Invariant Measure for Diffusions with Jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established

Efficiency and Information Aggregation in Auctions with Costly Information

Consider an auction in which $k$ identical objects are sold to $n>k$ bidders who each have a value for one object which can have both private and common components to it. Private information concerning the common component of the object is not exogenously given, but rather endogenous and bidders face a cost to becoming informed. If the cost of information is not prohibitively high, then the equilibrium price in a uniform price auction will not aggregate private information, in contrast to the costless information case. Moreover, for a wide class of auctions if the cost of information is not prohibitively high then the objects can only be allocated in a weakly efficient sense, and then only if the equilibrium proportion of endogenously informed agents is vanishing as the economy grows. In spite of these results, it is shown that there is a mechanism for which there exist equilibria and for which (weak) efficiency is achieved as the economy grows in the face of endogenous information acquisition.Auctions, Efficiency, Information Acquisition, Information Aggregation

Increase of entropy under convolution and self-similar sets with overlaps

Abstract. Sets that consist of finitely many smaller-scale copies of itself are known as self-similar. Due to the likely irregularity in their structure, the size of these sets is often measured in the form of dimension. The existence of tools that can be used to calculate this quantity depends greatly on whether the cylinders of which the set consists of are sufficiently separated from each other. If this is the case, the dimension of the set is known to equal its similarity dimension, a quantity that is relatively easy to calculate. There is a long-standing open conjecture stating that, for a general set on the real line, the only case in which the dimension of the set does not equal its similarity dimension, is when at some scale there is an exact overlap among the cylinders of the set. The main result in this thesis is a step towards showing that this is indeed the case; in the presence of an exact overlap, the distance between the cylinders of the set decreases exponentially. This result is due to M. Hochman and it appeared in his paper “On self-similar sets with overlaps and inverse theorems for entropy” (2012) and forms the basis of our discussion in Section 4. In Section 1, we analyse the growth of entropy of a probability measure under convolution. The main result of this section is a generalization of the Freiman theorem from additive combinatorics to the fractal regime, stating that if the entropy of a convolution measure is not too large, then one of the marginal measures has to be either locally uniform or locally atomic. This result is also due to Hochman and is one of the main tools used in proving the results of Section 4. In Sections 2 and 3, we introduce the concepts of dimension and the main tools required in understanding the structure of a self-similar set or measure with sufficient separation conditions in place. Most of the results here can be found in any text-book concerning fractal geometry, e.g. Falconer’s “Fractal Geometry” (1990)

An Asymptotic Analysis of Nearly Unstable inar (1) Models

This paper considers integer-valued autoregressive processes where the autoregression parameter is close to unity.We consider the asymptotics of this `near unit root' situation.The local asymptotic structure of the likelihood ratios of the model is obtained, showing that the limit experiment is Poissonian.This Poisson limit experiment is used to construct efficient estimators and tests.integer-valued times series;Poisson limit experiment;local-to-unity asymptotics

Large deviations and applications : the finite dimensional case

Includes bibliographical references (p. 96-98).Research supported by the Air Force Office of Scientific Research. AFOSR-89-0276B Research supported by the Army Research Office. DAAL03-86-K-171A. Dembo, O. Zeitouni

Formalizing Constructive Analysis: A comparison of minimal systems and a study of uniqueness principles.

Αυτή η διατριβή εξετάζει ορισμένες πλευρές της τυποποίησης και της αξιωματικοποίησης της κατασκευαστικής ανάλυσης. Η έρευνα στους κλάδους της κατασκευαστικής ανάλυσης που αντιστοιχούν στις διάφορες εκδοχές κατασκευαστικότητας διεξάγεται σε μια πλειάδα τυπικών ή όχι συστημάτων, των οποίων οι σχέσεις είναι ασαφείς. Αυτό το πρόβλημα αποβαίνει κρίσιμο για την ανάπτυξη της σχετικά νέας περιοχής των κατασκευαστικών ανάστροφων μαθηματικών. Η εργασία αυτή συμβάλλει σε μια πιο καθαρή εικόνα. Το Μέρος 1 περιέχει μία ακριβή σύγκριση των δύο ευρύτερα χρησιμοποιούμενων συστημάτων που τυποποιούν τον κοινό πυρήνα της κατασκευαστικής, της ενορατικής, της αναδρομικής και της κλασικής ανάλυσης, των Μ και EL, των Kleene και Troelstra, αντιστοίχως. Αποδεικνύεται ότι το EL είναι ασθενέστερο από το M και ότι η διαφορά τους αποτυπώνεται από μια αρχή η οποία εγγυάται την ύπαρξη χαρακτηριστικής συνάρτησης για κάθε αποκρίσιμο κατηγόρημα φυσικών αριθμών. Με παρόμοια επιχειρήματα προκύπτουν συγκρίσεις για τα περισσότερα από τα χρησιμοποιούμενα ελαχιστικά συστήματα. Στην κατασκευαστική ανάλυση χρησιμοποιούνται διάφορες αρχές επιλογής, συνέχειας και άλλες. Στο Μέρος 2, μελετώνται σχέσεις μεταξύ πολλών από αυτές, στις εκδοχές τους με μία συνθήκη μοναδικότητας, ένα χαρακτηριστικό από το οποίο απορρέουν ενδιαφέρουσες ιδιότητες, καθώς και σχέσεις μεταξύ αυτών των αρχών και μη κατασκευαστικών λογικών αρχών, στο πνεύμα των ανάστροφων μαθηματικών.This dissertation investigates certain aspects of the formalization and axiomatization of constructive analysis. The research in the branches of constructive analysis corresponding to the various forms of constructivism is carried out in a multitude of formal or informal systems, whose relations are unclear. This problem becomes quite crucial for the development of the relatively new field of constructive reverse mathematics. This work contributes to a clearer picture. Part 1 contains a precise comparison of the two most widely used systems which formalize the common core of constructive, intuitionistic, recursive and classical analysis, namely Kleene's M and Troelstra's EL. It is shown that EL is weaker than M and that their difference is captured by a function existence principle asserting that every decidable predicate of natural numbers has a characteristic function. Applying similar arguments, comparisons of most of the used minimal systems are obtained. In constructive analysis, various forms of choice principles, continuity principles and many others are used. Part 2 studies relations between many of them, in their versions having a uniqueness condition, a feature from which interesting properties follow, as well as relations between these principles and non-constructive logical principles, in the spirit of reverse mathematics

