**Multi-species coalescents**by**Arno Siri Jegousse**, IIMAS, UNAM, México**Limit theorems for the partitioning process**by**Emmanuel Schertzer**, University of Paris, France**A stochastic model for reproductive isolation by mating preference**by**Helene Leman**, CIMAT, México**Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime**by**José David Campos**, UCR, Costa Rica**Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble**by**Gia Bao NGUYEN**, Universidad de Chile, Chile**Velocity estimates for random walks in a random environment with low disorder and a small local drift**by**Santiago Juan Saglietti****Asymptotics of Wasserstein Distances on discrete Spaces**by**Carla Tameling**, University of Goettingen, Alemania**Existence and consistency of the Wasserstein barycenter**by**Thibaut Le Gouic**, University of Marseille, France**Aggregation methods based on Wasserstein barycenters**by**Eustasio del Barrio**, Universidad de Valladolid, IMUVA, España**Glivenko-Cantelli for Monge-Kantorovich Empirical Processes**by**Marc Hallin**, Universit``é Libre de Bruxelles, Belgium**Sum rules via large deviations**by**Jan Nagel**, Technical University Munich, Germany**On the maximum of the characteristic polynomial of the Circular Beta Ensemble****Joseph Najnudel**, Universidad de Cincinnati, USA**Large deviations for random measures**by**Alain Rouault**, Université Versailles Saint Quentin, France**Compression and Conditional Emulation of Climate Model Output**by**Dorit Hammerling**, NCAR, Boulder, Colorado, USA**Comparison between spatio-temporal random processes and application to climate model data**by**Bo Li**, University of Illinois at Urbana-Champaign, USA**Relative Abundance Estimation by Merging Datasets Collected under Different Spatial Designs**by**Souparno Ghosh**, Texas Tech, USA**Abrupt convergence for generalized Ornstein-Uhlenbeck process**by**Juan Carlos Pardo Millán**, CIMAT, México**Fluctuation Theory of Markov additive processes and applications to stable processes**by**Víctor Rivero**, CIMAT, México**Heat content and spectral heat content asymptotics of second order**by**Luis Guillermo Acuña**, Universidad de Costa Rica, Costa Rica

**A stochastic model for reproductive isolation by mating preference**- Helene Leman, CIMAT, México

Summary: In this talk, we present and study a stochastic model of population where individuals are equivalent from ecological points of view, and differ only by their mating _x000D_ preference: two individuals with the same genotype have a higher probability to produce a viable offspring. The population is subdivided in two patches and individuals may migrate between them. We show that mating preferences by themselves, even if they are very small, are enough to entail reproductive isolation between patches.

**Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble**- Gia Bao NGUYEN, Universidad de Chile, Chile

Summary: Consider a system of N non-intersecting Brownian bridges, all starting from 0 at time t = 0 and returning to 0 at time t = 1. A result of K. Johansson implies that the suitably rescaled maximal height of the top path converges as N goes to infinity to the Tracy-Widom GOE distribution from random matrix theory. In this talk, I will present a result which shows that the squared maximal height of the top path in the case of finite N is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of Johansson's result, and provides an explanation of how the Tracy-Widom GOE distribution arises in the KPZ universality class. The result can also be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier. I will also discuss some results about the argmax of the top path, as well as related results for other models. This is joint work with Daniel Remenik.

**Velocity estimates for random walks in a random environment with low disorder and a small local drift**- Santiago Juan Saglietti

Summary: We derive asymptotic estimates for the velocity of random walks in random environments which are perturbations of the simple symmetric random walk but posses a small local drift in a given direction. Our results are in the spirit of previous expansions obtained by Sabot and also complement previous results presented by Sznitman. Joint work with Clément Laurent, Alejandro Ramírez and Christophe Sabot.

**Existence and consistency of the Wasserstein barycenter**- Thibaut Le Gouic, Univ. Marseille, France

Summary: We will define a notion of barycenter in the Wasserstein space of a locally compact geodesic space using Fréchet mean for that purpose. We will then talk about conditions of existence of this barycenter and state consistency results. Finally, we will talk about how it applies in the framework of empirical measures.

