Rényi映射的广义斐波那契数和极值律
Generalized Fibonacci numbers and extreme value laws for the Rényi map
摘要：
本文证明了一个随机过程的极值律，它是由迭代Rényi映射$x \mapsto βx \pmod 1$得到的，其中假定$β>1$是整数. Haiman（2018）导出了阈值重复集的勒贝格测度的递推公式。我们展示了这个递推公式是如何与一个重新缩放的版本的$k$ -广义斐波那契序列。对于后者的序列，我们推导出一个比内公式，导致一个封闭形式的表达式的分布的部分最大值的随机过程。证明的极值法律是通过推导与Fibonacci序列相关的特征多项式的优势根的尖锐的界限。
英文摘要：
In this paper we prove an extreme value law for a stochastic process obtained by iterating the Rényi map $x \mapsto βx \pmod 1$, where we assume that $β>1$ is an integer. Haiman (2018) derived a recursion formula for the Lebesgue measure of threshold exceedance sets. We show how this recursion formula is related to a rescaled version of the $k$-generalized Fibonacci sequence. For the latter sequence we derive a Binet formula which leads to a closed-form expression for the distribution of partial maxima of the stochastic process. The proof of the extreme value law is completed by deriving sharp bounds for the dominant root of the characteristic polynomial associated with the Fibonacci sequence.
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