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Register a new userAlmost-sure Growth Rate of Generalized Random Fibonacci sequences
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Elise Janvresse
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We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (linear case) and $widetilde F_{n+2} = |lambda widetilde F_{n+1} pm widetilde F_{n}|$ (non-linear case), where each $pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<ple 1$). Our main result is that, when $lambda$ is of the form $lambda_k = 2cos (pi/k)$ for some integer $kge 3$, the exponential growth of $F_n$ for $0<ple 1$, and of $widetilde F_{n}$ for $1/k < ple 1$, is almost surely positive and given by $$ int_0^infty log x d u_{k, rho} (x), $$ where $rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ u_{k, rho}$ is an explicit probability distribution on $RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<ple 1$ in the easier case $lambdage 2$. Finally, we study the variations of the exponent as a function of $p$.
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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.
We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1$ and for $nge 1$, $F_{n+2} = F_{n+1} pm F_{n}$ (linear case) or $F_{n+2} = |F_{n+1} pm F_{n}|$ (non-linear case), where each sign is independent and either + with probability $p$ or - with probability $1-p$ ($0<ple 1$). Our main result is that the exponential growth of $F_n$ for $0<ple 1$ (linear case) or for $1/3le ple 1$ (non-linear case) is almost surely given by $$int_0^infty log x d u_alpha (x), $$ where $alpha$ is an explicit function of $p$ depending on the case we consider, and $ u_alpha$ is an explicit probability distribution on $RR_+$ defined inductively on Stern-Brocot intervals. In the non-linear case, the largest Lyapunov exponent is not an analytic function of $p$, since we prove that it is equal to zero for $0<ple1/3$. We also give some results about the variations of the largest Lyapunov exponent, and provide a formula for its derivative.
The generalized Fibonacci sequences are sequences ${f_n}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t in mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the $p$-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo $n$ in more depth and find an interpretation of the rank in terms of the order of an element in the multiplicative group of a finite field when $n$ is an odd prime. Finally, we study the distribution of the rank over different values of $s$ when $t = -1$ and suggest directions for further study involving the rank modulo prime powers of generalized Fibonacci sequences.
This work is devoted to almost sure and moment exponential stability of regime-switching jump diffusions. The Lyapunov function method is used to derive sufficient conditions for stabilities for general nonlinear systems; which further helps to derive easily verifiable conditions for linear systems. For one-dimensional linear regime-switching jump diffusions, necessary and sufficient conditions for almost sure and $p$th moment exponential stabilities are presented. Several examples are provided for illustration.
In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {mathcal T}_{infty}$: The $n$-th edge isoperimetric number $delta(n)$ is defined to be $min_{|S|=n, S subset V({mathcal T}_{infty})} |(S,bar{S})|$, where $(S,bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${pm (2^d - 1): $d$ is a positive integer}$. The quantity of interest is $tau(n)$, the minimum number of coins necessary to make change for $n$ cents. (3) Certain Meta-Fibonacci sequences: The Tanny sequence is defined by $T(n)=T(n{-}1{-}T(n{-}1))+T(n{-}2{-}T(n{-}2))$ and the Conolly sequence is defined by $C(n)=C(n{-}C(n{-}1))+C(n{-}1{-}C(n{-}2))$, where the initial conditions are $T(1) = C(1) = T(2) = C(2) = 1$. These are well-known meta-Fibonacci sequences. The main result that ties these three together is the following: $$ delta(n) = tau(n) = n+ 2 + 2 min_{1 le k le n} (C(k) - T(n-k) - k).$$ Apart from this, we prove several other results which bring out the interconnections between the above three concepts.
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