A COLLECTION OF CENTRAL LIMIT TYPE RESULTS IN GENERALIZED ZECKENDORF DECOMPOSITIONS
A COLLECTION OF CENTRAL LIMIT TYPE RESULTS IN GENERALIZED ZECKENDORF DECOMPOSITIONS
复制标题
广义ZECKENDORF分解的中心极限类型结果的集合
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Steven J. Miller
中科院分区：
文献类型：
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作者：
Steven J. Miller
Zeckendorf定理指出，如果Fibonacci数的索引为F1 = 1，F2 = 2，F3 = 3，F4 = 5，. . .，则每个正整数都可以唯一地写成不相邻的Fibonacci数之和。这个结果可以推广到某些类的线性递归关系{Gn}与适当的概念分解。对于许多分解，已知在M ∈ [Gn，Gn+1）的分解中的和数的分布在n → ∞时收敛到高斯。这项工作讨论了一个更一般的方法来证明这种渐近高斯行为，也绕过了以前的方法中的技术障碍。该方法的动机是由二项式an，k =（n k）。二项式满足递归an，k = an−1，k + an−1，k−1，并且众所周知具有如下性质：由Pr[Xn = k] = an，k/ ∑∞ i=0 an，i在n → ∞时收敛于高斯分布。这种新方法证明，适当的二维递归表现出类似的渐进高斯行为。由此，我们可以重新证明由许多线性递归关系给出的分解中的和项数是渐近高斯的，并证明了对任意非负整数g，当n → ∞时，M ∈ [Gn，Gn+1）的分解中大小为g的间隙数也收敛于高斯.
Zeckendorf’s Theorem states that if the Fibonacci numbers are indexed as F1 = 1, F2 = 2, F3 = 3, F4 = 5, . . . , then every positive integer can be written uniquely as the sum of non-adjacent Fibonacci numbers. This result can be generalized to certain classes of linear recurrence relations {Gn} with appropriate notions of decompositions. For many decompositions, the distribution of the number of summands in the decomposition of an M ∈ [Gn, Gn+1) is known to converge to a Gaussian as n → ∞. This work discusses a more general approach to proving this kind of asymptotic Gaussian behavior that also bypasses technical obstructions in previous approaches. The approach is motivated by the binomials an,k = ( n k ) . The binomials satisfy the recursion an,k = an−1,k + an−1,k−1 and are well known to have the property that the random variables {Xn}n=1 given by Pr[Xn = k] = an,k/ ∑∞ i=0 an,i converge to a Gaussian as n → ∞. This new approach proves that appropriate two-dimensional recurrences exhibit similar asymptotic Gaussian behavior. From this, we can reprove that the number of summands in decompositions given by many linear recurrence relations is asymptotically Gaussian and additionally prove that for any non-negative integer g, the number of gaps of size g in the decomposition of an M ∈ [Gn, Gn+1) also converges to a Gaussian as n → ∞.
广义 Zeckendorf 分解间隙的中心极限定理
DOI：
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发表时间：
2019
期刊：
The Fibonacci quarterly
影响因子：
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作者：
Li, Ray;Miller, Steven J.
通讯作者：
Miller, Steven J.