arXiv subjects

Łukasz Dębowski

Publications and source records attributed to Łukasz Dębowski.

At least 19 recordsLinked to original sources

arXiv2026

Vocabulary Growth Fundamentals: Bernstein Functions and Hausdorff Sequences

We survey the theory of vocabulary growth founded in the setting of stochastic processes. In particular, we model the expected number of types through Bernstein functions and Hausdorff sequences. These classes of mathematical objects, defined by alternating signs of their derivatives or differences, can be related to continuous-time Poisson point processes and discrete-time IID processes, respectively. Building on previous accounts of the vocabulary growth, we integrate the broader theories of Bernstein functions and Hausdorff sequences and connect them with recently developed hapax rate models. In particular, we prove that the logistic hapax rate model has a non-negative spectrum and hence it defines a Bernstein function, thereby solving an earlier posed problem. We also analyze the limitations of the Bernstein--Hausdorff theory of the vocabulary growth by considering its generalizations under stationary and Weibull renewal processes.

math.PR↗

arXiv2026

Corrections of Zipf's and Heaps' Laws Derived from Hapax Rate Models

The article introduces corrections to Zipf's and Heaps' laws based on systematic models of the proportion of hapaxes, i.e., words that occur once. The derivation rests on two assumptions. The first one is the standard urn model which predicts that marginal frequency distributions for shorter texts look as if word tokens were sampled blindly from a given longer text. The second assumption posits that the hapax rate is a simple function of the text length. Four such functions are discussed: the constant model, the cancelation model, the linear model, and the logistic model. As a simple illustration, it is shown that the logistic model yields the best fit for a sample of 14 texts in English. The need and the availability of more complex mixture models that reflect two-regime vocabularies for larger corpora is also discussed.

cs.CL↗

arXiv2026

Multiperiodic Processes: Ergodic Sources with a Sublinear Entropy

Several explicit stochastic processes are known to satisfy Hilberg's law, a power-law growth of block entropy conjectured for natural language and recently connected to the neural scaling law. Existing examples either possess a positive Shannon entropy rate, are non-ergodic, or require comparatively involved constructions. We introduce multiperiodic processes, a new class of stationary ergodic processes over the natural numbers generated by random shifts of deterministic multiperiodic sequences. Under mild conditions, multiperiodic processes have vanishing Shannon entropy rate and, under a suitable parameterization, they satisfy both Zipf's law for symbol frequencies and Hilberg's law for block entropy. Since multiperiodic processes are not mixing, we identify the open problem of constructing an elementary strongly mixing source with vanishing entropy rate and Hilberg's law.

cs.IT↗

arXiv2026

From Zipf's Law to Neural Scaling through Heaps' Law and Hilberg's Hypothesis

We inspect the deductive connection between the neural scaling law and Zipf's law -- two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model -- such as a large language model -- changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all four statistical laws.

cs.IT↗

arXiv2025

Repetition and recurrence times: Dual statements and summable mixing rates

By an analogy to the duality between the recurrence time and the longest match length, we introduce a quantity dual to the maximal repetition length, which we call the repetition time. Extending prior results, we sandwich the repetition time in terms of unconditional and conditional min-entropies. The upper bound holds if the mixing rate $ϕ(n)$ is summable, whereas the lower bound only assumes stationarity. Our reasoning makes a repeated use of dualities between so-called times and so-called counts that generalize the duality of the recurrence time and the longest match length. We also discuss the analogy of these results with the Wyner-Ziv/Ornstein-Weiss theorem, which sandwiches the recurrence time in terms of Shannon entropies.

cs.IT↗

arXiv2025

From Letters to Words and Back: Invertible Coding of Stationary Measures

Motivated by problems of statistical language modeling, we consider probability measures on infinite sequences over two countable alphabets of a different cardinality, such as letters and words. We introduce an invertible mapping between such measures, called the normalized transport, that preserves both stationarity and ergodicity. The normalized transport applies so called self-avoiding codes that generalize comma-separated codes and specialize bijective stationary codes. The normalized transport is also connected to the usual measure transport via underlying asymptotically mean stationary measures. It preserves the ergodic decomposition. The normalized transport and self-avoiding codes arise in the problem of successive recurrence times. We show that successive recurrence times are ergodic for an ergodic measure, which strengthens a result by Chen Moy from 1959. We also relate the entropy rates of processes linked by the normalized transport.

math.PR↗

arXiv2025

Local Grammar-Based Coding Revisited

In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we invoke a simple harmonic bound on ranked probabilities, which reminds Zipf's law and simplifies universality proofs for minimal local grammar-based codes. Second, we refine known bounds on the vocabulary size, showing its partial power-law equivalence with mutual information and redundancy. These bounds are relevant for linking Zipf's law with the neural scaling law for large language models. Third, we develop a framework for universal codes with fixed infinite vocabularies, recasting universal coding as matching ranked patterns that are independent of empirical data. Finally, we analyze grammar-based codes with finite vocabularies being empirical rank lists, proving that that such codes are also universal. These results extend foundations of universal grammar-based coding and reaffirm previously stated connections to power laws for human language and language models.

cs.IT↗

arXiv2023

Corrections to "Universal Densities Exist for Every Finite Reference Measure"

In the article "Universal Densities Exist for Every Finite Reference Measure" (IEEE Trans. Inform. Theory, vol. 69, no. 8, pp. 5277--5288, 2023) we neglected to mention relevant contributions of Boris Ryabko. We cited a source by him that contains a construction of the universal density that we claimed to be our own invention without checking the source after drafting the article. Our article was motivated by a preprint by Feutrill and Roughan, about which we had learned when reviewing the PhD thesis by Andrew Feutrill. Whereas we were not allowed to contact Feutrill and Roughan besides the review form, we developed some ideas of theirs further, ignoring that we stepped into the area previously researched by Ryabko. Our published results exceed those by Ryabko but the article should have been refocused to report Ryabko's contributions. In this note, we detail our citation mistakes.

cs.IT↗

arXiv2023

Universal Densities Exist for Every Finite Reference Measure

As it is known, universal codes, which estimate the entropy rate consistently, exist for stationary ergodic sources over finite alphabets but not over countably infinite ones. We generalize universal coding as the problem of universal densities with respect to a fixed reference measure on a countably generated measurable space. We show that universal densities, which estimate the differential entropy rate consistently, exist for finite reference measures. Thus finite alphabets are not necessary in some sense. To exhibit a universal density, we adapt the non-parametric differential (NPD) entropy rate estimator by Feutrill and Roughan. Our modification is analogous to Ryabko's modification of prediction by partial matching (PPM) by Cleary and Witten. Whereas Ryabko considered a mixture over Markov orders, we consider a mixture over quantization levels. Moreover, we demonstrate that any universal density induces a strongly consistent Cesàro mean estimator of conditional density given an infinite past. This yields a universal predictor with the $0-1$ loss for a countable alphabet. Finally, we specialize universal densities to processes over natural numbers and on the real line. We derive sufficient conditions for consistent estimation of the entropy rate with respect to infinite reference measures in these domains.

cs.IT↗

arXiv2022

There Are Fewer Facts Than Words: Communication With A Growing Complexity

We present an impossibility result, called a theorem about facts and words, which pertains to a general communication system. The theorem states that the number of distinct words used in a finite text is roughly greater than the number of independent elementary persistent facts described in the same text. In particular, this theorem can be related to Zipf's law, power-law scaling of mutual information, and power-law-tailed learning curves. The assumptions of the theorem are: a finite alphabet, linear sequence of symbols, complexity that does not decrease in time, entropy rate that can be estimated, and finiteness of the inverse complexity rate.

