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# Stochastic models seminar 2019-2020

Stochastic models seminar 2019-2020

Stochastic Models Seminar meets at  the department of Mathematics and Statistics at University of Helsinki. The topics touch on various  random phenomena and they depend on the interests of the participants. Often the seminar talks work as a 'reading seminar', where we study topics of general interest.

The seminar meets  on Thursdays  14-16  in Exactum, room C124. It is run by prof. Kostya Izyurov (konstantin.izyurov@helsinki.fi), docent Ilkka Norros (ilkka.norros@elisanet.fi)  and prof.Eero Saksman (eero.saksman@helsinki.fi). Usually the talks are in English.

## Forthcoming talks:

Thursday12.12.2019 C124  14.15-15.15 o'clock

Julien Barral (Paris 13)

Dimensions of statistically self-affine Sierpinski sponges

## Previous talks:

Thursday 21.11.2019 C124 14-16 o'clock (joint talk with Mathematical physics seminar)

Anton Nazarov (Saint Petersburg State University)

Limit shape for the $so(2n+1)$ Lie algebras in the infinite rank limit and an electrostatic problem.

Abstract:

We consider a tensor power of the spinor representation of the Lie algebra $so(2n+1)$. Tensor product decomposition into irreducible representations leads to the appearance of probability measure on the set of dominant integral weights. We consider the behavior of this measure in the limit of infinite tensor power and infinite rank of the algebra.

We show that in this limit the measure is concentrated in the single weight, thus the limit shape phenomenon is observed. The coordinates of this weight can be seen as the positions of charged particles on the line. This particles repulse each other but are confined in a closed interval by an external potential. We show that the limit shape is described by the density function of these particles.

Charge density function is the solution of a variational problem for the equilibrium measure, thus our result is comparable to various results in theory of random matrices and orthogonal polynomials.

Our result is similar to famous Vershik-Kerov-Logan-Shepp limit shape for the Plancherel measure on Young diagrams, since the limit shape for Young diagrams can be obtained from tensor product decomposition of $sl(n)$-representations.

Thursday 14.11.2019 C124 14-16 o'clock (joint talk with Mathematical physics seminar)

Mikhail Basok (Saint Petersburg State University)

Tau-functions à la Dubédat and cylindrical events in the double-dimer model

Double-dimer model on a given graph is obtained by sampling two independent dimer configurations taken uniformly at random: this produces a number of loops and double edges, removing the latter one obtains what is called double dimer loop ensemble. Given a simply-connected domain on the complex plane consider its approximation by a sequence of Temperley domains on a square grid, where the step of the grid tends to zero. It was predicted by Kenyon that the corresponding sequence of double dimer loop ensembles converges to Conformal Loop Ensemble with parameter 4 (CLE(4)) sampled in the original domain. Recently this conjecture was supported by a breakthrough work of Dubedat: in this work a large family of observables, called topological correlators, was constructed and it was shown that their values for double-dimer loop ensembles converges to their values for CLE(4). Topological correlators carry a lot of topological information about loops by their construction and it was reasonable to expect that their values determine the probability measure on families of loops uniquely. As it turned out, values of topological correlators determine probabilities of a large class of events called cylindrical, which, in particular, implies the claim above: topological correlators do characterize a measure. In this talk we will discuss the construction of topological correlators and the machinery developed to extract concrete probabilities from their values. Based on a joint work with Dmitry Chelkak (Paris).

Thursday 10.10.2019 C124 14-16 o'clock

Kamalakshya Mahatab:

Resonance method and Riemann zeta function

Thursday 3.10.2019 C124 14-16 o'clock

Ilkka Norros:

On Eratosthenes sieve

Thursday 4.4.2019 C124 14-16 o'clock

Lauri Viitasaari:

Gaussian fluctuations for the stochastic heat equation with colored noise II

Thursday 28.3.2019 C124 14-16 o'clock

Lauri Viitasaari:

Gaussian fluctuations for the stochastic heat equation with colored noise I

Abstract: In this talk we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show that the spatial average of the solution over an Euclidean ball is close to a Gaussian distribution, when the radius of the ball tends to infinity. Our central limit theorem is described in the total variation distance, using Malliavin calculus and Stein's method. We also provide a functional central limit theorem and analogous result in 1-dimensional case when the noise is white in both time and space.

