**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 1.2.2018 at 16-18 C124

Lauri Viitasaari

**Linear backward stochastic differential equations driven by Gaussian processes II**

Thursday 8.2.2018 at 16-18 C124

Lauri Viitasaari

**Linear backward stochastic differential equations driven by Gaussian processes III**

## Previous talks:

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):

I**mportance 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

**Projections of Mandelbrot measures and phase transitions**

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

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

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

**Randomized Linear Algebra: Applications in Computational Mechanics, part II**

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

**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.

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

**On Kolmogorov complexity**[KOLMOGOROV-sarjaa]

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

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

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

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

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

**Real tree of fractional Brownian motion**

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

On the uniqueness of the multiplicative Gaussian chaos

On the uniqueness of the multiplicative Gaussian chaos

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

**Kolmogorov's contributions to logic**[KOLMOGOROV-sarjaa]

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

Optimal Berry-Esseen bounds on the Poisson space

Optimal Berry-Esseen bounds on the Poisson space

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

**[KOLMOGOROV-sarjaa]**

Divergent Fourier series

Divergent Fourier series

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

**Integrals on real trees**

**Torstai/Thursday, 17.9. 2015 klo**

**16.30**-18 C124 (**HUOM muuttunut aika ja paikka!**)**Berestycki's approach to Gaussian chaos**

## Aiempia esitelmiä kaudella 2014-15

Torstai/Thursday, 28.5.2015 klo 15-17 C129

**Lecture 6 of the course 'Probabilistic approximations'**

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

**Lecture 4 of the course 'Probabilistic approximations**

Torstai/Thursday, 9.4.2015 klo 15-17 C129

**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

**Decoupling on the Wiener space and applications to Backward Stochastic Differential Equations (BSDEs)**

**Lecture 2 of the course 'Probabilistic approximations'**

Torstai/Thursday, 5.3.2015 klo 15-17 C129

**"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 C129Teemu Roos

Learning Bayesian networks by Minimum Description Length

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

Data Stream Model

Torstai/Thursday, 28.11.2013 klo 15-17 CK111

**Martingale approach to repairable systems**

**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

**Rumor spreading and first-passage times in a large population of active and passive individuals**

Torstai/Thursday, 3.10.2013 klo 15-17 CK111

**Stochastic process models for repairable systems**

Torstai/Thursday, 26.9.2013 klo 15-17 CK111

**Noncommutative probability and quantum random walk**

Torstai/Thursday, 19.9.2013 klo 15-17 CK111

**Eigenvalues of random matrices and Gaussian Free Fields**

Torstai/Thursday, 29.8.2013 klo 15-17 C129

**On quadratic functionals of fractional processes**