Geodesic
Frequency distributions in bioinformatics: the development
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2010), pp. 3-22

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The mathematical investigation of large-scale biomolecular sequences is being carried out by analyzing the properties of events arising in such sequences. This survey is devoted to discussion of results in this field. Based on general empirical facts being fulfilled for all frequency distributions, we discuss the axiomatics suggested by J. Astola and E. Danielyan. The axiomatics postulates the regular variation of frequency distribution with asymptotically constant slowly varying component, the form of its shape, and the stability by parameters. The verification of the axiomatics fulfillment for well-known frequency distributions is done. The paper describes also methods of construction of new parametric families of frequency distributions. These methods are: usage of stationary distributions of birth-death process, special functions, stable densities, etc. The problem of stability by parameters is formulated the results on stability by parameters in terms of various classical metrics are given. The conditions of regular variation for different families of frequency distributions are formulated.
Keywords: frequency distributbiomion, olecular sequence, regular variation, convexity, stability by parameters, asymptotic expansion. 
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J. T. Astola; E. A. Danielyan; S. K. Arzumanyan. Frequency distributions in bioinformatics: the development. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2010), pp. 3-22. http://geodesic.mathdoc.fr/item/UZERU_2010_3_a0/