The notion of a normal bit sequence was introduced by Borel in 1909; it was the first definition of an individual random object. Normality is a weak notion of randomness requiring only that all $2^n$ factors (substrings) of arbitrary length~$n$ appear with the same limit frequency $2^{-n}$. Later many stronger definitions of randomness were introduced, and in this context normality found its place as ``randomness against a finite-memory adversary''. A quantitative measure of finite-state compressibility was also introduced (the finite-state dimension) and normality means that the finite state dimension is maximal (equals~$1$). Recently Nandakumar, Pulari and S (2023) introduced the notion of relative finite-state dimension for a binary sequence with respect to some other binary sequence (treated as an oracle), and the corresponding notion of conditional (relative) normality. (Different notions of conditional randomness were considered before, but not for the finite memory case.) They establish equivalence between the block frequency and the gambling approaches to conditional normality and finite-state dimensions. In this note we revisit their definitions and explain how this equivalence can be obtained easily by generalizing known characterizations of (unconditional) normality and dimension in terms of compressibility (finite-state complexity), superadditive complexity measures and gambling (finite-state gales), thus also answering some questions left open in the above-mentioned paper.

翻译：Borel于1909年引入正规比特序列的概念，这是对个体随机对象的首次定义。正规性是一种较弱的随机性概念，仅要求任意长度n的所有$2^n$个因子（子串）以相同极限频率$2^{-n}$出现。此后人们提出了许多更强的随机性定义，在此背景下，正规性被定位为"对抗有限记忆敌手的随机性"。定量衡量有限状态可压缩性的指标（有限状态维数）也被提出，正规性即指该维数达到最大值（等于$1$）。近年来，Nandakumar、Pulari与S（2023）引入了关于二元序列相对于另一序列（视为预言机）的相对有限状态维数概念，以及相应的条件（相对）正规性概念。（此前虽已有条件随机性的不同定义，但均未涉及有限记忆情形。）他们建立了条件正规性与有限状态维数在块频率与博弈方法间的等价关系。本文重新审视其定义，阐明如何通过推广已知的（无条件）正规性与维数在可压缩性（有限状态复杂度）、超可加复杂度测度及博弈（有限状态博弈）中的刻画方法，简便推导出这种等价关系，从而也回应了上述论文中遗留的若干开放问题。