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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, the `!random Fibonacci sequence`! is a `F33f`_`[stochastic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stochastic]`_`f analogue of the `F33f`_`[Fibonacci sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fibonacci_sequence]`_`f defined by the `F33f`_`[recurrence relation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Recurrence_relation]`_`f f n = f n − − 1 ± ± f n − − 2 {\\displaystyle f_{n}=f_{n-1}\\pm f_{n-2}} , where the signs + or − are chosen `F33f`_`[at random`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bernoulli_distribution]`_`f with equal probability 1 2 {\\displaystyle {\\tfrac {1}{2}}} , `F33f`_`[independently`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Independence_(probability_theory)]`_`f for different n {\\displaystyle n} . By a `F33f`_`[theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Theorem]`_`f of `F33f`_`[Harry Kesten`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harry_Kesten]`_`f and `F33f`_`[Hillel Furstenberg`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hillel_Furstenberg]`_`f, random recurrent sequences of this kind grow at a certain `F33f`_`[exponential rate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_growth]`_`f, 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 A078416 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f), a `F33f`_`[mathematical constant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematical_constant]`_`f that was later named Viswanath's constant.`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]
>>Contents
• `F0af`_`[Description`#description]`_`f
• `F0af`_`[Growth rate`#growth-rate]`_`f
• `F0af`_`[Generalization`#generalization]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[External links`#external-links]`_`f
-─
>>Description
A random Fibonacci sequence is an `F33f`_`[integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f `F33f`_`[random sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_sequence]`_`f given by the numbers f n {\\displaystyle f_{n}} for `F33f`_`[natural numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_number]`_`f n {\\displaystyle n} , where f 1 = f 2 = 1 {\\displaystyle f_{1}=f_{2}=1} and the subsequent terms are chosen randomly according to the random recurrence relation f n = { f n − − 1 + f n − − 2 , with probability 1 2 ; f n − − 1 − − f n − − 2 , with probability 1 2 . {\\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}}} An instance of the random Fibonacci sequence starts with 1,1 and the value of the each subsequent term is determined by a `F33f`_`[fair coin`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fair_coin]`_`f 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 `F33f`_`[Fibonacci sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fibonacci_sequence]`_`f (`*F`*`*n`*), 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , … … . {\\displaystyle 1,1,2,3,5,8,13,21,34,55,\\ldots .} If the signs alternate in minus-plus-plus-minus-plus-plus-... pattern, the result is the sequence 1 , 1 , 0 , 1 , 1 , 0 , 1 , 1 , 0 , 1 , … … . {\\displaystyle 1,1,0,1,1,0,1,1,0,1,\\ldots .}
However, such patterns occur with vanishing probability in a random experiment. In a typical run, the terms will not follow a predictable pattern: 1 , 1 , 2 , 3 , 1 , − − 2 , − − 3 , − − 5 , − − 2 , − − 3 , … … for the signs + , + , + , − − , − − , + , − − , − − , … … . {\\displaystyle 1,1,2,3,1,-2,-3,-5,-2,-3,\\ldots {\\text{ for the signs }}+,+,+,-,-,+,-,-,\\ldots .}
Similarly to the deterministic case, the random Fibonacci sequence may be profitably described via `F33f`_`[matrices`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_(mathematics)]`_`f: ( f n − − 1 f n ) = ( 0 1 ± ± 1 1 ) ( f n − − 2 f n − − 1 ) , {\\displaystyle {f_{n-1} \\choose f_{n}}={\\begin{pmatrix}0&1\\\\\\pm 1&1\\end{pmatrix}}{f_{n-2} \\choose f_{n-1}},}
where the signs are chosen independently for different `*n`* with equal probabilities for + or −. Thus ( f n − − 1 f n ) = M n M n − − 1 … … M 3 ( f 1 f 2 ) , {\\displaystyle {f_{n-1} \\choose f_{n}}=M_{n}M_{n-1}\\ldots M_{3}{f_{1} \\choose f_{2}},} where (`*M`*`*k`*) is a sequence of `F33f`_`[independent identically distributed random matrices`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Independent_and_identically-distributed_random_variables]`_`f taking values `*A`* or `*B`* with probability 1/2: A = ( 0 1 1 1 ) , B = ( 0 1 − − 1 1 ) . {\\displaystyle A={\\begin{pmatrix}0&1\\\\1&1\\end{pmatrix}},\\quad B={\\begin{pmatrix}0&1\\\\-1&1\\end{pmatrix}}.}
>>Growth rate
`F33f`_`[Johannes Kepler`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Johannes_Kepler]`_`f discovered that as `*n`* increases, the ratio of the successive terms of the Fibonacci sequence (`*F`*`*n`*) `F33f`_`[approaches`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Limit_of_a_sequence]`_`f the `F33f`_`[golden ratio`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Golden_ratio]`_`f φ φ = ( 1 + 5 ) / 2 , {\\displaystyle \\varphi =(1+{\\sqrt {5}})/2,} which is approximately 1.61803. In 1765, `F33f`_`[Leonhard Euler`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Leonhard_Euler]`_`f published an explicit formula, known today as the `F33f`_`[Binet formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binet_formula]`_`f, F n = φ φ n − − ( − − 1 / φ φ ) n 5 . {\\displaystyle F_{n}={{\\varphi ^{n}-(-1/\\varphi )^{n}} \\over {\\sqrt {5}}}.}
It demonstrates that the Fibonacci numbers grow at an exponential rate equal to the golden ratio `*φ`*.
