Lancaster Probability Days 2017

Tuesday 28 March - Thursday 30 March

This meeting will bring together probabilists from across the UK and is supported by the Institute for Mathematics and its Applications (IMA) and the Royal Statistical Society (RSS). It will be of particular interest to those working in the following areas: Applications of probability to data science, Probabilistic models in risk and Planar random growth.

Each of these topics will have one day of the meeting devoted to it, with talks from three speakers in the afternoon. The mornings are reserved for discussions and collaborative work.

Topics

Applications of probability to data science

Two current topics in data science are adaptive and large-scale methods. Adaptive methods are methods that have minimal modelling assumptions while providing performance guarantees often on par with those of parametric methods with highly rigid assumptions. Such methods work well together with large scale methods that are able to make use of large amounts of data to infer complex models. Recent approaches to adaptivity and large-scale data processing will be presented in this workshop.

Probabilistic models in risk

Various advanced methods from probability theory are useful in the study of risk models. Recently good progress has been made in the understanding of Cramer-Lundberg risk model with the premium rate depending on the current level of risk reserve. Another current topic of research in risk theory is one related to models with heavy-tailed claims which model the occurrence of catastrophic events. These topics will be discussed in the workshop.

Planar random growth

Many examples of growth processes that occur in nature, for example, cancer tumours, mineral deposits, and lightning strikes, exhibit complex stochastic behaviour. Although these models have generated substantial interest amongst mathematician and physicists over the last 50 years, it is only recently that researchers have begun to understand the long-range interactions that take place in such systems. In this workshop, we will discuss the main challenges that exist and the new techniques that are in the process of being developed.

Schedule

Tuesday 28th: Applications of probability to data science

13:00 - 13:00: Coffee and Tea
13:30 - 14:30: Steffen Grünewälder
14:45 - 15:45: Heather Battey
15:45 - 16:15: Coffee and Tea
16:15 - 17:15: Natalia Bochkina
18:30: Conference dinner

Wednesday 29th: Probabilistic models in risk

13:00 - 13:00: Coffee and Tea
13:30 - 14:30: Dmitry Korshunov
14:45 - 15:45: Vitali Wachtel
15:45 - 16:15: Coffee and Tea
16:15 - 17:15: Denis Denisov

Thursday 30th: Planar random growth

13:00 - 13:00: Coffee and Tea
13:30 - 14:30: Amanda Turner
14:45 - 15:45: James Norris
15:45 - 16:15: Coffee and Tea
16:15 - 17:15: Vittoria Silvestri

The local organising committee consists of Alexander Belton, Natasha Blitvic, Steffen Grünewälder, Dmitry Korshunov, Amanda Turner and Dirk Zeindler (Chair).

Please direct any enquiries to the organisers or to probability@lancaster.ac.uk.

Speakers and Abstracts

Heather Battey

Exploring and exploiting new structured classes of covariance and inverse covariance matrices

An estimate of a covariance or inverse covariance (precision) matrices is an essential ingredient to many multivariate statistical procedures. When the dimension, p, of the covariance matrix is large relative to the sample size, the sample covariance matrix is inconsistent in non-trivial matrix norms, and its non-invertibility renders many techniques in multivariate analysis impossible. Structural assumptions are necessary in order to restrain the estimation error, even if this comes at the expense of some approximation error if the structural assumptions fail to hold. I will introduce new structured model classes for the estimation of large covariance and precision matrices. These model classes result from imposing sparsity in the domain of the matrix logarithm. I will first present a probabilistic result regarding the structure induced on the eigenvectors and eigenvalues of the covariance and inverse covariance matrices corresponding to a random matrix logarithm drawn from a particular sparse class. I will then consider an enlarged class of sparse matrix logarithms and exploit the structure of the class to define estimators of covariance and precision matrices with quantifiable asymptotic properties

Natalia Bochkina

Adaptive Bayesian density estimation based on a gamma mixture.

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local Holder densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness, the rates are adaptive to the smoothness.

One of the novel features of the paper is that these results hold for densities with polynomial tails. Similar results are obtained using a hierarchical Bayesian model based on the mixture of inverse gamma densities which can be used to estimate adaptively densities with very heavy tails, including Cauchy density. This is joint work with Judith Rousseau (Universite Paris Dauphine, France).

