KFU. Card publication. Calculation of Precise Constants in a Probability Model of Zipf's Law Generation and Asymptotics of Sums of Multinomial Coefficients

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CALCULATION OF PRECISE CONSTANTS IN A PROBABILITY MODEL OF ZIPF'S LAW GENERATION AND ASYMPTOTICS OF SUMS OF MULTINOMIAL COEFFICIENTS

Form of presentation

Articles in international journals and collections

Year of publication

2017

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Authors, employees of KFU

Bochkarev Vladimir Vladimirovich, author

Lerner Eduard Yulevich, author

Bibliographic description in the original language

Bochkarev, V. Calculation of Precise Constants in a Probability Model of Zipf's Law Generation and Asymptotics of Sums of Multinomial Coefficients / V. Bochkarev, E. Lerner // International Journal of Mathematics and Mathematical Sciences. - 2017. - Vol. 2017.

Annotation

Let $\omega_0, \omega_1,\ldots, \omega_n$ be a full set of outcomes (symbols) and let positive $p_i$, $i=0,\ldots,n$, be their probabilities ($\sum_{i=0}^n p_i=1$). Let us treat $\omega_0$ as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the non-increasing order of their probabilities. Let $p(r)$ be the probability of the $r$th word in this list. We prove that if at least one of ratios $\log p_i/\log p_j$, $i,j\in\{ 1,\ldots,n\}$, is irrational, then the limit $\lim_{r\to\infty} p(r)/r^{-1/\gamma}$ exists and differs from zero; here $\gamma$ is the root of the equation $\sum_{i=1}^n p_i^\gamma=1$. The limit constant can be expressed (rather easily) in terms of the entropy of the distribution~$(p_1^\gamma,\ldots,p_n^\gamma)$.

Keywords

Power law, Pascal pyramid, Entropy, Stirling formula, Gaussian approximation, Uniform distribution of sequences, Monkey model

The name of the journal

International Journal of Mathematics and Mathematical Sciences

URL

https://www.hindawi.com/journals/ijmms/2017/9143747/

Please use this ID to quote from or refer to the card

https://repository.kpfu.ru/eng/?p_id=158657&p_lang=2

Resource files

File name	Size (MB)	Format
9143747.pdf	2,46	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bochkarev Vladimir Vladimirovich	ru_RU
dc.contributor.author	Lerner Eduard Yulevich	ru_RU
dc.date.accessioned	2017-01-01T00:00:00Z	ru_RU
dc.date.available	2017-01-01T00:00:00Z	ru_RU
dc.date.issued	2017	ru_RU
dc.identifier.citation	Bochkarev, V. Calculation of Precise Constants in a Probability Model of Zipf's Law Generation and Asymptotics of Sums of Multinomial Coefficients / V. Bochkarev, �. Lerner // International Journal of Mathematics and Mathematical Sciences. - 2017. - Vol. 2017.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=158657&p_lang=2	ru_RU
dc.description.abstract	International Journal of Mathematics and Mathematical Sciences	ru_RU
dc.description.abstract	Let $\omega_0, \omega_1,\ldots, \omega_n$ be a full set of outcomes (symbols) and let positive $p_i$, $i=0,\ldots,n$, be their probabilities ($\sum_{i=0}^n p_i=1$). Let us treat $\omega_0$ as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the non-increasing order of their probabilities. Let $p(r)$ be the probability of the $r$th word in this list. We prove that if at least one of ratios $\log p_i/\log p_j$, $i,j\in\{ 1,\ldots,n\}$, is irrational, then the limit $\lim_{r\to\infty} p(r)/r^{-1/\gamma}$ exists and differs from zero; here $\gamma$ is the root of the equation $\sum_{i=1}^n p_i^\gamma=1$. The limit constant can be expressed (rather easily) in terms of the entropy of the distribution~$(p_1^\gamma,\ldots,p_n^\gamma)$.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Power law	ru_RU
dc.subject	Pascal pyramid	ru_RU
dc.subject	Entropy	ru_RU
dc.subject	Stirling formula	ru_RU
dc.subject	Gaussian approximation	ru_RU
dc.subject	Uniform distribution of sequences	ru_RU
dc.subject	Monkey model	ru_RU
dc.title	Calculation of Precise Constants in a Probability Model of Zipf's Law Generation and Asymptotics of Sums of Multinomial Coefficients	ru_RU
dc.type	Articles in international journals and collections	ru_RU