cs.IT↗

arXiv2021

Universal Coding and Prediction on Ergodic Martin-Löf Random Points

Suppose that we have a method which estimates the conditional probabilities of some unknown stochastic source and we use it to guess which of the outcomes will happen. We want to make a correct guess as often as it is possible. What estimators are good for this? In this work, we consider estimators given by a familiar notion of universal coding for stationary ergodic measures, while working in the framework of algorithmic randomness, i.e, we are particularly interested in prediction of Martin-Löf random points. We outline the general theory and exhibit some counterexamples. Completing a result of Ryabko from 2009 we also show that universal probability measure in the sense of universal coding induces a universal predictor in the prequential sense. Surprisingly, this implication holds true provided the universal measure does not ascribe too low conditional probabilities to individual symbols. As an example, we show that the Prediction by Partial Matching (PPM) measure satisfies this requirement with a large reserve.

math.PR↗

arXiv2020

Bounds for Algorithmic Mutual Information and a Unifilar Order Estimator

Inspired by Hilberg's hypothesis, which states that mutual information between blocks for natural language grows like a power law, we seek for links between power-law growth rate of algorithmic mutual information and of some estimator of the unifilar order, i.e., the number of hidden states in the generating stationary ergodic source in its minimal unifilar hidden Markov representation. We consider an order estimator which returns the smallest order for which the maximum likelihood is larger than a weakly penalized universal probability. This order estimator is intractable and follows the ideas by Merhav, Gutman, and Ziv (1989) and by Ziv and Merhav (1992) but in its exact form seems overlooked despite some nice theoretical properties. In particular, we can prove both strong consistency of this order estimator and an upper bound of algorithmic mutual information in terms of it. Using both results, we show that all (also uncomputable) sources of a finite unifilar order exhibit sub-power-law growth of algorithmic mutual information and of the unifilar order estimator. In contrast, we also exhibit an example of unifilar processes of a countably infinite order, with a deterministic pushdown automaton and an algorithmically random oracle, for which the mentioned two quantities grow as a power law with the same exponent. We also relate our results to natural language research.

cs.IT↗

arXiv2020

On a Class of Markov Order Estimators Based on PPM and Other Universal Codes

We investigate a class of estimators of the Markov order for stationary ergodic processes which form a slight modification of the constructions by Merhav, Gutman, and Ziv in 1989 as well as by Ryabko, Astola, and Malyutov in 2006 and 2016. All the considered estimators compare the estimate of the entropy rate given by a universal code with the empirical conditional entropy of a string and return the order for which the two quantities are approximately equal. However, our modification, which we call universal Markov orders, satisfies a few attractive properties, not shown by the mentioned authors for their original constructions. Firstly, the universal Markov orders are almost surely consistent, without any restrictions. Secondly, they are upper bounded asymptotically by the logarithm of the string length divided by the entropy rate. Thirdly, if we choose the Prediction by Partial Matching (PPM) as the universal code then the number of distinct substrings of the length equal to the universal Markov order constitutes an upper bound for the block mutual information. Thus universal Markov orders can be also used indirectly for quantification of long memory for an ergodic process.

cs.IT↗

arXiv2017

Is Natural Language a Perigraphic Process? The Theorem about Facts and Words Revisited

As we discuss, a stationary stochastic process is nonergodic when a random persistent topic can be detected in the infinite random text sampled from the process, whereas we call the process strongly nonergodic when an infinite sequence of independent random bits, called probabilistic facts, is needed to describe this topic completely. Replacing probabilistic facts with an algorithmically random sequence of bits, called algorithmic facts, we adapt this property back to ergodic processes. Subsequently, we call a process perigraphic if the number of algorithmic facts which can be inferred from a finite text sampled from the process grows like a power of the text length. We present a simple example of such a process. Moreover, we demonstrate an assertion which we call the theorem about facts and words. This proposition states that the number of probabilistic or algorithmic facts which can be inferred from a text drawn from a process must be roughly smaller than the number of distinct word-like strings detected in this text by means of the PPM compression algorithm. We also observe that the number of the word-like strings for a sample of plays by Shakespeare follows an empirical stepwise power law, in a stark contrast to Markov processes. Hence we suppose that natural language considered as a process is not only non-Markov but also perigraphic.

