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In number theory, the Embree-Trefethen constant is a threshold value labelled β*.## Literature

## See also

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For a fixed real β, consider the recurrence $x\_\{n+1\}=x\_n\; pm\; beta\; x\_\{n-1\}$ where the sign in the sum is chosen at random for each n independently with equal probabilities for "+" and "−".

It can be proven that for any choice of β, the limit

- $sigma(beta)\; =\; lim\_\{n\; to\; infty\}\; (|x\_n|^\{1/n\})$

exists almost surely. In informal words, the sequence behaves exponentially with probability one, and σ(β) can be interpreted as its almost sure rate of exponential growth.

We have

- σ < 1 for 0 < β < β* = 0.70258 approximately,

- σ > 1 for β* < β,

Regarding values of σ, we have:

- σ(1) = 1.13198824... (Viswanath's constant), and
- σ(β*) = 1.

The constant is named after applied mathematicians Mark Embree and Lloyd N. Trefethen.

- Embree, M., and L.N. Trefethen (1999): Growth and decay of random Fibonacci sequences. Proceedings of the Royal Society London A 455(July):2471-2485

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Last updated on Thursday April 10, 2008 at 09:29:38 PDT (GMT -0700)

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Last updated on Thursday April 10, 2008 at 09:29:38 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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