We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise, forecasting systems, and study their properties. The richer mathematical structure that thus arises lets us, amongst other things, better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every sequence of binary outcomes has an associated filter of intervals it is random for. It may happen that none of these intervals is precise—a single real number—which justifies the title of this paper. We illustrate this by showing that randomness associated with non-stationary precise forecasting systems can be captured by a constant interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision. But imprecise randomness can't always be explained away as a result of oversimplification: we show that there are sequences that are random for a constant interval forecast, but never random for any computable (more) precise forecasting system. We also show that the set of sequences that are random for a non-vacuous interval forecasting system is meagre, as it is for precise forecasting systems.

我们使用博弈论概率的马丁格尔理论方法将不精确性纳入随机性研究中。特别是，我们定义了几个与区间而非精确预测系统相关的随机性概念，并研究了它们的特性。由此产生的更丰富的数学结构让我们能够更好地理解现有结果并将其放置在精确极限上。当我们关注恒定区间预测时，我们发现每个二元结果序列都有一个相关的区间过滤器，它是随机的。可能这些区间都不是精确的——一个单一的实数——这证明了这篇论文的标题是正确的。我们通过展示与非平稳精确预测系统相关的随机性可以通过恒定间隔预测来说明这一点，那么它必须不那么精确：因此，模型简单性的增加是以精度的损失为代价的。但由于过于简单化，不精确的随机性并不总是能被解释掉：我们表明，对于恒定间隔预测，有些序列是随机的，但对于任何可计算（更）精确的预测系统来说，它们绝不是随机的。我们还表明，非空区间预测系统的随机序列集是微不足道的，就像精确预测系统一样。