We study a deliberately simple, fully non-linguistic model of text: a sequence of independent draws from a finite alphabet of letters plus a single space symbol. A word is defined as a maximal block of non-space symbols. Within this symbol-level framework, which assumes no morphology, syntax, or semantics, we derive several structural results. First, word lengths follow a geometric distribution governed solely by the probability of the space symbol. Second, the expected number of words of a given length, and the expected number of distinct words of that length, admit closed-form expressions based on a coupon-collector argument. This yields a critical word length k* at which word types transition from appearing many times on average to appearing at most once. Third, combining the exponential growth of the number of possible strings of length k with the exponential decay of the probability of each string, we obtain a Zipf-type rank-frequency law p(r) proportional to r^{-alpha}, with an exponent determined explicitly by the alphabet size and the space probability. Our contribution is twofold. Mathematically, we give a unified derivation linking word lengths, vocabulary growth, critical length, and rank-frequency structure in a single explicit model. Conceptually, we argue that this provides a structurally grounded null model for both natural-language word statistics and token statistics in large language models. The results show that Zipf-like patterns can arise purely from combinatorics and segmentation, without optimization principles or linguistic organization, and help clarify which phenomena require deeper explanation beyond random-text structure.

We study minimax strategies for the online prediction problem with expert advice.

We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB).

All experiments were run on at least three random seeds.

A.1 Proof of Proposition 1 and 3 To prove Proposition 1, we first need the following lemma: Readers may refer to [47] for the proof of this lemma. Let's first consider the left handside, The first inequality is due to information processing inequality. The compactness assumption in Proposition 2 seems restrictive, since BNNs with Gaussian priors on weights will break the compactness assumption. Indeed, the assumptions in proposition 2 are merely sufficient conditions. In this section, we discuss the non-parametric counter part of Proposition 2, i.e., is the grid functional KL between a parametric model and a Gaussian process is still finite?

The frequency of the preferred order for a noun phrase formed by demonstrative, numeral, adjective and noun has received significant attention over the last two decades. We investigate the actual distribution of the preferred 24 possible orders. There is no consensus on whether it can be well-fitted by an exponential or a power law distribution. We find that an exponential distribution is a much better model. This finding and other circumstances where an exponential-like distribution is found challenge the view that power-law distributions, e.g., Zipf's law for word frequencies, are inevitable. We also investigate which of two exponential distributions gives a better fit: an exponential model where the 24 orders have non-zero probability or an exponential model where the number of orders that can have non-zero probability is variable. When parsimony and generalizability are prioritized, we find strong support for the exponential model where all 24 orders have non-zero probability. This finding suggests that there is no hard constraint on word order variation and then unattested orders merely result from undersampling, consistently with Cysouw's view.

We study minimax strategies for the online prediction problem with expert advice.