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