With the prevalence of internet, it is gradually becoming a trend to use the network as a tool of computer-added education. However, it is used to present the computer-added education with static state of the word, but it is just convenient to read for the user and there are no difference with traditional textbook. As the growing up of WWW and the development of Java, the interactive computer-added education is becoming a trend in the future and it can promote the effect of teaching basic statistics with the application of this new media. The instructor can take advantage of HTML by combining with Java Applets to achieve the display of interactive education through WWW. In this paper, we will use six examples of Java Applets about statistical computer-added education to help student easily to learn and to understand some abstract statistical concepts. The key methods to reach the goal are visualization and simulation with the display of graphics or games. Finally, we will discuss how to use the Applets and how to add the Java Applets into your homepage easily.

We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute its unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero and infinity and derive analogues of classical probabilistic results such as the central limit theorem and large deviations principle.

Le sujet principal de ce mémoire est l'étude de la distribution asymptotique de la fonction f_m qui compte le nombre de diviseurs premiers distincts parmi les nombres premiers $p_1,...,p_m$. Au premier chapitre, nous présentons les sept résultats qui seront démontrés au chapitre 4. Parmi ceux-ci figurent l'analogue du théorème d'Erdos-Kac et un résultat sur les grandes déviations. Au second chapitre, nous définissons les espaces de probabilités qui serviront à calculer les probabilités asymptotiques des événements considérés, et éventuellement à calculer les densités qui leur correspondent. Le troisième chapitre est la partie centrale du mémoire. On y définit la promenade aléatoire qui, une fois normalisée, convergera vers le mouvement brownien. De là, découleront les résultats qui formeront la base des démonstrations de ceux chapitre 1. / The main topic of this masters thesis is the study of the asymptotic distribution of the fonction f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$. The first chapter provides the seven main results which will later on be proved in chapter 4. Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations. In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities. The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of chapiter 1.

Cette thèse se compose de plusieurs travaux portant sur l'énumération et le comportement asymptotique de structures combinatoires apparentées aux partitions d'entiers. Un premier travail s'intéresse aux partitions d'entiers bipartites, qui constituent une généralisation bidimensionnelle des partitions d'entiers. Des équivalents du nombre de partitions sont obtenus dans le régime critique où l'un des entiers est de l'ordre du carré de l'autre entier et au delà de ce régime critique. Ceci complète les résultats établis dans les années cinquante par Auluck, Nanda et Wright. Le deuxième travail traite des chaînes polygonales à sommets entiers dans le plan. Pour un modèle statistique introduit par Sinaï, une représentation intégrale exacte de la fonction de partition est donnée. Ceci conduit à un équivalent du nombre de chaînes joignant deux points distants qui fait intervenir les zéros non triviaux de la fonction zêta de Riemann. Une analyse combinatoire détaillée des chaînes convexes est présentée. Elle permet de montrer l'existence d'une forme limite pour les chaînes convexes aléatoires ayant peu de sommets, répondant ainsi à une question ouverte de Vershik. Un troisième travail porte sur les zonotopes à sommets entiers en dimension supérieure. Un équivalent simple est donné pour le logarithme du nombre de zonotopes contenus dans un cône convexe et dont les extrémités sont fixées. Une loi des grands nombres est établie et la forme limite est caractérisée par la transformée de Laplace du cône. / This thesis consists of several works dealing with the enumeration and the asymptotic behaviour of combinatorial structures related to integer partitions. A first work concerns partitions of large bipartite integers, which are a bidimensional generalization of integer partitions. Asymptotic formulæ are obtained in the critical regime where one of the numbers is of the order of magnitude of the square of the other number, and beyond this critical regime. This completes the results established in the fifties by Auluck, Nanda, and Wright. The second work deals with lattice convex chains in the plane. In a statistical model introduced by Sinaï, an exact integral representation of the partition function is given. This leads to an asymptotic formula for the number of chains joining two distant points, which involves the non trivial zeros of the Riemann zeta function. A detailed combinatorial analysis of convex chains is presented. It makes it possible to prove the existence of a limit shape for random convex chains with few vertices, answering an open question of Vershik. A third work focuses on lattice zonotopes in higher dimensions. An asymptotic equality is given for the logarithm of the number of zonotopes contained in a convex cone and such that the endings of the zonotope are fixed. A law of large numbers is established and the limit shape is characterized by the Laplace transform of the cone.

Weakly nonstationary processes appear in many challenging problems related to the physics of complex systems. An interesting question is how to quantify the rate of convergence to Gaussian behavior of rescaled heteroscedastic comming from economics time series with stationary first moments but nonstationary multifractal long-range correlated second moments and also time series generated from fractionated brownian motion where the series correlation is dependent of a parameter. Here it is used the approach Which uses a recently proposed extension of the Lévy sections theorem. It was analyzed the statistical and multifractal properties of heteroscedastic time series and found that the Lévy sections approach provides a faster convergence to Gaussian behavior relative to the convergence of traditional partial sums of variables. To understand this transition it is used several statistical tests to provide enough data on convergence behavior. It was also observed that the rescaled signals retain multifractal properties even after reaching what appears to be the stable Gaussian regime. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Processos não-estacionários com interações fracas aparecem como problemas desafiadores em sistemas complexos em física. Uma questão interessante é como quantificar a taxa de convergência para o comportamento gaussiano em séries temporais heteroscedásticas, sem uma variância única em toda a série, provenientes de sistemas financeiros, reescaladas com os primeiros momentos estacionários mas com uma multifractalidade não estacionária e segundos momentos que possuem uma correlação do longo alcance e verificar o mesmo mecanismo também em séries temporais geradas a partir de um movimento Browniano Fracionado onde a correlação da série depende de um parâmetro ajustável. Aqui é usada uma extensão do teorema das seções de Lévy. Analisando as propriedades estatísticas e multifractais de uma série temporal heteroscedástica e encontrando que as seções de Lévy fornece uma convergência mais rápida para o comportamento gaussiano relativo à convergência das tradicionais somas de variáveis, o teorema do limite central. Para entender essa transição foram utilizados vários testes estatísticos que forneceram dados suficientes sobre o comportamento de convergência. Também observou-se que os sinais reescalados mantêm suas propriedades multifractais mesmo depois de atingirem um regime que parece ser um regime gaussiano.

Nous étudions la mesure spectrale des transformations stationnaires, puis nous l’utilisons pour étudier le théorème ergodique et le théorème limite central. Nous étudions également les martingales avec une nouvelle preuve du théorème central limite, sans analyse de Fourier. Pour le théorème limite central pour marches aléatoires dans un environnement aléatoire sur la dimension 1, on donne deux méthodes pour l’obtenir: approximation pour une martingale et méthode des moments. La méthode des martingales fait résoudre l’équation de Dirichlet (I - P)h = 0, alors que celle des moments résoudre l’équation de Poisson (I - P)h = f. Enfin, nous pouvons utiliser la deuxième méthode pour prouver la relation d’Einstein pour des diffusions réversibles dans un environnement aléatoire dans une dimension. / We study the spectral measure for stationary transformations, and then apply to Ergodic theorem and Central limit theorem. We study also martingale process with a new proof of the central limit theorem without Fourier analysis. For the central limit theorem for random walks in random environment, we give two methods to obtain it: martingale approximation and moments. The method of martingales solves Dirichlet’s equation (I - P)h = 0, and the method of moments solves Poisson’s equation (I - P)h = f. Finally, we can use the second method to prove the Einstein relation for reversible diffusions in random environment in one dimension.

