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

TL;DR

This paper investigates the growth rate of the expected value of generalized random Fibonacci sequences with unbalanced coin flips and a parameter lambda, establishing exponential growth conditions and providing algebraic formulas for specific cases.

Contribution

It extends previous methods to analyze the expected growth rate of generalized random Fibonacci sequences, including cases with unbalanced coin probabilities and specific lambda values.

Findings

Expected value grows exponentially for lambda >= 2 and 0 < p <= 1.

Exponential growth occurs for p > (2 - lambda_k)/4 when lambda = 2 cos(pi/k).

Provides algebraic expressions for growth rates in special cases.

Abstract

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 = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_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.