Properties of partial-sum discrete probability distributions
Michaela Koščová
PhD thesis advisor: Ján Mačutek

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PhD thesis - Full text

Abstract: The dissertation thesis focuses on two main topics, namely invariance of partial summation and iterated partial summation. 1) For each discrete probability distribution there exists one and only one summation under which the distribution is invariant. If the summations are parametrized, the parent distribution remains invariant in some cases, but sometimes we obtain another distribution as the descendant. We suggest a division of discrete distributions into two families which reflect the behaviour with respect to the parametrization. Some examples of distributions from both families are shown, and the necessary and sufficient condition for a distribution to belong to one of the two families is proven. 2) Partial summations can be applied iteratively. It is shown that the limit distribution exists for a wide spectrum of partial summations when applied to a parent with a finite support. The existence of such limit is proved by the power method which was originally developed in matrix theory to find eigenvalues and eigenvectors.

References
[1] Michaela Koščová, Ján Mačutek, Emmerich Kelih: A data-based classification of Slavic languages: indices of qualitative variation applied to grapheme frequencies. Journal of Quantitative Linguistics. - Vol. 23, No. 2 (2016), s. 177-190.
[2] Ján Mačutek, Ján, Chromý, Michaela Koščová: Menzerath-Altmann law and prothetic /v/ in spoken Czech. Journal of Quantitative Linguistics. - Roč. 26, č. 1 (2019), 66-80.
[3] Michaela Koščová, Radoslav Harman, Ján Mačutek: Iterated partial summations applied to finite-support discrete distributions. Mathematica Slovaca. - Roč. 70, č. 2 (2020), s. 489-496