Thursday 21.3.2019 C124 14-16 o'clock

Sirkka-Liisa Eriksson:

Hyperbolic harmonic functions and hyperbolic  Brownian motion

Thursday 7.3.2019 C124 14-16 o'clock

Hannu Reittu:

An Ising model for graph community detection

Thursday 17.1.2019 C124 14-16 o'clock

Diu Tran (University of Jyväskylä):

"Contributions to the asymptotic study of Hermite driven processes"

Abstract: Let $Z^{q, H}$ denote a Hermite process of order $q \geq 1$ and self-similarity parameter $H \in (\frac12, 1)$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. When $q = 1$, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as $q ⩾ 2$.
In our seminar, I would like to discuss first the asymptotic behaviour of quadratic functionals of Hermite-driven long-memory moving average processes. Then, we apply these results to estimate drift parameters for a Vasicek-type model driven by Hermite process of the form $dXa(bXt)dt+dZ^{q, H}$. For all possible values of $H$ and $q$, we prove strong consistency and we analyze the asymptotic fluctuations. Some open questions will be mentioned in the end of the talk.

Thursday 29.11.2018 C124 14-16 o'clock

Sigurdur Hafstein:

"Numerical computation of Lyapunov functions for SDE"

Abstract: When studying the stability of equilibria in deterministic ODEs the Lyapunov stability theory, whose centerpiece is the Lyapunov function, play a major role.   Khasminskii developed a corresponding theory for SDEs.  For nonlinear ODEs the construction of a Lyapunov function remains a very hard problem and for nonlinear SDEs the construction is much harder.  Indeed, even for linear SDEs the construction is very hard. We discuss the theory developed by Khasminskii and some recent numerical approaches to compute Lyapunov functions for SDEs.

Tuesday 13.11.2018 C124 14-16 o'clock (NOTE THE TIME AND PLACE! - joint with Geometric and Functional analysis seminar)

Eero Saksman:

"Elementary introduction to probabilistic number theory (part I)"

Abstract:

This is a first part of series of talks (aimed especially for students) which aim to give an introduction to basics of probabilistic number theory. No previous knowledge on number theory is assumed (and actually only a little amount of probability is needed).

Friday 5.10. 2018 at 14.00 - 15.00 D123 (NOTE THE TIME AND ROOM!)

Bertrand Duplantier (Institute for Theoretical Physics, Paris-Saclay University):

"Complex Generalized Integral Means Spectrum of Whole-Plane SLE"

Abstract: "We describe recent advances in the multifractality of the harmonic measure of Schramm-Loewner Evolution (SLE). A generalized notion of integral means spectrum is introduced, that involves the averaged mixed moments of both the SLE conformal map and its derivative. It yields a dual description of interior and exterior whole-plane SLE. Integrability loci exist in the moment plane, with four different forms of the spectrum separated by phase transition lines. These forms can be extended to complex moments. Based on joint works with Dmitry Belyaev, Ilia Binder, Xuan Hieu Ho, Thanh Binh Le, and Michel Zinsmeister."

Thursday 3.5. 2018 at 16-18 C124 Joint session with Mathematical physics seminar

Jean-Pierre Eckmann (University of Geneva)

"Breathers as Metastable States for the Discrete NLS equation" (Abstract)

Thursday 12.4. 2018 at 16-18 C124

Dario Gasbarra

Maximum entropy inversion of Laplace Transform in Diffusion-Magnetic Resonance Imaging

Thursday 15.3. and 22.3. 2018 at 16-18 C124

Harri Hakula

Data-driven sampling on manifold Part I-II

Abstract: In this seminar talk we examine the problem of generating realizations of a random vector with values in a finite-dimensional Euclidean space that are statistically consistent with a dataset of observations of this vector. In particular we are interested in applications related to uncertainty quantification and numerical solution of stochastic PDEs. The central question is how to discover and characterize the geometry and the structure of the dataset. Two approaches are discussed: The recent work by Soize and Ghanem and our ongoing project with Pauliina Ilmonen.

