On an Asymptotic Series of Ramanujan

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $\phi(X)$ when the distribution of $X$ belongs to a suitable family indexed by a convolution parameter. Examples include the problem of inverse moments for distribution families such as the binomial or the negative binomial.Comment: To appear, Ramanujan

A preferential attachment model with random initial degrees

In this paper, a random graph process ${G(t)}_{t\geq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{t\geq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+\delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $\tau=\min\{\tau_{W}, \tau_{P}\}$, where $\tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{t\geq 1}$ and $\tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.Comment: In the published form of the paper, the proof of Proposition 2.1 is incomplete. This version contains the complete proo

Cut Points and Diffusions in Random Environment

In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.Comment: 44 pages; accepted for publication in "Journal of Theoretical Probability

Combined burden and functional impact tests for cancer driver discovery using DriverPower

The discovery of driver mutations is one of the key motivations for cancer genome sequencing. Here, as part of the ICGC/TCGA Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium, which aggregated whole genome sequencing data from 2658 cancers across 38 tumour types, we describe DriverPower, a software package that uses mutational burden and functional impact evidence to identify driver mutations in coding and non-coding sites within cancer whole genomes. Using a total of 1373 genomic features derived from public sources, DriverPower's background mutation model explains up to 93% of the regional variance in the mutation rate across multiple tumour types. By incorporating functional impact scores, we are able to further increase the accuracy of driver discovery. Testing across a collection of 2583 cancer genomes from the PCAWG project, DriverPower identifies 217 coding and 95 non-coding driver candidates. Comparing to six published methods used by the PCAWG Drivers and Functional Interpretation Working Group, DriverPower has the highest F1 score for both coding and non-coding driver discovery. This demonstrates that DriverPower is an effective framework for computational driver discovery

Searching for planar signatures in WMAP

We search for planar deviations of statistical isotropy in the Wilkinson Microwave Anisotropy Probe (WMAP) data by applying a recently introduced angular-planar statistics both to full-sky and to masked temperature maps, including in our analysis the effect of the residual foreground contamination and systematics in the foreground removing process as sources of error. We confirm earlier findings that full-sky maps exhibit anomalies at the planar ($l$) and angular ($\ell$) scales $(l,\ell)=(2,5),(4,7),$ and $(6,8)$, which seem to be due to unremoved foregrounds since this features are present in the full-sky map but not in the masked maps. On the other hand, our test detects slightly anomalous results at the scales $(l,\ell)=(10,8)$ and $(2,9)$ in the masked maps but not in the full-sky one, indicating that the foreground cleaning procedure (used to generate the full-sky map) could not only be creating false anomalies but also hiding existing ones. We also find a significant trace of an anomaly in the full-sky map at the scale $(l,\ell)=(10,5)$, which is still present when we consider galactic cuts of 18.3% and 28.4%. As regards the quadrupole ($\ell=2$), we find a coherent over-modulation over the whole celestial sphere, for all full-sky and cut-sky maps. Overall, our results seem to indicate that current CMB maps derived from WMAP data do not show significant signs of anisotropies, as measured by our angular-planar estimator. However, we have detected a curious coherence of planar modulations at angular scales of the order of the galaxy's plane, which may be an indication of residual contaminations in the full- and cut-sky maps.Comment: 15 pages with pdf figure

Tail probabilities of St. Petersburg sums, trimmed sums, and their limit

We provide exact asymptotics for the tail probabilities $\mathbb{P} \{S_{n,r} > x\}$ as $x \to \infty$, for fix $n$, where $S_{n,r}$ is the $r$-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the $r$-trimmed limits.Comment: 24 pages, 2 figure

Increased DNA methylation variability in type 1 diabetes across three immune effector cell types

The incidence of type 1 diabetes (T1D) has substantially increased over the past decade, suggesting a role for non-genetic factors such as epigenetic mechanisms in disease development. Here we present an epigenome-wide association study across 406,365 CpGs in 52 monozygotic twin pairs discordant for T1D in three immune effector cell types. We observe a substantial enrichment of differentially variable CpG positions (DVPs) in T1D twins when compared with their healthy co-twins and when compared with healthy, unrelated individuals. These T1D-associated DVPs are found to be temporally stable and enriched at gene regulatory elements. Integration with cell type-specific gene regulatory circuits highlight pathways involved in immune cell metabolism and the cell cycle, including mTOR signalling. Evidence from cord blood of newborns who progress to overt T1D suggests that the DVPs likely emerge after birth. Our findings, based on 772 methylomes, implicate epigenetic changes that could contribute to disease pathogenesis in T1D.This work was funded by the EU-FP7 project BLUEPRINT (282510) and the Wellcome Trust (99148). We thank all twins for taking part in this study; Kerra Pearce and Mark Kristiansen (UCL Genomics) for processing the Illumina Infinium HumanMethylation450 BeadChips; Rasmus Bennet for technical assistance; and Laura Phipps for proofreading the manuscript. The BMBF Pediatric Diabetes Biobank recruits patients from the National Diabetes Patient Documentation System (DPV), and is financed by the German Ministry of Education and Research within the German Competence Net Diabetes Mellitus (01GI1106 and 01GI1109B). It was integrated into the German Center for Diabetes Research in January 2015. We thank the Swedish Research Council and SUS Funds for support. We gratefully acknowledge the participation of all NIHR Cambridge BioResource volunteers, and thank the Cambridge BioResource staff for their help with volunteer recruitment. We thank members of the Cambridge BioResource SAB and Management Committee for their support of our study and the NIHR Cambridge Biomedical Research Centre for funding. The Cardiovascular Epidemiology Unit is supported by the UK Medical Research Council (G0800270), BHF (SP/09/002), and NIHR Cambridge Biomedical Research Centre. Research in the Ouwehand laboratory is supported by the NIHR, BHF (PG-0310-1002 and RG/09/12/28096) and NHS Blood and Transplant. K.D. is funded as a HSST trainee by NHS Health Education England. M.F. is supported by the BHF Cambridge Centre of Excellence (RE/13/6/30180). A.D., E.L., L.C. and P.F. receive additional support from the European Molecular Biology Laboratory. A.K.S. is supported by an ADA Career Development Award (1-14-CD-17). B.O.B. and R.D.L. acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) and European Federation for the Study of Diabetes, respectively

Reconstructing cell cycle and disease progression using deep learning

We show that deep convolutional neural networks combined with nonlinear dimension reduction enable reconstructing biological processes based on raw image data. We demonstrate this by reconstructing the cell cycle of Jurkat cells and disease progression in diabetic retinopathy. In further analysis of Jurkat cells, we detect and separate a subpopulation of dead cells in an unsupervised manner and, in classifying discrete cell cycle stages, we reach a sixfold reduction in error rate compared to a recent approach based on boosting on image features. In contrast to previous methods, deep learning based predictions are fast enough for on-the-fly analysis in an imaging flow cytometer