La première partie de cette thèse porte sur l’étude de la marche aléatoire sur le tore Td := Rd/Zd définie par une mesure de probabilité SLd(Z). Pour étudier le Théorème Central Limite et la loi du logarithme itéré, nous appliquons la méthode de Gordin qui consiste à se ramener à des martingales. Pour cela, nous utilisons un résultat de Bourgain, Furmann, Lindenstrauss et Mozes nous permettant de résoudre l’équation de Poisson pour des points ayant de bonnes propriétés diophantiennes. Dans la deuxième partie, nous étudions la marche sur Rd\{0} définie par l’action de SLd(R) et nous montrons un résultat de vitesse de convergence dans le théorème de renouvellement de Guivarc’h et Le Page. / The first part of this thesis deals with the random walk on the torus Td := Rd/Zd defined by a robability measure on SLd(Z). To study the Central Limit Theorem and the Law of the Iterated Logarithm, we apply Gordin’s method. To do so, we use a result proved by Bourgain, Furmann, Lindenstrauss and Mozes to solve Poisson’s equation at point’s having good diophantine properties.In the second part, we study the walk on Rd \ {0} defined by the action of SLd(R) and we prove a result about the rate of convergence in Guivarc’h and Le Page’s renewal theorem.

Der Beitrag stellt eine Verallgemeinerung der Ergebnisse dar, die in den Informationen/07; 1976,05 veröffentlicht wurden. Sei F_n(x) die Verteilungsfunktion der Summe X_1+X_2+...+X_n, wobei X_1, X_2, ...,X_n unabhängige und nicht notwendig identisch verteilte Zufallsgrößen mit endlichen absoluten Momenten c_m, m>2, sind, und sei Phi die standardisierte Normalverteilungsfunktion. Es werden absolute Konstanten L_m derart berechnet, dass wir Fehlerabschätzungen im unleichmäßigen zentralen Grenzwertsatz explizit angeben können. Als Spezialfall ergibt sich die ungleichmäßige Fehlerschranke von A.BIKELIS (1966) im Fall der Existenz dritter absoluter Momente. Weiterhin werden Grenzwertsätze unter Voraussetzung einseitiger Momente betrachtet. Es werden einige Literaturhinweise angegeben.:1. Grenzwertsätze für verschieden verteilte Zufallsgrößen S. 1 2. Grenzwertsätze unter Voraussetzung einseitiger Momente S. 6 3. Beweise zum Abschnitt 1 S. 7 4. Beweise zum Abschnitt 2 S. 14 Literatur S. 16 / The paper is a generalization of the results, published by the author in Informationen/07; 1976,05. Let F_n(x) be the cdf of X_1+X_2+...+X_n, where X_1, X_2, ...,X_n are non iid random variables with m-th absolute moment c_m, m>2, and Phi the cdf of the unit normal law. Explicit universal constants L_m are computed such that we have some error estimates in the nonuniform central limit theorem. A special case is the nonuniform error bound by A.BIKELIS (1966) in the case of existence of third absolute moments. Furthermore limit theorems with assumption of onesided moments are considered. Some references are given.:1. Grenzwertsätze für verschieden verteilte Zufallsgrößen S. 1 2. Grenzwertsätze unter Voraussetzung einseitiger Momente S. 6 3. Beweise zum Abschnitt 1 S. 7 4. Beweise zum Abschnitt 2 S. 14 Literatur S. 16

Es wird eine Folge unabhängiger zentrierter Zufallsgrößen betrachtet, die absolute Momente der Ordnung m, 2<m<3, besitzen mögen. Dann gelten für die normierte Verteilungsfunktion der Zufallssumme X_1+X_2+...+X_n der zentrale Grenzwertsatz und insbesondere eine ungleichmäßige Fehlerabschätzung von A.BIKELIS (1966). In der vorliegenden Note werden die analytische Struktur der in dieser Fehlerabschätzung auftretenden Konstanten L=L(m) genauer untersucht sowie dazu erzielte numerische Resultate vorgelegt. Abschließend werden einige Literaturhinweise angegeben. Der Fall m=3 wurde bereits in der Dissertation (TU Dresden 1977) des Autors untersucht. / We consider a sequence of centered and independent random variables with moments of order m, 2<m<3. Now the central limit theorem for the distribution function of the normed sum X_1+X_2+...+X_n and especially a nonuniform error estimate by A.BIKELIS (1966) hold. In this paper the analytical structure of the appearing constant L=L(m) of the error bound and numerical results are presented. Finally some references are given. The case m=3 was already studied in the thesis (Dissertation TU Dresden, 1977) by the author.

