Predicting single-cell transcription dynamics even when the central limit theorem fails

Central Limit Theorem in Multitype Branching Random Walk

A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

I-13: Transcriptome Dynamics of Human and Mouse Preimplantation Embryos Revealed by Single Cell RNA-Sequencing

Background: Mammalian preimplantation development is a complex process involving dramatic changes in the transcriptional architecture. However, it is still unclear about the crucial transcriptional network and key hub genes that regulate the proceeding of preimplantation embryos. Materials and Methods: Through single-cell RNAsequencing (RNA-seq) of both human and mouse preimplantation embryos, ...

Central and Local Limit Theories in Markov Dependent Random Variables

Vibration of Road Vehicles with Non linear Suspensions

In order to investigate the effects of non-linear springs in vibrating behavior of vehicles, the independent suspension of conventional vehicles could be modeled as a non-linear single degree of freedom system. The equation of motion for the system would be a non-linear third order ordinary differential equation, when considering the elasticity of rubber bushings in joints of shock absorber. It...