-mer-based methods are widely used in bioinformatics, but there are many gaps in our understanding of their statistical properties. Here, we consider the simple model where a sequence (e.g., a genome or a read) undergoes a simple mutation process through which each nucleotide is mutated independently with some probability , under the assumption that there are no spurious -mer matches. How does this process affect the -mers of ? We derive the expectation and variance of the number of mutated -mers and of the number of islands (a maximal interval of mutated -mers) and oceans (a maximal interval of nonmutated -mers). We then derive hypothesis tests and confidence intervals (CIs) for given an observed number of mutated -mers, or, alternatively, given the Jaccard similarity (with or without MinHash). We demonstrate the usefulness of our results using a few select applications: obtaining a CI to supplement the Mash distance point estimate, filtering out reads during alignment by Minimap2, and rating long-read alignments to a de Bruijn graph by Jabba.
