:02:42Z

<math display=block>; some copyediting

Show changes

LaundryPizza03: /* Generalization */ now redirected here

2022-02-21T23:34:10Z

Generalization: now redirected here

← Previous revision		Revision as of 23:34, 21 February 2022
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	==Generalization==		==Generalization==
			[[Mark Embree]] and [[Nick Trefethen]] showed in 1999 that the sequence

	The [[Embree–Trefethen constant]] describes the qualitative behavior of the random sequence with the recurrence relation

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

	~~for different values of ''β''. [[Mark Embree]] and [[Nick Trefethen]] showed in 1999 that this sequence~~ decays almost surely if ''β'' is less than a critical value {{math\|''β''* ≈ 0.70258}}, 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''<sub>min</sub> ≈ 0.36747}} approximately equal to {{math\|''σ''(''β''<sub>min</sub>) ≈ 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>		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''<sub>min</sub> ≈ 0.36747}} approximately equal to {{math\|''σ''(''β''<sub>min</sub>) ≈ 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>

	==References==		==References==

LaundryPizza03: added Category:Stochastic processes using HotCat

2022-02-21T05:25:30Z

added Category:Stochastic processes using HotCat

← Previous revision		Revision as of 05:25, 21 February 2022
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	[[Category:Mathematical constants]]		[[Category:Mathematical constants]]
	[[Category:Number theory]]		[[Category:Number theory]]
			[[Category:Stochastic processes]]

LaundryPizza03: /* Related work */ elaborate

2022-02-21T00:31:09Z

Related work: elaborate

← Previous revision		Revision as of 00:31, 21 February 2022
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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]].		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]].

	==~~Related work~~==		==Generalization==

	The [[Embree–Trefethen constant]] describes the qualitative behavior of the random sequence with the recurrence relation		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>		: <math> f_n=f_{n-1}\pm \beta f_{n-2}</math>

	for different values of β.<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>		for different values of ''β''. [[Mark Embree]] and [[Nick Trefethen]] showed in 1999 that this sequence decays almost surely if ''β'' is less than a critical value {{math\|''β''* ≈ 0.70258}}, 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''<sub>min</sub> ≈ 0.36747}} approximately equal to {{math\|''σ''(''β''<sub>min</sub>) ≈ 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>

	==References==		==References==

Citation bot: Alter: title, pages. Add: s2cid. Formatted dashes. | Use this bot. Report bugs. | Suggested by Neko-chan | Category:Mathematical constants | #UCB_Category 84/89

2021-11-04T23:10:37Z

Alter: title, pages. Add: s2cid. Formatted dashes. | Use this bot. Report bugs. | Suggested by Neko-chan | Category:Mathematical constants | #UCB_Category 84/89

← Previous revision		Revision as of 23:10, 4 November 2021
Line 1:		Line 1:
	{{short description\|Randomized mathematical sequence based upon the Fibonacci sequence}}		{{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''<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…{{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 }}</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 \| year = 2006 \|arxiv=math.NT/0510159}}</ref>		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…{{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 ==		== Description ==
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	: <math> f_n=f_{n-1}\pm \beta f_{n-2}</math>		: <math> f_n=f_{n-1}\pm \beta f_{n-2}</math>

	for different values of β.<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 }}</ref>		for different values of β.<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>

	==References==		==References==

Leschnei: /* top */ removed bold face

2021-03-01T13:59:19Z

top: removed bold face

← Previous revision		Revision as of 13:59, 1 March 2021
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	{{short description\|Randomized mathematical sequence based upon the Fibonacci sequence}}		{{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''<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…{{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 }}</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 \| year = 2006 \|arxiv=math.NT/0510159}}</ref>		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…{{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 }}</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 \| year = 2006 \|arxiv=math.NT/0510159}}</ref>

	== Description ==		== Description ==

← Previous revision		Revision as of 22:04, 26 January 2023
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	{{Short description\|Randomized mathematical sequence based upon the Fibonacci sequence}}		{{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]] <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>		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>

	==Description==		==Description==
	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		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>		<math display=block>
	f_n = \begin{cases}		f_n = \begin{cases}
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	\end{cases}		\end{cases}
	</math>		</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''<sub>''n''</sub>},		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''<sub>''n''</sub>),
	<math display=block> 1,1,2,3,5,8,13,21,34,55,\ldots. </math>		<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		If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence
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	\text{ for the signs } +, +, +, -, -, +, -, -, \ldots.</math>		\text{ for the signs } +, +, +, -, -, +, -, -, \ldots.</math>

	Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via matrices:		Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via [[matrix (mathematics)\|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 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		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 display=block>{f_{n-1} \choose f_{n}} = M_{n}M_{n-1}\ldots M_3{f_{1} \choose f_{2}},</math>
	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:		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:
	<math display=block> A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad		<math display=block> A=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, \quad
	B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}. </math>		B=\begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}. </math>

	==Growth rate==		==Growth rate==
	[[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]],		[[Johannes Kepler]] discovered that as ''n'' increases, the ratio of the successive terms of the Fibonacci sequence (''F''<sub>''n''</sub>) [[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>		<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''.		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 ~~(mathematics)\|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:		In 1960, [[Hillel Furstenberg]] and [[Harry Kesten]] showed that for a general class of random 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 [[nth root\|''n''th root]] of \|''f''<sub>''n''</sub>\| 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 display=block> \sqrt[n]{\|f_n\|} \to 1.1319882487943\dots \text{ as } n \to \infty. </math>