Thursday 8.2.2018 at 16-18 C124

Lauri Viitasaari

Linear backward stochastic differential equations driven by Gaussian processes III

Thursday 1.2.2018 at 16-18 C124

Lauri Viitasaari

Linear backward stochastic differential equations driven by Gaussian processes II

Thursday 25.1.2018 at 16-18 C124

Lauri Viitasaari

Linear backward stochastic differential equations driven by Gaussian processes I

Abstract: We discuss linear backward stochastic differential equations (BSDEs) driven by general Gaussian processes. We introduce a generalised concept of Skorokhod type integrals that we use in our setup. With the help of our integral, we define the concept of mild solution to the BSDE that allows different notions of integral. As one of our main result, we show that the solution to the linear BSDE exists even in the mild sense only in the case of martingales. That is, we show that once your underlying Gaussian process is not a martingale, one can always find a terminal value such that the mild solution does not exist. In order to establish our result, we define concepts of indefinite Wiener integral and a quasi-conditional expectation operator. We analyse the basic properties of these operators, and show how they are connected to BSDEs. In particular, we show that if the mild solution exists, then the terminal value is in the domain of the quasi-conditional expectation operator. With this result at hand, we obtain our non-existence result by showing that there are elements not belonging to the domain of the quasi-conditional expectation operator.

Thursday 7.12.2017 at 16-17 C124

Marianna Bolla (Budapest University of Technology and Economics):

Spectral clustering and parameter testing

Abstract:

To recover the structure of large graphs and rectangular arrays (for example, microarrays, socal, economic, or communication networks) classical methods of cluster analysis may not be carried out on the whole object because of computational size limitations. In other situations, we want to compare graphs and contingency tables of different sizes. We show how spectra are applicable for clustering (partitioning) the vertices such that the induced subgraphs on them and the bipartite subgraphs between any pair of them exhibit regular behavior of information flow within or between the vertex subsets, or to find biclustering of a contingency table (e.g., microarray) such that clusters of equally functioning genes equally influence conditions of the same cluster. If the measure we want to minimize or maximize (e.g., balanced multiway cuts or the multiway discrepancy) is testable we can select a smaller part of the graph (by an appropriate randomization) and consistently estimate this so-called testable parameter from that part.

Thursday 27.11.2017 at 16-18 C124  (2 talks)

Ellen Powell (ETH):

Liouville measure as a multiplicative cascade via level sets of the Gaussian free field

Gauthier Lambert (University of Zurich):

Fluctuations of smooth eigenvalues statistics for random matrices

Abstracts:  (Powell) I will discuss a new construction of the subcritical and critical chaos measures associated with the 2d Gaussian free field (GFF). The approach is based on the theory of local sets for the GFF, and builds a strong link between multiplicative cascades and  Liouville measures. I will also talk in more detail about the critical case, where uniqueness is a harder problem and has led to other interesting questions. This talk is based on joint work with Juhan Aru and Avelio Sepúlveda.

(Lambert) The eigenvalues of random matrices are basic examples of point processes with strong correlations and their scaling limits are very different from that of independent points. After a short presentation of the so-called circular unitary ensemble (CUE) and Gaussian unitary ensemble (GUE), I am planning to discuss the law of large number and the central limit theorem for smooth linear statistics for both models. The method is elementary and the talk aims at a general audience.

Monday 27.11.2017 at 13.00-14.00 C124 (Note the time!)

Konrad Kolesko (University of Innsbruck and  Wrocklaw University):

"Convergence of complex Biggins martingales on the phase boundary"

Thursday 16.11.2017 at 16-18 C124

Ilkka Norros (VTT):

Graphon topologies II

Abstract: Graphons are limit objects of sequences of dense graphs. We consider topologies related to graphons.

Thursday 9.11.2017 at 16-18 C124

Ilkka Norros (VTT):

Graphon topologies

Thursday 26.10.2017 at 16-18 C124

Ilkka Norros (VTT):

Basics of graphons

Abstract: Graphons are limit objects of sequences of dense graphs. Let’s have a look at the basics of their theory.

