论文标题
rényi地图的广义斐波那契数和极值法律
Generalized Fibonacci numbers and extreme value laws for the Rényi map
论文作者
论文摘要
在本文中，我们证明了通过迭代rényi映射$ x \ mapstoβx\ pmod 1 $获得的随机过程的极值定律，我们假设$β> 1 $是整数。海曼（Haiman，2018）得出了Lebesgue阈值超出阈值的递归公式。我们展示了此递归公式与$ K $循环的斐波那契序列的重新定制版本有关。对于后一个序列，我们得出了一个BINET公式，该公式会导致封闭形式的表达，以分布随机过程的部分最大值。极值定律的证明是通过得出与斐波那契序列相关的特征多项式的主要词根来完成的。
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.
