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Genomic evolution can be viewed as string-editing processes driven bymutations. An understanding of the statistical properties resulting from thesemutation processes is of value in a variety of tasks related to biologicalsequence data, e.g., estimation of model parameters and compression. At thesame time, due to the complexity of these processes, designing tractablestochastic models and analyzing them are challenging. In this paper, we studytwo kinds of systems, each representing a set of mutations. In the firstsystem, tandem duplications and substitution mutations are allowed and in theother, interspersed duplications. We provide stochastic models and, viastochastic approximation, study the evolution of substring frequencies forthese two systems separately. Specifically, we show that $k$-mer frequenciesconverge almost surely and determine the limit set. Furthermore, we present amethod for finding upper bounds on entropy for such systems.

Evolution of $k$-mer Frequencies and Entropy in Duplication and Substitution Mutation Systems