Thursday 05.10.2017 at 16-18 C124

Matti Vihola (JYU):

Importance sampling type estimators based on approximate marginal Markov chain Monte Carlo

Abstract: We consider importance sampling (IS) type weighted estimators, based on Markov chain Monte Carlo (MCMC) which targets an approximate marginal of the target distribution. In the context of Bayesian latent variable models, the MCMC typically operates on the hyperparameters, and the subsequent weighting may be based on importance sampling or sequential Monte Carlo (SMC), but allows for estimators based on multilevel Monte Carlo as well. The IS approach provides a natural alternative to delayed acceptance (DA) pseudo-marginal/particle MCMC, and enjoys many benefits against DA, including a straightforward parallelisation and additional flexibility of the MCMC implementation. We discuss minimal conditions which ensure strong consistency of the suggested estimators, and provide central limit theorems with expressions for asymptotic variances. Our experimental results with state-space models, using Laplace approximations and time-discretised diffusions, are promising and show that IS type approach can provide substantial speedup against an analogous DA scheme, and is often competitive even without parallelisation. This is joint work with Jouni Helske and Jordan Franks.

Thursday 28.9.2017 at 16-18 C124

Lauri Viitasaari:

Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes with applications to stochastic differential equations

Abstract: In this talk we study the existence of pathwise Stieltjes integrals for Hölder continuous integrator and integrand having infinite p-variation for all values of p, and we discuss a notion of sufficient variability for the integrand which ensuresthe existence of the integral in a pathwise sense. We also show that the integral can be defined as a limit of Riemann–Stieltjes sums for a large class of processes, and provide new estimates on the accuracy of numerical approximations of such integrals. In the end of the talk we discuss applications to stochastic differential equations. In particular, we provide existence and uniqueness result and the convergence rate for the Euler scheme for a new class of stochastic differential equations.

Thursday 21.9.2017 at 16-18 C124

Dario Gasbarra:

New moments criteria for convergence towards normal product/tetilla laws.

Abstract (https://arxiv.org/abs/1708.07681) We consider, in the classical probability, the distribution F_∞ ∼ N 1 × N 2 where N 1 , N 2 are two independent standard  normal random variables, and in the setting of free probability, F_∞ ∼ (S 1 S 2 + S 2 S 1 ) / 2 known as tetilla law, where S 1 , S 2 are freely independent normalized semicircular random variables. We provide new characterization of F_∞ within the second Wiener (Wigner) chaos. Our characterizations can be seen as the classical moments matching problem. More precisely, we show that for any generic element F in the second Wiener (Wigner) chaos with variance one the laws of  F and F_∞ coincide if and only if E (F^4 ) = 9 (resp. φ(F^4 ) = 2.5), and E (F^{2r} ) = ((2r − 1)! ! )^ 2 (resp. φ(F^{2r} ) = 2r )) for some r ≥ 3, where φ is the relevant tracial state. We use our moments criteria to study the non central limit theorems within the second Wiener (Wigner) chaos with target random  variable F_∞ . Our results can be seen as a slight generalization of some findings in Nourdin & Poly , Azmoodeh, et. al, in the classical probability, and of Deya & Nourdin in the free probability setting who proved the characterization result with r=3.

Thursday 14.9.2017 at 16-18 C124

Ilkka Norros:

Coding of stochastic block models again

Thursday 16.3.2017 klo 14-16 C122

Ilkka Norros:

Coding of stochastic block models II

Thursday 9.3.2017 klo 14-16 C124 (note the room!)

Ilkka Norros:

Coding of stochastic block models

Thursday 23.2.2017 klo 14-16 B321 (note the room!(note that we have 2 speakers!)

