**Assessing time-varying causal interactions and treatment effects with applications to mobile health**by**Susan Murphy**, University of Michigan, USA**Stable processes through the theory of self-similar Markov****processes**by**Andreas****E. Kyprianou,**University of Bath, UK**Robust inference in functional data analysis**by**Graciela****Boente**, Universidad de Buenos Aires, Argentina**An introduction to stochastic games**by**Onésimo Hernández Lerma**, CINVESTAV, México**Random walk on sparse random directed graphs**by**Pietro Caputo**, Università Roma Tre, Italy**Natural parametrization and the loop-erased random****walk**by**Gregory****Lawler, University of Chicago**, University of Chicago, USA**The Langevin MCMC: theory and methods**by**Eric Moulines**, Ecole Polytechnique, France******Universal randomness in****2D**by**Scott****Sheffield**, Massachusetts Institute of Technology, USA

*Assessing time-varying causal interactions and treatment effects with applications to mobile health*******Susan****Murphy**, University of Michigan, USA

Abstract: A critical question in the development of mobile health interventions is in which contexts, is it most useful to push treatments to the user. This question concerns interaction effects between time-varying context (location, stress, time of day, mood, ambient noise, etc.) and a time-varying treatment on user behavior. In this talk we discuss the micro-randomized trial design and associated causal data analyses for use in assessing these types of interactions. We illustrate this approach with mobile health intervention data.

*Stable processes through the theory of self-similar Markov processes*******Andreas E. Kyprianou**, University of Bath, United Kingdom

Abstract: Stable processes are considered to be a prototype class of Markov processes with path discontinuities for which one may examine phenomena that do not appear in the setting of diffusions. Historically they have been examined using stochastic analysis, potential analysis, Fourier analysis, scaling properties as well as through the fluctuation theory of Levy processes. In this talk we will introduce the notion of their representation as self-similar Markov processes and explain how we can gain new insights into path properties of stable processes by examining their so-called Lampert-Kiu decomposition.

*Robust inference in functional data analysis*******Graciela Boente Boente**, Universidad de Buenos Aires and CONICET, Argentina

Abstract: Functional data analysis provides modern analytical tools for data that are recorded as images or as a continuous phenomenon over a period of time. Because of the intrinsic nature of these data, they can be viewed as realizations of random functions often assumed to be in a Hilbert space such as \(L^2({\cal{I}})\), with \(\cal{I}\) a real interval or a finite dimensional Euclidean set.

In particular, functional principal components or functional canonical correlation are statistical procedures developed to reduce the dimensionality retaining as much information as possible with respect to the measure of interest. To be more precise, the first *q* functional principal components provide the best *q*-dimensional approximation to random elements in Hilbert spaces, while functional canonical correlation is a tool to quantify correlations between pairs of observed random curves for which a sample is available.

We will discuss some approaches leading to obtain estimators of the principal directions less sensitive to atypical observations. In particular, the robust procedures developed to estimate the principal directions can be used to construct diagnostic measures. If possible, we will also discuss methods to provide robust inferences for the canonical functions and for the quantities of interest under a semi-linear functional model.

*An introduction to stochastic games,**To the memory of Lloyd S. Shapley (1923-2016)*******Onésimo Hernández Lerma**, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México

Abstract:This talk is dedicated to the memory of Lloyd S. Shapley who passed away on March 12, 2016. Shapley introduced many important concepts in game theory, including the notion of stochastic games [1]. Our talk is an introductory, nontechnical presentation of some concepts in cooperative and noncooperative game theory, and some recent results on stochastic games.

[1] L.S. Shapley. Stochastic games. Proc. Nat. Acad. Sci. USA, 39 (1953). 1095-1100.

*Random walk on sparse random directed graphs*******Pietro Caputo**, Università Roma Tre, Italia

Abstract: A random walk on a finite graph exhibits cutoff if its distance from stationarity remains close to the initial value for a certain number of iterations and then abruptly drops to near zero on a much shorter time scale. Originally discovered in the context of card shuffling by Aldous and Diacon is in 1986, this remarkable phenomenon is now rigorously established for man y reversible chains. Here we consider the non-reversible case of random walks on sparse random directed graphs, for which even the stationary distribution is far from being understood. We establish the cutoff phenomenon, determine its time window and prove that the cutoff profile approaches a universal gaussian shape. Moreover, we determine an explicit recursive equation characterizing the stationary distribution. This is joint work with Charles Bordenave and Justin Salez.

*Natural parametrization and the loop-erased random walk***Gregory Lawler**, University of Chicago, USA

Abstract: The loop-erased random walk (LERW) in two dimensions is obtained by erasing loops from simple random walk. It was shown by Schramm, Werner, and myself that the scaling limit of this path is a Schramm-Loewner evolution (SLE). This result requires a special parametrization of the curve in terms of "capacilty". However, one for many applications one would prefer to scale the curve in a more natural way by the (normalized) number of steps of the path. After reviewing LERW and SLE, I will discuss work in the last few years that has made this "natural parametrization" rigorous. This includes joint work with Fredrik Viklund, Christian Benes, and Mohammad Rezaei.

*The Langevin MCMC: theory and methods*******Eric Moulines**, Ecole Polytechnique, France

Abstract: In machine learning literature, a large number of problems amount to simulate a density which is log-concave (at least in the tails) and perhaps non smooth. Most of the research efforts so far has been devoted to the Maximum A posteriori problem, which amounts to solve a high-dimensional convex (perhaps non smooth) program. The purpose of this lecture is to understand how we can use ideas which have proven very useful in machine learning community to solve large scale optimization problems to design efficient sampling algorithms, with convergence guarantees (and possibly "usable" convergence bounds).

In high dimension, only first order method (exploiting exclusively gradient information) is a must. Most of the efficient algorithms know so far may be seen as variants of the gradient descent algorithms, most often coupled with "partial updates" (coordinates descent algorithms). This of course suggests to study methods derived from Euler discretization of the Langevin diffusion, which may be seen as a noisy version of the gradient descent. Partial updates may in this context as "Gibbs steps" where some components are frozen. This algorithm may be generalized in the non-smooth case by "regularizing" the objective function. The Moreau-Yosida inf-convolution algorithm is an appropriate candidate in such case, because it does not modify the minimum value of the criterion while transforming a non smooth optimization problem in a smooth one. We will prove convergence results for these algorithms with explicit convergence bounds both in Wasserstein distance and in total variation.

*Universal randomness in 2D***Scott Sheffield**, Massachusetts Institute of Technology, USA

Abstract: I will introduce several universal and canonical random objects that are (at least in some sense) two dimensional or planar, along with discrete analogs of these objects. In particular, I will introduce probability measures on the space of paths, the space of trees, the space of surfaces, and the space of growth processes. I will argue that these are in some sense the most natural and symmetric probability measures on the corresponding spaces. I will then describe several surprising relationships between these canonical objects. Many of these ideas have been historically motivated by physics, especially string theory, conformal field theory, gauge theory, and statistical mechanics.

© 2016 Universidad de Costa Rica.

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Tel. (506) 2511-6606