cs.IT↗

arXiv2017

Maximal Repetition and Zero Entropy Rate

Maximal repetition of a string is the maximal length of a repeated substring. This paper investigates maximal repetition of strings drawn from stochastic processes. Strengthening previous results, two new bounds for the almost sure growth rate of maximal repetition are identified: an upper bound in terms of conditional Rényi entropy of order $γ>1$ given a sufficiently long past and a lower bound in terms of unconditional Shannon entropy ($γ=1$). Both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time. We also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity. In particular, it is shown that a power-law logarithmic growth of maximal repetition with respect to the string length, recently observed for texts in natural language, may hold only if the conditional Rényi entropy rate given a sufficiently long past equals zero. According to this observation, natural language cannot be faithfully modeled by a typical hidden Markov process, which is a class of basic language models used in computational linguistics.

cs.IT↗

arXiv2016

Regular Hilberg Processes: An Example of Processes with a Vanishing Entropy Rate

A regular Hilberg process is a stationary process that satisfies both a hyperlogarithmic growth of maximal repetition and a power-law growth of topological entropy, which are a kind of dual conditions. The hyperlogarithmic growth of maximal repetition has been experimentally observed for texts in natural language, whereas the power-law growth of topological entropy implies a vanishing Shannon entropy rate and thus probably does not hold for natural language. In this paper, we provide a constructive example of regular Hilberg processes, which we call random hierarchical association (RHA) processes. Our construction does not apply the standard cutting and stacking method. For the constructed RHA processes, we demonstrate that the expected length of any uniquely decodable code is orders of magnitude larger than the Shannon block entropy of the ergodic component of the RHA process. Our proposition supplements the classical result by Shields concerning nonexistence of universal redundancy rates.

cs.IT↗

arXiv2016

Consistency of the Plug-In Estimator of the Entropy Rate for Ergodic Processes

A plug-in estimator of entropy is the entropy of the distribution where probabilities of symbols or blocks have been replaced with their relative frequencies in the sample. Consistency and asymptotic unbiasedness of the plug-in estimator can be easily demonstrated in the IID case. In this paper, we ask whether the plug-in estimator can be used for consistent estimation of the entropy rate $h$ of a stationary ergodic process. The answer is positive if, to estimate block entropy of order $k$, we use a sample longer than $2^{k(h+ε)}$, whereas it is negative if we use a sample shorter than $2^{k(h-ε)}$. In particular, if we do not know the entropy rate $h$, it is sufficient to use a sample of length $(|X|+ε)^{k}$ where $|X|$ is the alphabet size. The result is derived using $k$-block coding. As a by-product of our technique, we also show that the block entropy of a stationary process is bounded above by a nonlinear function of the average block entropy of its ergodic components. This inequality can be used for an alternative proof of the known fact that the entropy rate a stationary process equals the average entropy rate of its ergodic components.

cs.IT↗

arXiv2015

A Preadapted Universal Switch Distribution for Testing Hilberg's Conjecture

Hilberg's conjecture about natural language states that the mutual information between two adjacent long blocks of text grows like a power of the block length. The exponent in this statement can be upper bounded using the pointwise mutual information estimate computed for a carefully chosen code. The bound is the better, the lower the compression rate is but there is a requirement that the code be universal. So as to improve a received upper bound for Hilberg's exponent, in this paper, we introduce two novel universal codes, called the plain switch distribution and the preadapted switch distribution. Generally speaking, switch distributions are certain mixtures of adaptive Markov chains of varying orders with some additional communication to avoid so called catch-up phenomenon. The advantage of these distributions is that they both achieve a low compression rate and are guaranteed to be universal. Using the switch distributions we obtain that a sample of a text in English is non-Markovian with Hilberg's exponent being $\le 0.83$, which improves over the previous bound $\le 0.94$ obtained using the Lempel-Ziv code.

cs.IT↗