Alain Durmus (Paris Telecom):

"Optimal scaling and convergence of Markov chain"

Monte Carlo methods

AND

Matti Vihola (JY):

"Unbiased estimators and multilevel Monte Carlo"

Abstracts:

DURMUS: Sampling over high-dimensional space has become a prerequisite in applications of Bayesian statistics, for example to machine learning problems. The most common methods for dealing with this problem are Markov Chain Monte Carlo methods.  In this talk, I will present new insights on the computational complexity of these algorithms.  First, I will discuss the optimal scaling problem of the random walk Metropolis algorithm applied to densities which are differentiable in Lp mean but which may be irregular at some points (like Laplace type densities for example) and / or are supported on an interval. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of (Roberts, Gelman, Gilks, 1997). In the second part of my talk, we will present a method based on the Euler discretization of the  Langevin diffusion with either constant or decreasing stepsizes. We will give several new results establishing the convergence to stationarity under different conditions on the log-density. A particular attention of these bounds with respect to the dimension of the state space will be paid.

VIHOLA: Multilevel Monte Carlo (MLMC) is commonly applied, for instance, in numerical approximation of expectations with respect to diffusions. The unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related to MLMC. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes as special cases. New lower variance estimators are proposed, which are stratified versions of earlier unbiased schemes. Under general conditions, essentially when MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are asymptotically as efficient as MLMC, both in terms of variance and cost.

Thursday 12.1.2017 klo 16-18 C124   (joint with the geomeric analysis  seminar: note the new time !)

Meng Wu (University of Oulu)

On a conjecture of Furstenberg about intersections of Cantor sets

Abstract: Two compact sets E,F of the real line are said to be strongly transverse if for each u and t, the Hausdorff dimension (dim) of the intersection of E and uF+t is bounded by dim(E)+dim(F)-1 or 0, whichever is larger. In the late 60's, Furstenberg conjectured that two closed sets E,F of [0,1] are strongly transverse if E is invariant under multiplication by 2 (mod 1) and F is invariant under multiplication by 3 (mod 1). In this talk, we will recall some recent progress regarding this conjecture and present a solution.

Thursday 24.11.2016 at 14-16 in D123

Konstantin Izyurov

Local sets of the Gaussian free field

The Gaussian free field is one the simplest statistical filed theories; it is a Gaussian process that can be viewed as a natural generalization of a Brownian motion to multi-dimensional time. A local set of a GFF is an analog of a stopping time, enabling some sort of a strong Markov property for the field. The theory of local sets is an essential part in the study of relation between GFF and Stochastic Loewner evolution, due to Schramm, Dubédat, Sheffield and Miller. In this talk, I will go through basic facts about the local sets, following lecture notes by Wendelin Werner.

Thursday 17.11.2016  at 14-16  in D123
Ilkka Norros

Statistical models and measurements

Thursday 3.11.2016  at 14-16  in D123
Hannu Reittu

Regular decomposition revisited

Thursday 6.10.2016  at 14-16  in D123
Christian Webb

Multiplicative chaos in number theory and random matrix theory

Abstract: I will discuss the statistical behavior of the Riemann zeta function on the critical line. More precisely, I will talk about how in the vicinity of a random point on the critical line, the behavior of the zeta function can be described in terms of a random generalized function known as a Gaussian multiplicative chaos distribution. I will also discuss connections between the Riemann zeta function and random matrix theory in this setting. This is joint work with Eero Saksman.

Thursday 22. 9. 2016  at 16-18  in C124
Janne Junnila

On non-Gaussian multiplicative chaos

## Talks in the past (2015-16)

Torstai/Thursday, 28.4. 2016  klo 16-18 C124
Ilkka Norros

A category of Markov processes

Torstai/Thursday, 7.4. 2016  klo 16-18 C124
Antti Kupiainen

Kolmogorov and Turbulence (KOLMOGOROV-sarjaa)

Torstai/Thursday, 17.3. 2016  klo 16-18 C124

Ilkka Norros

Stability and instability of file-sharing systems

Torstai/Thursday, 25.2. 2016  klo 16-18 C124

Julien Barral (Paris 13):

Projections of Mandelbrot measures and phase transitions

Torstai/Thursday, 11.2. 2016  klo 16-18 C124

Kari Eloranta
Kolmogorov and measurable dynamics: K and h  [KOLMOGOROV-sarjaa]

Torstai/Thursday, 4.2. 2016  klo 16-18 C124

Harri Hakula
Randomized Linear Algebra: Applications in Computational Mechanics, part II

Torstai/Thursday, 28.1. 2016  klo 16-18 C124

Harri Hakula
Randomized Linear Algebra: Applications in Computational Mechanics

Abstract: Data analysis has been a driving force behind recent advances in randomized linear algebra. The methods for computing partial matrix decompositions efficiently use random sampling to identify a subspace that captures most of the action of a matrix. In fact these methods rely on the concentration of measure phenomenon: the random input leads to very small variance on the quantities of interest.

