Polya urns via the contraction method

We propose an approach to analyze the asymptotic behavior of P\'olya urns based on the contraction method. For this, a new combinatorial discrete time embedding of the evolution of the urn into random rooted trees is developed. A decomposition of these trees leads to a system of recursive distributional equations which capture the distributions of the numbers of balls of each color. Ideas from the contraction method are used to study such systems of recursive distributional equations asymptotically. We apply our approach to a couple of concrete P\'olya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behavior.Comment: minor revision; accepted for publication in Combinatorics, Probability & Computing (Special issue dedicated to the memory of Philippe Flajolet

Approximating Perpetuities

A perpetuity is a real valued random variable which is characterised by a distributional fixed-point equation of the form X=AX+b, where (A,b) is a vector of random variables independent of X, whereas dependencies between A and b are allowed. Conditions for existence and uniqueness of solutions of such fixed-point equations are known, as is the tail behaviour for most cases. In this work, we look at the central area and develop an algorithm to approximate the distribution function and possibly density of a large class of such perpetuities. For one specific example from the probabilistic analysis of algorithms, the algorithm is implemented and explicit error bounds for this approximation are given. At last, we look at some examples, where the densities or at least some properties are known to compare the theoretical error bounds to the actual error of the approximation. The algorithm used here is based on a method which was developed for another class of fixed-point equations. While adapting to this case, a considerable improvement was found, which can be translated to the original method.Als Perpetuity wird vor allem in der Versicherungs- und Finanzmathematik eine reellwertige Zufallsvariable X bezeichnet, deren Verteilung implizit durch eine stochastische Fixpunktgleichung der Form X=AX+b charakterisiert ist. Dabei ist (A,b) ein Vektor von Zufallsvariablen, der unabhängig von X ist, Abhängigkeiten zwischen A und b sind jedoch erlaubt. Bedingungen für die Existenz und Eindeutigkeit von Lösungen solcher Fixpunktgleichungen sind bereits seit längerem bekannt. Für eine große Klasse solcher Perpetuities existieren Tail-Abschätzungen. Ziel dieser Arbeit ist es, den zentralen Bereich solcher Verteilungen zu untersuchen. Dazu wird ein Algorithmus für die Approximation der Verteilungsfunktionen und gegebenenfalls der Dichten von einer möglichst großen Klasse solcher Perpetuities entwickelt. Um für diese Approximationen explizite Fehlerschranken anzugeben, muss der Stetigkeitsmodul der approximierten Funktion abgeschätzt werden. