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

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.
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Submitted on : Tuesday, April 15, 2008 - 3:33:21 PM
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Elise Janvresse, Benoît Rittaud, Thierry de la Rue. Growth rate for the expected value of a generalized random Fibonacci sequence. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2009, 42, pp.085005. ⟨10.1088/1751-8113/42/8/085005⟩. ⟨hal-00273537⟩

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