**Aggregation methods based on Wasserstein barycenters**- Eustasio del Barrio, Universidad de Valladolid, IMUVA, España

Summary: We develop a general theory to address a consensus-based combination of estimations in a parallelized or distributed estimation setting. Taking into account the possibility of very discrepant estimations instead of a full consensus we consider a "wide consensus" procedure. The approach is based on the consideration of trimmed barycenters and trimmed *k*-barycenters in the Wasserstein space of probability distributions on \(\mathbb{\lbrace R}^d\) with finite second order moments. We provide existence and consistency results as well as characterizations of barycenters of probabilities that belong to (non necessarily elliptical) location and scatter familes. On these families the effective computation of barycenters and distances can be addressed through a consistent iterative algorithm. Since once a shape has been chosen these computations just depend on the locations and scatters the theory can be applied to cover with great generality a wide consensus approach for location and scatter estimation or in cluster analysis.

**Sum rules via large deviations**- Jan Nagel, Technical University Munich, Germany

Summary: We show a large deviation principle for the weighted spectral measure of random matrices corresponding to a general potential. Unlike for the empirical eigenvalue distribution, the speed reduces to n and the rate function contains a contribution of eigenvalues outside of the limit support. As an application, this large deviation principle yields a probabilistic proof of the celebrated Killip-Simon sum rule: a remarkable relation between the entries of a Jacobi-operator and its spectral measure.

**Large deviations for random measures**- Alain Rouault, Université Versailles Saint Quentin, France

Summary: In relation with the course of Barry Simon and the talk of Jan Nagel, we prove that the large deviations method may also provide probabilistic proofs of known sum rules for matrix-valued mesures (Damanik-Killip-Simon (2010) for MOPRL, Delsarte-Genin-Kamp (1978) for MOPUC) and allow to state new ones. The coefficients (Jacobi or Verblunsky) are matrices, which induces issues of non-commutativity.

**Compression and Conditional Emulation of Climate Model Output**- Dorit Hammerling, NCAR, Boulder, Colorado, USA

Summary: Numerical climate model simulations runs at high spatial and temporal resolutions generate massive quantities of data. As our computing capabilities continue to increase, storing all of the generated data is becoming a bottleneck, and thus is it important to develop methods for representing the full datasets by smaller compressed versions. We propose a statistical compression and decompression algorithm based on storing a set of summary statistics as well as a statistical model describing the conditional distribution of the full dataset given the summary statistics. The statistical model can be used to generate realizations representing the full dataset, along with characterizations of the uncertainties in the generated data. Thus, the methods are capable of both compression and conditional emulation of the climate models. Considerable attention is paid to accurately modeling the original dataset, particularly with regard to the inherent spatial nonstationarity in global temperature fields, and to determining the statistics to be stored, so that the variation in the original data can be closely captured, while allowing for fast decompression and conditional emulation on modest computers.

**Comparison between spatio-temporal random processes and application to climate model data**- Bo Li, University of Illinois at Urbana-Champaign, USA

Summary: Comparing two spatio-temporal processes are often a desirable exercise. For example, assessments of the difference between various climate models may involve the comparisons of the synthetic climate random fields generated as simulations from each model. We develop rigorous methods to compare two spatio-temporal random processes both in terms of moments and in terms of temporal trend, using the functional data analysis approach. A highlight of our method is that we can compare the trend surfaces between two random processes, which are motivated by evaluating the skill of synthetic climate from climate models in terms of capturing the pronounced upward trend of real-observational data. We perform simulations to evaluate our methods and then apply the methods to compare different climate models as well as to evaluate the synthetic temperature fields from model simulations, with respect to observed temperature fields.

**On the maximum of the characteristic polynomial of the Circular Beta Ensemble**- Joseph Najnudel, Universidad de Cincinnati, USA

Summary: In this talk, we present our result on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it gives an improvement and a generalization of the previous recent results by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2.

**Asymptotics of Wasserstein Distances on discrete Spaces**- Carla Tameling, University of Goettingen, Alemania

Summary: We derive distributional limits for empirical Wasserstein distances of probability measures supported on discrete sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and Hadamard directional differentiability. A careful calibration of the norm is needed in order to combine differentiability and weak convergence of the empirical process. We give an explicit form of the limiting distribution for ultra-metric spaces.

Based on this we illustrate how Wasserstein based inference can be used in large scale problems. An application from nanoscale microscopy is given. This is joint work with Axel Munk and Max Sommerfeld.

**Glivenko-Cantelli for Monge-Kantorovich Empirical Processes**- Marc Hallin, Université Libre de Bruxelles, Belgium

Summary: A data-driven center-outward ordering of \(\mathbb{R}^d\) based on measure transportation ideas has been introduced in Chernozhukov, Galichon, Hallin and Henry (2016, *Annals of Statistics*, in press) under the name of Monge-Kantorovich ranks. A Glivenko-Cantelli theorem is established for the corresponding empirical process.

