Approximations for sums of three-valued 1-dependent symmetric random variables

The sum of symmetric three-point 1-dependent nonidentically distributed random variables is approximated by a compound Poisson distribution. The accuracy of approximation is estimated in the local and total variation norms. For distributions uniformly bounded from zero,the accuracy of approximation is of the order O(n–1). In the general case of triangular arrays of identically distributed summands, the accuracy is at least of the order O(n–1/2). Nonuniform estimates are obtained for distribution functions and probabilities. The characteristic functionmethod is used.

Quantile normalization of single-cell RNA-seq read counts without unique molecular identifiers

Single-cell RNA-seq (scRNA-seq) profiles gene expression of individual cells. Unique molecular identifiers (UMIs) remove duplicates in read counts resulting from polymerase chain reaction, a major source of noise. For scRNA-seq data lacking UMIs, we propose quasi-UMIs: quantile normalization of read counts to a compound Poisson distribution empirically derived from UMI datasets. When applied to ground-truth datasets having both reads and UMIs, quasi-UMI normalization has higher accuracy than alternatives such as census counts. Using quasi-UMIs enables methods designed specifically for UMI data to be applied to non-UMI scRNA-seq datasets.

On coincidences of tuples in a q-ary tree with random labels of vertices

Let all vertices of a complete q-ary tree of finite height be independently and equiprobably labeled by the elements of some finite alphabet. We consider the numbers of pairs of identical tuples of labels on chains of subsequent vertices in the tree. Exact formulae for the expectations of these numbers are obtained, convergence to the compound Poisson distribution is proved. For the size of cluster composed by pairs of identically labeled chains we also obtain exact formula for the expectation.

On coincidences of tuples in a binary tree with random labels of vertices

Let all vertices of a complete binary tree of finite height be independently and equiprobably labeled by the elements of some finite alphabet. We consider the numbers of pairs of identical tuples of labels on chains of subsequent vertices in the tree. Exact formulae for the expectations of these numbers are obtained. Convergence to the compound Poisson distribution is proved.