From Fibonacci numbers to central limit type theorems
From Fibonacci numbers to central limit type theorems
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从斐波那契数到中心极限型定理
DOI：
10.1016/j.jcta.2012.03.014
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Yinghui Wang
中科院分区：
文献类型：
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作者：
Steven J. Miller;Yinghui Wang
Zeckendorf的一个美丽的定理指出，每个整数都可以唯一地写成非连续Fibonacci数的和[公式：见正文]。Lekkerkerker（1951-1952）[13]证明了[Fn，Fn+1）中整数的平均被加数为n/（φ2+1），其中φ为黄金平均数。这一点已被概括为：给定非负整数c1，c2，.，cL，其中c1，cL>0和递归序列[公式：H1=1，Hn+1= c1 Hn + c2 Hn −1+ cnH 1 +1（1 <$n<L）和Hn+1= c1 Hn + c2 Hn −1 + cnHn +1−L（n <$L），每个正整数可以唯一地写为∑ aiHi在ai的自然约束下，[Hn，Hn+1）的大小为n，并且当n→∞时，被加数的分布收敛于高斯分布。以前的方法使用数论或遍历理论。我们把这个问题转化为一个组合问题。除了重新推导这些结果之外，我们的方法还推广到其他问题（在续集论文中（Gaudet et al.，预印本[2]）我们展示了这种视角如何使我们能够确定被加数之间的间隙分布）。例如，已知每个整数都可以唯一地写成±Fn的和，使得每两个相同（相反）符号的项的索引至少相差4（3）。负被加数的存在引入了以前问题中没有的复杂性和特征。我们证明了正和负被加数的分布收敛于一个具有可计算的负相关的二元正态分布，即−（21−2φ）/（29+2φ）<$−0.551058。
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers [Formula: see text] . Lekkerkerker (1951–1952) [13] proved the average number of summands for integers in [Fn,Fn+1) is n/(φ2+1), with φ the golden mean. This has been generalized: given non-negative integers c1,c2,…,cLwith c1,cL>0 and recursive sequence [Formula: see text] with H1=1, Hn+1=c1Hn+c2Hn−1+⋯+cnH1+1 (1⩽n<L) and Hn+1=c1Hn+c2Hn−1+⋯+cLHn+1−L(n⩾L), every positive integer can be written uniquely as ∑aiHiunder natural constraints on the aiʼs, the mean and variance of the numbers of summands for integers in [Hn,Hn+1) are of size n, and as n→∞ the distribution of the number of summands converges to a Gaussian. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to other problems (in the sequel paper (Gaudet et al., preprint [2]) we show how this perspective allows us to determine the distribution of gaps between summands). For example, it is known that every integer can be written uniquely as a sum of the ±Fnʼs, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely −(21−2φ)/(29+2φ)≈−0.551058.