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 semi-line 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.
