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Date: 25-1-2021
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Consider the Fibonacci-like recurrence
(1) |
where , , and each sign is chosen independently and at random with probability 1/2. Surprisingly, Viswanath (2000) showed that
(2) |
(OEIS A078416) with probability one. This constant is sometimes known as Viswanath's constant.
Considering the more general recurrence
(3) |
the limit
(4) |
exists for almost all values of . The critical value such that is given by
(5) |
(OEIS A118288) and is sometimes known as the Embree-Trefethen constant.
Since Fibonacci numbers can be computed as products of Fibonacci Q-matrices, this same constant arises in the iterated multiplication of certain pairs of random matrices (Bougerol and Lacrois 1985, pp. 11 and 157).
REFERENCES:
Batista Oliveira, J. and De Figueiredo, L. H. "Interval Computation of Viswanath's Constant." Reliab. Comput. 8, 131-138, 2002.
Bougerol, P. and Lacrois, J. Random Products of Matrices With Applications to Infinite-Dimensional Schrödinger Operators. Basel, Switzerland: Birkhäuser, 1985.
Devlin, K. "Devlin's Angle: New Mathematical Constant Discovered: Descendent of Two Thirteenth Century Rabbits." March 1999. https://www.maa.org/devlin/devlin_3_99.html.
Embree, M. and Trefethen, L. N. "Growth and Decay of Random Fibonacci Sequences." Roy. Soc. London Proc. Ser. A, Math. Phys. Eng. Sci. 455, 2471-2485, 1999.
Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 227-228, 2002.
Michon, G. P. "Final Answers: Numerical Constants." https://home.att.net/~numericana/answer/constants.htm#viswanath.
Peterson, I. "Fibonacci at Random: Uncovering a New Mathematical Constant." Sci. News 155, 376, June 12, 1999. https://sciencenews.org/sn_arc99/6_12_99/bob1.htm.
Sloane, N. J. A. Sequences A078416 and A118288 in "The On-Line Encyclopedia of Integer Sequences."
Viswanath, D. "Random Fibonacci Sequences and the Number 1.13198824...." Math. Comput. 69, 1131-1155, 2000.
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