In computational mechanics the interest lies in the smallest eigenvalues of generalized eigenvalue problems which is exactly opposite to data analysis where the focus is on largest singular values. Remarkably, for realistic classes of symmetric and positive definite systems the randomized algorithms perform better than the state-of-the-art general purpose eigensolvers. This result appears to be new.

Torstai/Thursday, 14.1. 2016  klo 16-18 C124

Teemu Roos
On Kolmogorov complexity  [KOLMOGOROV-sarjaa]

Torstai/Thursday, 3.12. 2015  klo 16-18 C124

Dario Gasbarra
Asymptotic clustering for the eigenvalues of  a random matrix with isotropic Gaussian noise

Torstai/Thursday, 19.11. 2015  klo 16-18 C124

Paavo Salminen
"Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" [KOLMOGOROV-sarjaa]

Torstai/Thursday, 12.11. 2015  klo 16-18 C124

Ilkka Norros
Real tree of fractional Brownian motion

Torstai/Thursday, 05.11. 2015  klo 16-18 C124

Janne Junnila
On the uniqueness of the multiplicative Gaussian chaos

Torstai/Thursday, 29.10. 2015  klo 16-18 C124

Jan von Plato
Kolmogorov's contributions to logic [KOLMOGOROV-sarjaa]

Torstai/Thursday, 22.10. 2015  klo 16-18 C124

Ehsan Azmoodeh
Optimal Berry-Esseen bounds on the Poisson space

Torstai/Thursday, 8.10. 2015  klo 16-18 C124

Eero Saksman
Divergent Fourier series
[KOLMOGOROV-sarjaa]

Torstai/Thursday, 1.10. 2015  klo 16-18 C124

Ilkka Norros
Integrals on real trees

Torstai/Thursday, 17.9. 2015  klo 16.30-18 C124  (HUOM muuttunut aika ja paikka!)
Eero Saksman
Berestycki's approach to Gaussian chaos

## Aiempia esitelmiä kaudella 2014-15

Torstai/Thursday, 28.5.2015  klo 15-17 C129

Ehsan Azmoodeh
Lecture 6 of the course 'Probabilistic approximations'

Torstai/Thursday, 30.4.2015  klo 14-16 C129 (HUOM aika!)

Ehsan Azmoodeh
Lecture 4 of the course 'Probabilistic approximations

Torstai/Thursday, 9.4.2015  klo 15-17 C129

Giancarlo Pastor

The Candès-Tao Theory of Compressed Sensing: Essentials, Strategies, Systems, ... , and Proofs

ABSTRACT:

The last decade has seen the growth of a new theory called "compressed sensing". Due to this theory it is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This lecture follows the progress of Candès and Tao to state the essentials of compressed sensing. Recent efforts from industry to implement this theory to massive data acquisition systems will be also covered. In the last part, the lecture reformulates the "surprising" classical recovery strategy of compressed sensing in terms of sparsity and entropy.

Torstai/Thursday, 19.3.2015  klo 15-17 C129

Stefan Geiss (Jyväskylä)
Decoupling on the Wiener space and applications to Backward Stochastic Differential Equations (BSDEs)
ABSTRACT: We introduce a decoupling technique on the Wiener space that is used to describe path-dependent fractional smoothness and that yields to a general class of an-isotropic Besov spaces containing the classical spaces obtained by real interpolation. This approach is used to derive variational $L_p$-estimates for the solutions to quadratic BSDEs. The techniques are partially based on harmonic analysis.

