Limit theorems for sequences of random trees

We consider a random tree and introduce a metric in the space of trees to define the ""mean tree"" as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.

Billiards in a General Domain with Random Reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain D R(d) until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord ""picked at random"" in D, and we study the angle of intersection of the process with a (d - 1) -dimensional manifold contained in D.

Hypothesis testing in an errors-in-variables model with heteroscedastic measurement errors

In many epidemiological studies it is common to resort to regression models relating incidence of a disease and its risk factors. The main goal of this paper is to consider inference on such models with error-prone observations and variances of the measurement errors changing across observations. We suppose that the observations follow a bivariate normal distribution and the measurement errors are normally distributed. Aggregate data allow the estimation of the error variances. Maximum likelihood estimates are computed numerically via the EM algorithm. Consistent estimation of the asymptotic variance of the maximum likelihood estimators is also discussed. Test statistics are proposed for testing hypotheses of interest. Further, we implement a simple graphical device that enables an assessment of the model`s goodness of fit. Results of simulations concerning the properties of the test statistics are reported. The approach is illustrated with data from the WHO MONICA Project on cardiovascular disease. Copyright (C) 2008 John Wiley & Sons, Ltd.

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119-130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model. (C) 2007 Elsevier B.V. All rights reserved.

Geodesics and Almost Geodesic Cycles in Random Regular Graphs

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d(G)(u, v) is at least d(C)(u, v) - e(n). Let omega(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= log(d-1)log(d-1) n+omega(n) and vertical bar C vertical bar =2 log(d-1) n+O(omega(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 115-136, 2011

Asymmetric Ramsey Properties of Random Graphs Involving Cliques

Consider the following problem: Forgiven graphs G and F(1),..., F(k), find a coloring of the edges of G with k colors such that G does not contain F; in color i. Rodl and Rucinski studied this problem for the random graph G,,, in the symmetric case when k is fixed and F(1) = ... = F(k) = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p <= bn(-beta) for some constants b = b(F,k) and beta = beta(F). This result is essentially best possible because for p >= Bn(-beta), where B = B(F, k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n(-beta(F1,..., Fk)) for arbitrary F(1), ..., F(k). In this article we address the case when F(1),..., F(k) are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G(n,p) with p <= bn(-beta) for some constant b = b(F(1),..., F(k)), where beta = beta(F(1),..., F(k)) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F,,..., Fk) such that for p >= Bn(-beta) the random graph G(n,p) a.a.s. does not have a valid k-edge-coloring provided the so-called KLR-conjecture holds. (C) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 34, 419-453, 2009

Network genealogy of 195-bp satellite DNA supports the superimposed hybridization hypothesis of Trypanosoma cruzi evolutionary pattern

Trypanosoma cruzi is highly diverse genetically and has been partitioned into six discrete typing units (DTUs), recently re-named T. cruzi I-VI. Although T. cruzi reproduces predominantly by binary division, accumulating evidence indicates that particular DTUs are the result of hybridization events. Two major scenarios for the origin of the hybrid lineages have been proposed. It is accepted widely that the most heterozygous TcV and TcVI DTUs are the result of genetic exchange between TcII and TcIII strains. On the other hand, the participation of a TcI parental in the current genome structure of these hybrid strains is a matter of debate. Here, sequences of the T. cruzi-specific 195-bp satellite DNA of TcI, TcII, Tat, TcV, and TcVI strains have been used for inferring network genealogies. The resulting genealogy showed a high degree of reticulation, which is consistent with more than one event of hybridization between the Tc DTUs. The data also strongly suggest that Tat is a hybrid with two distinct sets of satellite sequences, and that genetic exchange between TcI and TcII parentals occurred within the pedigree of the TcV and TcVI DTUs. Although satellite DNAs belong to the fast-evolving portion of eukaryotic genomes, in >100 satellite units of nine T. cruzi strains we found regions that display 100% identity. No DTU-specific consensus motifs were identified, inferring species-wide conservation. (C) 2010 Elsevier B.V. All rights reserved.

Keratinocyte growth factor: a new mesothelial targeted therapy to reduce postoperative pericardial adhesions

Background: Several methods have been utilized to prevent pericardial and retrosternal adhesions, but none of them evaluated the mesothelial regenerative hypothesis. There are evidences that the mesothelial trauma reduces pericardial fibrinolytic capability and induces an adhesion process. Keratinocyte growth factor (KGF) has proven to improve mesothelial cells proliferation. This study investigated the influence of keratinocyte growth factor in reducing post-surgical adhesions. Methods: Twelve pigs were operated and an adhesion protocol was employed. Following a stratified randomization, the animals received a topical application of KGF or saline. At 8 weeks, intrapericardial adhesions were evaluated and a severity score was established. The time spent to dissect the adhesions and the amount of sharp dissection used, were recorded. Histological sections were stained with sirius red and morphometric analyses were assessed with a computer-assisted image analysis system. Results: The severity score was lower in the KGF group than in the control group (11.5 vs 17, p = 0.005). The dissection time was lower in the KGF group (9.2 +/- 1.4 min vs 33.9 +/- 9.2 min, p = 0.004) and presented a significant correlation with the severity score (r = 0.83, p = 0.001). A significantly less sharp dissection was also required in the KGF group. Also, adhesion area and adhesion collagen were significantly tower in the KGF group than in the control group. Conclusion: The simulation of pericardial cells with KGF reduced the intensity of postoperative adhesions and facilitated the re-operation. This study suggests that the mesothelial regeneration is the new horizon in anti-adhesion therapies. (C) 2008 European Association for Cardio-Thoracic Surgery. Published by Elsevier B.V. All rights reserved.