Ensaios em análise assintótica de regressão não-paramétrica

This work is composed of three essays in the eld of nonparametric inference, all closely inter-related. The rst essay aims to stablish uniform convergence rates under mixing conditions for the local linear estimator under a xed-design setting of the form t/T, t ∈ {1, . . . , T}, T ∈ N. It was found that the order of the weak and the strong uniform convergence is the same as that of stablished by Hansen (2008) and Kristensen (2009) for the random design setting. The second essay studies the asymptotic properties of the estimators derived from reversing the three-step procedure of Vogt and Linton (2014). Weak uniform convergence rates was given to the trend and the periodic sequence estimators. Furthermore, the consistency of the fundamental period estimator and the asymptotic normality of the trend estimator was also stablished. The last study investigates the nite sample behavior of the estimators considered in the second essay. A plug-in type bandwith was proposed for the trend estimator. From our simulation results, the plug-in bandwidth performed well and the period estimator showed to be quite robust with respect to di erent bandwidth choices. The study was complemented with two applications, one in climatology and the other in economics.Este trabalho é composto por três ensaios na área de inferência não-paramétrica, bastante inter-relacionados. O primeiro ensaio visa estabelecer ordens de convergência uniforme sob condições mixing para o estimador linear local quando a estrutura de pontos é xa e da forma t/T, t ∈ {1, . . . , T}, T ∈ N. A ordem encontrada para as convergências uniforme, em probabilidade e quase certa, é a mesma daquela estabelecida por Hansen (2008) e Kristensen (2009) para o caso de estrutura de pontos aleatórios. O segundo ensaio estuda as propriedades assintóticas de estimadores obtidos ao se inverter o esquema de estimação em três etapas de Vogt e Linton (2014). Foram fornecidas as ordens de convergência uniforme em probabilidade para os estimadores da função de tendência e da sequência periódica. Além disso, a consistência do estimador do período fundamental e a normalidade assintótica do estimador de tendência também foram estabelecidas. O último estudo investiga o comportamento em amostras nitas dos estimadores considerados no segundo ensaio. Foram propostas janelas para o estimador de tendência do tipo plug-in. Para as simulações realizadas, a janela plug-in mostrou bom desempenho e o estimador do período revelou-se bastante robusto em resposta à diferentes escolhas de janelas. O estudo foi complementado com duas aplicações, uma em climatologia e outra em economia

高次元・スパースな設定における確率過程の統計モデルに対するDantzig selector

早大学位記番号:新8131早稲田大

Asymptotic inference for monstationary fractionally integrated processes

This paper studies the asymptotic of nonstationary fractionally integrated (NFI) multivariate processes with memory parameter d> 112. We provide conditions to establish a functional central limit theorem and weak convergence of stochastic integrals for NFI processes under the assumptions of these results are given. More specifically, we obtain the rates of convergence and limiting distributions of the OLS estimators of cointegrating vectors in triangular representations. Further, we extend Sims, Stock and Watson's (1990) analysis on estimation and hypothesis testing in vector autoregressions with integrated processes and deterministic components to the more general fractional framework. We show how their main conclusions remain valid when dealing with NFI processes. That is, whenever a block of coefficients can be written as coefficients on zero mean 1(0) regressors in a model that includes a constant term, they will have a joint asymptotic normal distribution, so that the corresponding restrictions can be tested using standard asymptotic chi-square distribution theory. Otherwise, in general, the associated statistics will have nonstandard limiting distributions