**Abrupt convergence for generalized Ornstein-Uhlenbeck process**- Juan Carlos Pardo Millán, CIMAT, México

Summary: In this talk, we study the cut-off phenomenon for a family of d-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes. Under some suitable conditions on the drift matrix and the Lévy measure of the driven processes present a profile cut-off with respect to the total variation distance. This is a joint work with Gerardo Barrera.

**Fluctuation Theory of Markov additive processes and applications to stable processes**- Víctor Rivero, CIMAT, México

Summary: As a generalisation of Lamperti’s transformation, it has been proved by Alili, Chaumont, Grackzyk and Zak that a stable process in dimension d can be seen as the exponential of a Markov additive process (MAP) time changed. In this talk, we aim at describing the so called upward, respectively downward, ladder height processes associated to this MAP. We will provide a precise description in the case where d=1, and, in general, how the characteristics of this process could be derived from known identities for stable processes. This is based on a ongoing collaboration with Kyprianou, Satitkanitkul and Sengul.

**Multi-species coalescents**- Arno Siri Jegousse, IIMAS, UNAM, México

Summary: In this talk, we define and characterise a two-level coalescent process (genealogical tree) of gene lineages within species lineages permitting simultaneous gene and species merger. The construction of this multi-species coalescent is closely related to exchangeable coalescents introduced mainly by Pitman or Sagitov in the late 90's. Thus we provide a Poissonian construction of the process. We also give a few foresights on the small-time behaviour of the number of blocks of the coalescent in the special case of a Kingman coalescent within a Kingman coalescent.

**Relative Abundance Estimation by Merging Datasets Collected under Different Spatial Designs**- Souparno Ghosh, Texas Tech, USA

Summary: Different wildlife agencies use different sampling protocols to estimate relative abundance of target species in different U.S. states. Consequently, the relative abundance of the species cannot be estimated reliably if its range spans multiple states. We propose a model based approach that can combine data collected under varied spatial sampling protocols to perform inference on the relative abundance of the target species over its entire range. Temporal extensions to establish long-term monitoring protocol is also discussed.

**Heat content and spectral heat content asymptotics of second order**- Luis Guillermo Acuña, Universidad de Costa Rica, Costa Rica

Summary: Let \(\bf{X}\) be an \(\alpha\)--stable process in \(\mathbb{R}^d\), \(0\leq \alpha \leq 2\) with transition densities denoted by $\left\{ p_t^{(\alpha)}(x,y): x,y\in \mathbb{R}^d, t\geq 0 \right\}$. Consider $\Omega\subset \mathbb{R}^d$ and open bounded set with finite measure. In the talk, we will discuss the behavior of the heat content over $\Omega$ defined by

$$\int_{\Omega}\int_{\Omega}p_t^{(\alpha)}(x,y)dxdy$$

as $t\downarrow 0$ and its connection to spectral heat content defined by

$$\int_{\Omega}\int_{\Omega}p_t^{D,\alpha}(x,y)dxdy,$$ where $\left\{ p_t^{D,\alpha}(x,y): x,y\in \Omega, t\geq 0 \right\}$ correspond to the transition densities of the $\alpha$--stable killed process of $X$ upon exiting $\Omega$. Open problems will be given during the talk.

**Limit theorems for the partitioning process**- Emmanuel Schertzer, University of Paris, France

Summary: In this talk, I will consider a haploid Wright--Fisher model with recombination, where each haplotype is a mosaic of its two parental chromosomes. Starting with uniformly colored and distinct chromosomes, each individual of the population at time t is a composite (or partition) of the colors originally present in the ancestral population. The partitioning process at time t is then defined as the color partition of a sampled chromosome. In particular, as time goes to infinity, it provides a description of the haplotype that will eventually fix in the population.

I will present some recent results on the partitioning process at stationarity. In particular, I will discuss the description of a typical color cluster, and a law of large numbers for the number of clusters inside a large portion of the chromosome.

**Asymptotic expansion of the invariant measure for ballistic random walk in the low disorder regime**- José David Campos, UCR, Costa Rica

Summary: We consider a random walk in random environment in the low disorder regime on ${\mathbb Z}^d$. That is, the probability that the random walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\epsilon \xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):|e|_1=1\}:x\in{\mathbb Z}^d\}$ are i.i.d. and $\epsilon>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $\epsilon$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $\epsilon$ for the invariant measure of random perturbations of the simple symmetric random walk in dimensions $d=2$.

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