Gál--Koksma 引理的精确普适归一化
Exact universal normalizations for the Gál--Koksma lemma
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中文总结 AI 辅助
该文解决了Gál--Koksma引理在连续块二阶矩假设下的普适归一化问题，给出充要可和性判据，并确定临界对数与迭代对数阈值。
中文摘要 AI 辅助
Gál--Koksma 引理是一种标准工具，用于将连续块上的二次均值估计转化为部分和的几乎处处界，且无需独立性、混合性或正交性假设。一个自然的开放问题是确定仅凭该假设必然成立的精确普适增长归一化。在抽象连续块二阶矩假设下的相应普适归一化问题已得到解决，其特征是精确刻画了哪些非递减归一化在整个容许类上一致有效。所得充要可和性判据即使对有界精确中心系统（具有常数优控函数和精确线性块方差）也是尖锐的，并确定了临界对数和迭代对数阈值。
英文摘要
The Gál--Koksma lemma is a standard tool for converting quadratic-mean estimates on consecutive blocks into almost-everywhere bounds for partial sums, without assumptions of independence, mixing, or orthogonality. A natural open problem is to determine exactly which universal growth normalizations are forced by this hypothesis alone. The corresponding universal normalization problem under the abstract consecutive-block second-moment hypothesis is resolved by characterizing exactly which non-decreasing normalizations are valid uniformly over the entire admissible class. The resulting necessary-and-sufficient summability criterion is sharp even for bounded exactly centred systems with constant majorants and exact linear block variance, and determines the critical logarithmic and iterated-logarithmic thresholds.
