5 research outputs found
Width and mode of the profile for some random trees of logarithmic height
We propose a new, direct, correlation-free approach based on central moments
of profiles to the asymptotics of width (size of the most abundant level) in
some random trees of logarithmic height. The approach is simple but gives
precise estimates for expected width, central moments of the width and almost
sure convergence. It is widely applicable to random trees of logarithmic
height, including recursive trees, binary search trees, quad trees,
plane-oriented ordered trees and other varieties of increasing trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000187 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
On a variance related to the Ewens sampling formula
A one-parameter multivariate distribution, called the Ewens sampling formula, was introduced in 1972 to model the mutation phenomenon in genetics. The case discussed in this note goes back to Lynch’s theorem in the random binary search tree theory. We examine an additive statistics, being a sum of dependent random variables, and find an upper bound of its variance in terms of the sum of variances of summands. The asymptotically best constant in this estimate is established as the dimension increases. The approach is based on approximation of the extremal eigenvalues of appropriate integral operators and matrices
Profile of random trees: correlation and width of random recursive trees and binary search trees
The levels of trees are nodes with the same distance to the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp sign-changes when one level is fixed and the other one is varying. We also propose a new means for deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by the singularity analysis
Single-cell mutational burden distributions in birth-death processes
Genetic mutations are footprints of cancer evolution and reveal critical
dynamic parameters of tumour growth, which otherwise are hard to measure in
vivo. The mutation accumulation in tumour cell populations has been described
by various statistics, such as site frequency spectra (SFS) from bulk or
single-cell data, as well as single-cell division distributions (DD) and
mutational burden distributions (MBD). An integrated understanding of these
distributions obtained from different sequencing information is important to
illuminate the ecological and evolutionary dynamics of tumours, and requires
novel mathematical and computational tools. We introduce dynamical matrices to
analyse and unite the SFS, DD and MBD based on a birth-death process. Using the
Markov nature of the model, we derive recurrence relations for the expectations
of these three distributions. While recovering classic exact results in
pure-birth cases for the SFS and the DD through our new framework, we also
derive a new expression for the MBD as well as approximations for all three
distributions when death is introduced, confirming our results with stochastic
simulations. Moreover, we demonstrate a natural link between the SFS and the
single-cell MBD, and show that the MBD can be regenerated through the DD.
Surprisingly, the single-cell MBD is mainly driven by the stochasticity arising
in the DD, rather than the extra stochasticity in the number of mutations at
each cell division.Comment: 27 pages, 6 figure