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Stochastic models seminar 2017-2018

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.   A special topic (began fall 2015) is  overview of the most most important contributions of  Kolmogorov in science - talks on this topic will be given from time to time.

The seminar meets  on Thursdays  16-18  in Exactum, room C124. during fall 2017 (and at 14-16 starting spring 2018). It is run by prof. Kostya Izyurov (konstantin.izyurov@helsinki.fi), prof. Ilkka Norros (ilkka.norros@vtt.fi)  and prof.Eero Saksman (eero.saksman@helsinki.fi). Usually the talks are in English.

 


Forthcoming talks:

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.




Previous talks:

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
On stochastic Moduli of Quadrilaterals


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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