In 1960, `F33f`_`[Hillel Furstenberg`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hillel_Furstenberg]`_`f and `F33f`_`[Harry Kesten`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Harry_Kesten]`_`f showed that for a general class of `F33f`_`[random matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_matrix]`_`f products, the `F33f`_`[norm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Matrix_norm]`_`f 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 `F33f`_`[nth root`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nth_root]`_`f of |`*f`*`*n`*| converges to a constant value `*`F33f`_`[almost surely`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Almost_surely]`_`f`*, or with probability one: | f n | n → → 1.1319882487943 … … as n → → ∞ ∞ . {\\displaystyle {\\sqrt[{n}]{|f_{n}|}}\\to 1.1319882487943\\dots {\\text{ as }}n\\to \\infty .}
An explicit expression for this constant was found by Divakar Viswanath in 1999. It uses Furstenberg's formula for the `F33f`_`[Lyapunov exponent`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lyapunov_exponent]`_`f of a random matrix product and integration over a certain `F33f`_`[fractal measure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fractal]`_`f on the `F33f`_`[Stern–Brocot tree`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stern–Brocot_tree]`_`f. Moreover, Viswanath computed the numerical value above using `F33f`_`[floating point`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Floating_point]`_`f arithmetic validated by an analysis of the `F33f`_`[rounding error`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rounding_error]`_`f.
>>Generalization
`F33f`_`[Mark Embree`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mark_Embree]`_`f and `F33f`_`[Nick Trefethen`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nick_Trefethen]`_`f showed in 1999 that the sequence f n = ± ± f n − − 1 ± ± β β f n − − 2 {\\displaystyle f_{n}=\\pm f_{n-1}\\pm \\beta f_{n-2}}
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 `F33f`_`[fractal`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fractal]`_`f structure, with a global minimum near `*β`*min ≈ 0.36747 approximately equal to `*σ`*(`*β`*min) ≈ 0.89517.`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]
>>References
`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefviswanath1999`aViswanath, D. (1999). "Random Fibonacci sequences and the number 1.13198824..." `*Mathematics of Computation`*. `!69`! (231): 1131–1155. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1090/S0025-5718-99-01145-X.
`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefoliveirade-figueiredo2002`aOliveira, J. O. B.; De Figueiredo, L. H. (2002). "Interval Computation of Viswanath's Constant". `*Reliable Computing`*. `!8`! (2): 131. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1023/A:1014702122205. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 29600050.
`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefmakovermcgowan2006`aMakover, E.; McGowan, J. (2006). "An elementary proof that random Fibonacci sequences grow exponentially". `*Journal of Number Theory`*. `!121`!: 40–44. `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:math.NT/0510159. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.jnt.2006.01.002. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 119169165.
`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefembreetrefethen1999`a`F33f`_`[Embree, M.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mark_Embree]`_`f; `F33f`_`[Trefethen, L. N.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lloyd_N._Trefethen]`_`f (1999). "Growth and decay of random Fibonacci sequences" (PDF). `*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences`*. `!455`! (1987): 2471. `F33f`_`[Bibcode`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bibcode_(identifier)]`_`f:1999RSPSA.455.2471T. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1098/rspa.1999.0412. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 16404862.
>>External links
• `:reference-mathworld-random-fibonacci-sequence`a`:citerefweisstein`a`F33f`_`[Weisstein, Eric W.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_W._Weisstein]`_`f "Random Fibonacci Sequence". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.
• OEIS sequence A078416 (Decimal expansion of Viswanath's constant)
• Random Fibonacci Numbers. `F33f`_`[Numberphile`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Numberphile]`_`f's video about the random Fibonnaci sequence.
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