内函数的重对数律
Law of iterated logarithm for inner functions
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中文总结 AI 辅助
研究固定原点的非旋转内函数迭代线性组合的重对数律，利用内函数相关性质构造反向鞅，在温和系数条件及费勒型假设下，证明反向鞅差部分和的LIL并转移到内函数迭代线性组合，还确定了LIL尺度下的子序列极限点。
中文摘要 AI 辅助
在最近的一项工作中，为固定原点的非旋转内函数的迭代的线性组合建立了中心极限定理。本文在相同设置下证明了重对数律（LIL），对系数条件非常温和。还确定了LIL尺度下的所有子序列极限点。利用内函数的亚历山德罗夫 - 克拉克分解和保测性质构造接近内函数线性组合的反向鞅。在费勒型假设下证明了反向鞅差部分和的LIL，进而转移到内函数迭代的线性组合上。
英文摘要
In a recent work [\emph{Adv. Math.} 401 (2022), Paper No. 108318], a central limit theorem was established for the linear combinations of the iterates of a non-rotational inner function fixing the origin. In this paper, we prove the law of iterated logarithm (LIL) in the same setup, with a very mild condition on the coefficients. We also identify the full set of subsequential limit points at the LIL scale. Using the Aleksandrov--Clark decomposition and measure-preserving properties of the inner functions, one can construct a reverse martingale that is close to the linear combinations of inner functions. We prove the LIL for the partial sums of reverse martingale differences under a Feller-type assumption, which then transfers to the linear combinations of iterates of the inner functions.
