关于PLRS分解方差的渐近性态
On the Asymptotic Behavior of Variance of PLRS Decompositions
摘要：
正线性递归序列的形式为$H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$，其中每个$c_i \ge 0$和$c_1 c_L > 0$具有适当选择的初始条件。有一个合法分解的概念（粗略地说，给定序列中的项的和，我们不能使用递归关系来减少它），使得每个正整数都有一个唯一的合法分解使用序列中的项;这推广了Zeckendorf分解，它指出任何正整数都可以唯一地写为不相邻的Fibonacci数的和。以前的工作证明，不仅存在一个分解，但在合法的分解$m \in [H_n, H_{n+1})$ $K_n(m)$的数目收敛到高斯。使用部分分数和生成函数，很容易证明$n$中的均值和方差分别线性增长：$a n + b + o(1)$和$C n + d + o(1)$;困难在于证明$a$和$C$是正的。以前的方法依赖于对与生成函数和特征多项式相关的多项式的精细分析，并且在代数上是繁琐的。我们介绍新的，基本的技术，绕过这些问题。关键的见解是通过条件概率展开使用归纳和自举界限来显示方差是无界的，因此是$C > 0$（均值可以通过简单的计数参数轻松处理）。
英文摘要：
A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly, given a sum of terms in the sequence we cannot use the recurrence relation to reduce it) such that every positive integer has a unique legal decomposition using terms in the sequence; this generalizes the Zeckendorf decomposition, which states any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. Previous work proved not only that a decomposition exists, but that the number of summands $K_n(m)$ in legal decompositions of $m \in [H_n, H_{n+1})$ converges to a Gaussian. Using partial fractions and generating functions it is easy to show the mean and variance grow linearly in $n$: $a n + b + o(1)$ and $C n + d + o(1)$, respectively; the difficulty is proving $a$ and $C$ are positive. Previous approaches relied on delicate analysis of polynomials related to the generating functions and characteristic polynomials, and is algebraically cumbersome. We introduce new, elementary techniques that bypass these issues. The key insight is to use induction and bootstrap bounds through conditional probability expansions to show the variance is unbounded, and hence $C > 0$ (the mean is handled easily through a simple counting argument).
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