Denis Denisov

Tail asymptotics for the maximum and exit times of random walks via martingales.

In this talk, I will consider a one-dimensional random walk Sn. I will discuss a martingale approach to finding the tail asymptotics for the supremum supnSn and exit times taux:=min(n > 0: x+Sn <0). The talk is based on arXiv:1111.6810, arXiv:1403.7325 and some ongoing work.

Steffen Grünewälder

Compressed Empirical Measures

I will present results on compressed representations of expectation operators with a particular emphasis on expectations with respect to empirical measures. Such expectations are a cornerstone of non-parametric statistics and compressed representations are of great value when dealing with large sample sizes and computationally expensive methods. I will focus on a conditional gradient-like algorithm to generate such representations in infinite-dimensional function spaces. In particular, I will discuss extensions of classical convergence results to uniformly smooth Banach spaces (think Lp, 1 < p < 1, or various scales of Besov and Sobolev spaces); a counterexample to fast rates of convergence in norm when compact sets are used for approximations; workarounds based on slicing compact sets in suitable ways and a result about fast convergence when the norm convergence is replaced with a weaker form of convergence; results about the location of the representer of a probability measure inside the approximation set using smoothness assumptions on the kernel; and an application of these results to empirical processes. I will also shortly discuss how very similar techniques can be used to efficiently infer non-parametric regressors using large amounts of data.

Dmitry Korshunov

On subexponential tails for negatively driven compound renewal processes with application to two-dimensional ruin problem

We discuss subexponential tail asymptotics for the distribution of the maximum Mt:=\supu\in[0,t] Xu of a process Xt with negative drift for the entire range of t>0. We consider compound renewal processes with linear drift and Lévy processes being motivated by Cramér-Lundberg renewal risk process. These results allow to analyse the asymptotics of ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity and generic claim size is subexponential. (Particularly based on joint work with Sergey Foss (Edinburgh) and Zbignew Palmowski (Wroclaw).)

James Norris

Fluid limits in infinite dimensions

There is a general procedure to establish fluid limits for Markov process by computing the instantaneous drift and variance and using martingale estimates to show, given a small instantaneous variance, that the process itself is nearly deterministic.This works straightforwardly when only finitely many martingales are needed, as is the case for finite-dimensional processes. For function-valued or measure-valued processes, the same procedure can sometimes be adapted, relying on further ideas of compactness. I will sketch the general method and illustrate its use.

Vittoria Silvestri

The density of harmonic measure on the boundary of Hastings-Levitov clusters

The Hastings-Levitov models describe the growth of random sets (or clusters) in the complex plane as the result of the iterated composition of random conformal maps. The correlations between these maps are determined by the harmonic measure density profile on the boundary of the clusters. In this talk I will focus on the simplest case, that of i.i.d. conformal maps, and obtain a description of the local fluctuations of the harmonic measure density around its deterministic limit, showing that these are Gaussian. This is joint work with James Norris.

Amanda Turner

Scaling limits of Laplacian random growth models

The idea of using conformal mappings to represent randomly growing clusters has been around for almost 20 years. Examples include the Hastings-Levitov models for planar random growth, which cover physically occurring processes such as diffusion-limited aggregation (DLA), dielectric breakdown and the Eden model for biological cell growth, and more recently Miller and Sheffield's Quantum Loewner Evolution (QLE). In this talk, we will discuss ongoing work on a natural variation of the Hastings-Levitov family. For this model, we are able to prove that both singular and absolutely continuous scaling limits can occur. Specifically, we can show that for certain parameter values, under a sufficiently weak regularisation, the resulting cluster can be shown to converge to a randomly oriented one-dimensional slit, whereas under sufficiently strong regularisations, the scaling limit is a deterministically growing disk.

Vitali Wachtel

Cramer-Lundberg risk processes with level-dependent premium rates.

We consider a renewal risk model with a premium rate v(r), which depends on the current reserve r and converges, as r→∞, to the critical premium rate vc of the classical Cramer-Lundberg process. We study the asymptotic behaviour of ruin probabilities in this situation. It turns out that this behaviour is determined by the speed of convergence v(r) →vc.