This thesis project is an experimental study on how to approach quantitative portfolio credit risk modelling in Venture Debt portfolios. Facing a lack of applicable default data from ArK and publicly available sets, as well as seeking to capture companies that fail to service debt obligations before defaulting per se, we present an approach to risk modeling based on trends in revenue. The main framework revolves around driving a Monte Carlo simulation with Copluas to predict future revenue scenarios across a portfolio of early-stage technology companies. Three models for a random Gaussian walk, a Linear Dynamic System and an Autoregressive Integrated Moving Average (ARIMA) time series are implemented and evaluated in terms of their portfolio Value-at-Risk influence. The model performance confirms that modeling portfolio risk in Venture Debt is challenging, especially due to lack of sufficient data and thus a heavy reliance on assumptions. However, the empirical results for Value-at-Risk and Expected Shortfall are in line with expectations. The evaluated portfolio is still in an early stage with a majority of assets not yet in their repayment period and consequently the spread of potential losses within one year is very tight. It should further be recognized that the scope in terms of explanatory variables for sales and model complexities has been narrowed and simplified for computational benefits, transparency and communicability. The main conclusion drawn is that alternative approaches to model Venture Debt risk is fully possible, and should improve in reliability and accuracy with more data feeding the model. For future research it is recommended to incorporate macroeconomic variables as well as similar company analysis to better capture macro, funding and sector conditions. Furthermore, it is suggested to extend the set of financial and operational explanatory variables for sales through machine learning or neural networks. / Detta examensarbete är en experimentell studie för kvantitativ modellering av kreditrisk i Venture Debt-portföljer. Givet en brist på tillgänlig konkurs-data från ArK samt från offentligt tillgängliga databaser i kombination med ambitionen att inkludera företag som misslyckas med skuldförpliktelser innan konkurs per se, presenterar vi en metod för riskmodellering baserad på trender i intäkter. Ramverket för modellen kretsar kring Monte Carlo-simulering med Copluas för att estimera framtida intäktsscenarier över en portfölj med tillväxtbolag inom tekniksektorn. Tre modeller för en random walk, ett linjärt dynamiskt system och ARIMA- tidsserier implementeras och utvärderas i termer av deras inflytande på portföljens Value-at- Risk. Modellens prestationer bekräftar att modellering av portföljrisk inom Venture Debt är utmanande, särskilt på grund av bristen på tillräckliga data och därmed ett stort beroende av antaganden. Dock är de empiriska resultaten för Value-at-Risk och Expected Shortfall i linje med förväntningarna. Den utvärderade portföljen är fortfarande i ett tidigt skede där en majoritet av tillgångarna fortfarande befinner sig i en amorteringsfri period och följaktligen är spridningen av potentiella förluster inom ett år mycket snäv. Det bör vidare tillkännages att omfattningen i termer av förklarande variabler för intäkter och modellkomplexitet har förenklats för beräkningsfördelar, transparens och kommunicerbarhet. Den främsta slutsatsen som dras är att alternativa metoder för att modellera risker inom Venture Debt är fullt möjliga och bör förbättras i tillförlitlighet och precision när mer data kan matas in i modellen. För framtida arbete rekommenderas det att inkorporera makroekonomiska variabler samt analys av liknande bolag för att bättre fånga makro-, finansierings- och sektorsförhållanden. Vidare föreslås det att utöka uppsättningen av finansiella och operationella förklarande variabler för intäkter genom maskininlärning eller neurala nätverk.