Torstai/Thursday, 26.3.2015  klo 15-17 C129
Ehsan Azmoodeh
Lecture 2 of the course 'Probabilistic approximations'

Torstai/Thursday, 5.3.2015  klo 15-17 C129

Joonas Turunen
"A bijection between labelled plane trees and rooted and pointed planar quadrangulations"

ABSTRACT: The class of planar quadrangulations is an important subclass of the class of planar maps. In addition to their nice graph theoretical properties they also can be seen as discretizations of the two dimensional sphere, for instance. The latter property is a key observation in the theory of 2d quantum gravity, first conjectured by physicists and lately made rigorous by mathematicians.In this presentation I will show the construction of a bijection between the set of labelled plane trees with n vertices and an additional coding parameter and the set of rooted and pointed planar quadrangulations with n faces. The construction was first done by R. Cori and B. Vauquelin in 1981 and later simplified by G. Schaeffer. Thus the bijection is called the CVS bijection.

The CVS bijection provides a nice way to compute the number of rooted quadrangulations with n faces. This is a result of W.T. Tutte from 1963 obtained by a different method. Second, the bijection allows us to deal with the metric properties of a quadrangulation by the means of the labelling of the corresponding tree. This is needed in analysing the scaling limit of uniformly distributed random quadrangulations. Time permitting, I will briefly introduce these properties.

Torstai/Thursday, 26.2.2015 . klo 15-17 C129
Eric Moulines and Alain Durmus (Paris Telecom)
Moulines: "Subgeometric rates of convergence in Wasserstein distance for Markov chains"

Durmus: "New Langevin-based Metropolis algorithm "

ABSTRACTS: (Moulines) We provide sufficient conditions for the existence of the  invariant distribution and for subgeometric rates of convergence in Wasserstein distance for general state-space Markov chains which are (possibly) not irreducible.  Compared to (Butkovsky, 2013, AoAP) our approach is based on a purely probabilistic coupling construction which allows to retrieve rates of convergence matching those previously reported for convergence in total variation in (Douc, M., Soulier, 2007). Our results are applied to establish the subgeometric ergodicity in Wasserstein distance of non-linear autoregressive models and of the pre-conditioned Crank-Nicolson Markov chain Monte Carlo algorithm in Hilbert space.

(Durmus) The Metropolis Hastings algorithm provides a generic and efficient way to sample from a given distribution. Since the pioneering result of (Roberts, Gelman and Gilks 97'), there is an extensive work on optimal scaling of various Metropolis Hastings algorithms in different contexts. The interest of such results is to obtain an asymptotic number of steps needed by the algorithm to explore the state space in function of the dimension. The fewer this number is the better the algorithm is.

Maybe the most well known results are concerning the optimal scaling of the Random Walk Metropolis algorithm and the Metropolis Adjusted Langevin algorithm. Whereas the first needs O(d) steps to explore the space, where d is the dimension, the second on only needs O(d^(1/3)).We will present in this talk a new class of proposals for the Metropolis Hastings algorithm which leads to an exploration in O(d^(1/5)) number of steps. Also, in a second part, we will expose positive and negative ergodicity results for these new algorithms. Finally if time permits, we will discuss about practical solutions to reduce the computational cost of these algorithms and their link with the Hamiltoninan Monte Carlo algorithm. It is a joint work with Gareth O. Roberts, Gilles Vilmart, and Konstantinos Zygalakis.

Torstai/Thursday, 5.2.2015  ja 19.2. klo 15-17 C129
Ilkka Norros
Aldous' continuum random tree, part I and II

Torstai/Thursday, 29.1.2015  klo 15-17 C129
Pauliina Ilmonen
A Multivariate Hill Estimator

ABSTRACT: We propose an estimator of the tail index of regularly varying elliptical random vector. The estimator is based on the distance between a tail probability contour and the observations outside of this contour. The proposed estimator is a multivariate extension of the well-known univariate Hill estimator. We provide the multivariate estimator, we consider its asymptotical properties, and we present  large simulation study.  We illustrate the practical use of the new multivariate estimator by a real data example.

