Growth rate for the expected value of a generalized random Fibonacci sequence

الأصل · EN

A random Fibonacci sequence is defined by the relation gₙ = | gₙ₋₁ +/- gₙ₋₂ |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form gₙ = |λgₙ₋₁ +/- gₙ₋₂ |. When λ>=2 and 0 < p <= 1, we prove that the expected value of gₙ grows exponentially fast. When λ= λₖ = 2 cos(π/k) for some fixed integer k>2, we show that the expected value of gₙ grows exponentially fast for p>(2-λₖ)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author.