Embree–Trefethen constant

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In number theory, the Embree–Trefethen constant is a threshold value labelled β*.[1]

For a fixed positive number β, consider the recurrence relation

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 1, and

σ > 1 for β* < β,

so they grow exponentially.

Regarding values of σ, we have:

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


  1. ^ Embree, M.; Trefethen, L. N. (1999). "Growth and decay of random Fibonacci sequences" (PDF). Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 455 (1987): 2471. Bibcode:1999RSPSA.455.2471T. doi:10.1098/rspa.1999.0412.

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