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In the dissertation there is investigated a class of Markov chains defined by iterations of a function possessing a property of asymptotical homogeneity. Two problems are solved: 1) there are established rather general conditions under which the chain has unique stationary distribution; 2) for the chains evolving in a real line there are established conditions under which the stationary distribution of the chain is heavy-tailed. / Disertacijoje tirta Markovo grandinių klasė, kurios iteracijos nusakomos atsitiktinėmis asimptotiškai homogeninėmis funkcijomis, ir išspręsti du uždaviniai: 1) surastos bendros sąlygos, kurios garantuoja vienintelio stacionaraus skirstinio egzistavimą; 2) vienmatėms grandinėms surastos sąlygos, kurioms esant stacionarus skirstinys turi "sunkias" uodegas.

Disertacijoje tirta Markovo grandinių klasė, kurios iteracijos nusakomos atsitiktinėmis asimptotiškai homogeninėmis funkcijomis, ir išspręsti du uždaviniai: 1) surastos bendros sąlygos, kurios garantuoja vienintelio stacionaraus skirstinio egzistavimą; 1) vienmatėms grandinėms surastos sąlygos, kurioms esant stacionarus skirstinys turi "sunkias" uodegas. / In the dissertation there is investigated a class of Markov chains defined by iterations of a function possessing a property of asymptotical homogeneity. Two problems are solved: 1) there are established rather general conditions under which the chain has unique stationary distribution; 2) for the chains evolving in a real line there are established conditions under which the stationary distribution of the chain is heavy-tailed.

Šiame darbe nagrinėta tam tikra asimptotiškai homogeninė Markovo grandinė ir rasta jos stacionaraus skirstinio uodegos asimptotika. Nagrinėta grandinė negali būti ištirta šiuo metu žinomais metodais, todėl darbas turi praktinę reikšmę. Spręstas uždavinys aktualus sunkių uodegų analizėje. / In this work we have investigated some asymptotically homogeneous Markov chain and found asymptotics of the stationary distribution tail. To our best knowledge, considered chain cannot be investigated by means of existing methods, hence obtained results have practical value. Solved problem is relevant in heavy tail analysis.

<p>Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here. </p><p>Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.</p><p>One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.</p><p>Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.</p><p>In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models. </p><p>Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data. </p><p>The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.</p><p>Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.</p> / Dissertation

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The dependence structures in financial markets count among the most frequently discussed topics in the recent literature. However, no general consensus on modeling of the cross-market linkages has been reached. This thesis analyses the dependence structure and contagion in the financial markets in Central and Eastern Europe. Tail dependence, symmetry and dynamics of the dependence structure are examined. A conditional copula framework extended by recently developed dynamic generalized autoregressive score (GAS) model is used to capture the conditional time-varying joint distribution of stock market returns. Considering the Czech, Croatian, Hungarian, Austrian and Polish stock market indices over the 2005-2012 period, we find that time-varying Student's t GAS copula provides the best fit. The results show, that the degree of dependence increases substantially during the global financial crisis, having a direct impact on portfolio optimization.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund.

Our direct environment affects our lives directly. Christopher Alexander saw that we are able to feel or see if an object or structure is natural through the characteristics of them. He also saw that we generally feel better near these living, natural structures as it more closely resembles ourselves. Our bodies and our surroundings are made up of far more smaller than large things. When structures follow this pattern they are considered to be more natural, and when they move away from this pattern they are considered to be less natural and thus often boring or ugly. This scaling law is used to analyse the complex networks within cities. By analysing underlying structures instead of direct geometry it becomes possible to identify how living they are.This study applies these theories to analyse urban morphology within different cities. By identifying living structure within cities comparisons can be made between different types of cities. Specifically artificial and historical cities are analysed as they are counterparts in livingness. Following the identification of the living structure within these different types of cities an assessment can be made on what kind of an effect this has on our wellbeing based on Alexander’s theory. To see how living structure evolves over time a second analysis is performed which compares a city with its own evolution through time.Firstly natural cities and natural streets are identified in a bottom up approach based on the underlying structures of OpenStreetMap road data. Thereafter historical cities are compared with artificial cities because historical cities generally have living structure while artificial cities lack this. Then the developments of a historic city are identified and compared temporally. This research finds that current usage of concrete, steel and glass combined with very fast development speeds is detrimental to living structure within cities currently. Newer city developments should be performed in symbiosis with older city structures and the structure of the development should inhibit scaling as well as the buildings themselves. It is not sufficient to look only at geometry when managing cities, the importance of the fractal geometry, which is initially invisible must not be underestimated.