Words are sequences of letters over a finite alphabet. We study two intimately related topics for this object: quasi-randomness and limit theory. With respect to the first topic we investigate the notion of uniform distribution of letters over intervals, and in the spirit of the famous Chung–Graham–Wilson theorem for graphs we provide a list of word properties which are equivalent to uniformity. In particular, we show that uniformity is equivalent to counting 3-letter subsequences.

Inspired by graph limit theory we then investigate limits of convergent word sequences, those in which all subsequence densities converge. We show that convergent word sequences have a natural limit, namely Lebesgue measurable functions of the form $f:[0,1]\to [0,1]$. Via this theory we show that every hereditary word property is testable, address the problem of finite forcibility for word limits and establish as a byproduct a new model of random word sequences.

Along the lines of the proof of the existence of word limits, we can also establish the existence of limits for higher dimensional structures. In particular, we obtain an alternative proof of the result by Hoppen, Kohayakawa, Moreira, Ráth and Sampaio (Hoppen et al., 2013) establishing the existence of permutons.

单词是有限字母表上的字母序列。我们为此对象研究两个密切相关的主题：准随机性和极限理论。关于第一个主题，我们研究了字母在区间上的均匀分布的概念，本着著名的 Chung-Graham-Wilson 定理的精神，我们提供了一个等效于均匀性的单词属性列表。特别是，我们证明了一致性相当于对 3 个字母的子序列进行计数。

受图极限理论的启发，我们研究了收敛词序列的极限，即所有子序列密度都收敛的词序列。我们证明了收敛的词序列有一个自然的极限，即形式的 Lebesgue 可测函数$F：[0,1]\to [0,1]$. 通过这个理论，我们证明了每个遗传词的属性都是可测试的，解决了词限制的有限强制问题，并作为副产品建立了一个新的随机词序列模型。

沿着字数限制存在的证明，我们还可以建立更高维结构的限制存在。特别是，我们获得了 Hoppen、Kohayakawa、Moreira、Ráth 和 Sampaio（Hoppen 等，2013）确定置换存在的结果的替代证明。