广义随机Fibonacci序列期望值的增长率
Growth rate for the expected value of a generalized random Fibonacci sequence
摘要：
随机斐波那契序列由关系g_n =| g_{n-1} +/- g_{n-2}|，其中+/-符号通过为每个n投掷平衡硬币来选择。我们将这些序列推广到硬币不平衡的情况（用p表示a +的概率），其递推关系式为g_n =| λ g_{n-1} +/- g_{n-2}|.当λ>=2且0 < p <= 1时，我们证明了g_n的期望值是指数增长的。当λ= λ_k = 2 cos（π/k），k>2时，证明了当p>（2-λ_k）/4时，g_n的期望值是指数增长的，并给出了增长率的代数式.所涉及的方法扩展（和纠正）在以前的论文中介绍的第二作者。
英文摘要：
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.
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