Embree-Trefethen constant
- Embree-Trefethen constant
In number theory, the Embree-Trefethen constant is a threshold value labelled "β*".
For a fixed real β, consider the recurrence 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
:
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,so solutions to this recurrence decay exponentially as "n"→∞ with probability one, and:"σ" > 1 for "β*" < "β",so they grow exponentially.
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.
Literature
* 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
ee also
* Viswanath's constant
External links
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