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

Janvresse, Elise; Rittaud, Benoît; De La Rue, Thierry

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

Authors

Janvresse, Elise

Rittaud, Benoît

De La Rue, Thierry

Description

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-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 g_n = |λg_{n-1} +/- g_{n-2} |. When λ>=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When λ= λ_k = 2 cos(π/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-λ_k)/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.