Torstai/Thursday, 22.1.2015  klo 15-17 C129
Harri Hakula
Lorentz Gas Problem: Faint Echos of a Proof

Torstai/Thursday, 11.12.2014  klo 15-17 C129
Harri Hakula
Algorithm Based Fault Tolerant Computation

Torstai/Thursday, 27.11.2014  klo 17.00-18.00 C129 (Huom poikk. aika!)
Ehsan Azmoodeh (Luxenburg)
The fourth moment theorem: a revisit and generalization

Torstai/Thursday, 20.11.2014  klo 15-17 C129
Dario Gasbarra
Introduction to Malliavin Calculus in the Poisson space (part II)

Torstai/Thursday, 13.11.2014  klo 15-17 C129
Dario Gasbarra
Introduction to Malliavin Calculus in the Poisson space (part I)

Torstai/Thursday, 6.11.2014  klo 15-17 C129
Lasse Makkonen
Improving extreme value analysis

Torstai/Thursday, 23.10.2014  klo 15-17 C129
Hannu Reittu
On the correctness of the Regular decomposition algorithm of graph compression

Torstai/Thursday, 9.10.2014  klo 15-17 C129
Ilkka Norros
Random variables on finitely additive measures

Torstai/Thursday, 2.10.2014  klo 15-17 C129
Ilkka Norros
On the absolute continuity of finitely additive measures

Torstai/Thursday, 25.9.2014  klo 15-17 C129
Ilkka Norros
Proving ergodicity with help of zombies

Torstai/Thursday, 18.9.2014  klo 15-17 C129
Eero Saksman
The renewal theorem revisited: an analytical approach

## Esitelmiä kaudella 2013-14

Torstai/Thursday, 8.5.2014  klo 15-17 C129
Janne Junnila
Introduction to Kahane's multiplicative chaos

Torstai/Thursday, 3.4.2014  klo 15-17 C129

Simo Särkkä (Aalto)
Theory and Practice of Particle Filtering for State Space Models Bayesian networks

Torstai/Thursday, 27.3.2014  klo 15-17 C129
Teemu Roos
Learning Bayesian networks by Minimum Description Length

Torstai/Thursday, 20.3.2014  klo 15-17 C129
Yashar Memarian
The Gaussian Correlation Conjecture Proof

Torstai/Thursday, 27.2.2014  klo 15-17 C129
Hannu Reittu
Large  graph or matrix compression method inspired by Szemerédi's Regularity Lemma and Rissanen's Minimum Description Length Principle

Torstai/Thursday, 13.2.2014  klo 15-17 C129
Mikko Stenlund
A local limit theorem for random walks in balanced environments

Torstai/Thursday, 6.2.2014  klo 15-17 C129
Harri Hakula
Tensor trains and high-dimensional integration

Torstai/Thursday, 30.1.2014  klo 15-17 C129
Christian Hirsch:
Stationary Apollonian packings

David Neuhäuser:
Comparability, monotonicity and asymptotic properties of typical distances in connected random geometric graphs

Torstai/Thursday, 23.1.2014  klo 15-17 C129
Harri Hakula

Torstai/Thursday, 5.12.2013 klo 15-17 CK111
Helle Majander
Data Stream Model

Torstai/Thursday, 28.11.2013 klo 15-17 CK111
Ilkka Norros
Martingale approach to repairable systems

Torstai/Thursday, 21.11.2013 klo 15-17 CK111
Matti Leppäranta
Applications of stochastic models in research on natural water bodies

Torstai/Thursday, 31.10., 7.11., 14.11. klo 15-17 CK111
Dario Gasbarra
Combinatorics in free probability I-III

Torstai/Thursday, 17.10.2013 klo 15-17 CK111
Matti Vihola
Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms

Torstai/Thursday, 10.10.2013 klo 15-17 CK111
Lasse Leskelä
Rumor spreading and first-passage times in a large population of active and passive individuals

Torstai/Thursday, 3.10.2013 klo 15-17 CK111
Ilkka Norros
Stochastic process models for repairable systems
Torstai/Thursday, 26.9.2013 klo 15-17 CK111
Kay Schwieger
Noncommutative probability and quantum random walk

Torstai/Thursday, 19.9.2013 klo 15-17 CK111
Christian Webb
Eigenvalues of random matrices and Gaussian Free Fields

Torstai/Thursday, 29.8.2013 klo 15-17 C129
Katsuto Tanaka (Japan):
On quadratic functionals of fractional processes

Vanhemmat seminaarisivut
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