京都大学 / 新制・課程博士 / 博士(理学) / 甲第25091号 / 理博第4998号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 日野 正訓, 教授 COLLINSBenoit Vincent Pierre, 教授 楠岡 誠一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Ce manuscrit est dédié à l'étude de l'estimation paramétrique d'une famille de processus ponctuels appelée processus déterminantaux. Ces processus sont utilisés afin de générer et modéliser des configurations de points possédant de la dépendance négative, dans le sens où les points ont tendance à se repousser entre eux. Plus précisément, nous étudions les propriétés asymptotiques de divers estimateurs classiques de processus déterminantaux paramétriques, stationnaires et non-stationnaires, dans les cas où l'on observe une unique réalisation d'un tel processus sur une fenêtre bornée. Ici, l'asymptotique se fait sur la taille de la fenêtre et donc, indirectement, sur le nombre de points observés. Dans une première partie, nous montrons un théorème limite central pour une classe générale de statistiques sur les processus déterminantaux. Dans une seconde partie, nous montrons une inégalité de béta-mélange générale pour les processus ponctuels que nous appliquons ensuite aux processus déterminantaux. Dans une troisième partie, nous appliquons le théorème limite central obtenu à la première partie à une classe générale de fonctions estimantes basées sur des méthodes de moments. Finalement, dans la dernière partie, nous étudions le comportement asymptotique du maximum de vraisemblance des processus déterminantaux. Nous donnons une approximation asymptotique de la log-vraisemblance qui est calculable numériquement et nous étudions la consistance de son maximum. / This manuscript is devoted to the study of parametric estimation of a point process family called determinantal point processes. These point processes are used to generate and model point patterns with negative dependency, meaning that the points tend to repel each other. More precisely, we study the asymptotic properties of various classical parametric estimators of determinantal point processes, stationary and non stationary, when considering that we observe a unique realization of such a point process on a bounded window. In this case, the asymptotic is done on the size of the window and therefore, indirectly, on the number of observed points. In the first chapter, we prove a central limit theorem for a wide class of statistics on determinantal point processes. In the second chapter, we show a general beta-mixing inequality for point processes and apply our result to the determinantal case. In the third chapter, we apply the central limit theorem showed in the first chapter to a wide class of moment-based estimating functions. Finally, in the last chapter, we study the asymptotic behaviour of the maximum likelihood estimator of determinantal point processes. We give an asymptotic approximation of the log-likelihood that is computationally tractable and we study the consistency of its maximum.

<p>This thesis consists of four papers, all concerning random iteration of isometries. The papers are:</p><p>I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117.</p><p>II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript.</p><p>III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987.</p><p>IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript.</p><p>In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {<i>Z</i>n} ^∞_n=0, of the iterations corresponding to an initial point Z₀, “escapes to infinity" in the sense that <i>P</i>(<i>Z</i>n Є <i>K)</i> → 0, as <i>n</i> → ∞ for every bounded set <i>K</i>. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.</p><p>In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I.</p><p>In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of <b>R</b>n. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.</p><p>In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane. </p>

This thesis consists of four papers, all concerning random iteration of isometries. The papers are: I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117. II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript. III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987. IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript. In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point. In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I. In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of <b>R</b>n. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach. In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.

This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

Šiame magistriniame darbe yra nagrinėjami nepriklausomi vienodai pasiskirstę atsitiktiniai dydžiai, turintys visus absoliutinius baigtinius momentus. Magistrinio darbo tikslas - atlikti konvergavimo greičio į normalųjį dėsnį įvertinimą. Darbą sudaro aštuoni skyriai. Įvade aprašoma problema ir visi tyrimo parametrai. Antrasis skyrius skirtas teoriniai analizei. Šiame skyriuje pateikiamos svarbiausios teorinės žinios ir metodai, kurie bus taikomi magistrinio darbo uždaviniams bei tikslams įgyvendinti. Trečiame skyriuje nagrinėjami kumuliantai Bernulio schemos atveju, o ketvirtame - analizuojamas Čebyšovo asimptotinis skleidinys ir pasinaudojus matematiniu paketu Maple, grafiniu būdu, tyrinėjamas jo konvergavimas. Aproksimacijos normaliuoju dėsniu tikslumui įvertinti naudojamas charakteristinių funkcijų metodas, todėl penktasis skyrius yra skiriamas suglodinimo nelygybių patikslinimui. Šeštame skyriuje, pasinaudojus turimais rezultatais, realizuojamas magistrinio darbo tikslas, o septintame - patikrinamas absoliutinės paklaidos įvertis Bernulio schemos atveju. Išvados ir rezultatai glaustai išdėstomi aštuntame skyriuje. / This master thesis considers independiant and identically distributed random variables, having absolute finite moments. The main task is to determine error estimate of the normal approximation. The work consists of eight chapters. In the introduction are considered problems and all subjects of research. The second chapter is designed for the theory analysis. Here are placed the main theoretical studies and methods that are used to solve the aims of the master thesis. The third chapter is intended to deal with cumulants in case of the Bernoulli’s distribution, the fourth one - is analyzing the Čebyšova’s asymptotic expansion and it convergence with the help of the mathematical package Maple. The method of characteristic’s functions is used to find the remainder term of the normal approximation, so the fifth chapter is designed to specify smoothing inequalities. Based on these results, the main task of the master thesis was obtained and specified in the sixth chapter. In the seventh one the error estimate in case of Bernoulli’s distribution, was examined with a mathematical package Maple. The short conclusions and results are placed in the eighth chapter.

Masters of Commerce / If share price fluctuations follow a simple random walk then it implies that forecasting models based on historical growth rates have little ability to forecast acceptable share price movements over a certain period. The simple random walk description of share price dynamics is obtained when a large number of investors have equal probability to buy or sell based on their own opinion. This simple random walk description of the stock market is in essence the Efficient Market Hypothesis, EMT. EMT is the central concept around which financial modelling is based which includes the Black-Scholes model and other important theoretical underpinnings of capital market theory like mean-variance portfolio selection, arbitrage pricing theory (APT), security market line and capital asset pricing model (CAPM). These theories, which postulates that risk can be reduced to zero sets the foundation for option pricing and is a key component in financial software packages used for pricing and forecasting in the financial industry. The model used by Black and Scholes and other models mentioned above are Gaussian, i.e. they exhibit a random nature. This Gaussian property and the existence of expected returns and continuous time paths (also Gaussian properties) allow the use of stochastic calculus to solve complex Black- Scholes models. However, if the markets are not Gaussian then the idea that risk can be. (educed to zero can lead to a misleading and potentially disastrous sense of security on the financial markets. This study project test the null hypothesis - share prices on the JSE follow a random walk - by means of graphical techniques such as symmetry plots and Quantile-Quantile plots to analyse the test distributions. In both graphical techniques evidence for the rejection of normality was found. Evidenceleading to the rejection of the hypothesis was also found through nonparametric or distribution free methods at a 1% level of significance for Anderson-Darling and Runs test.

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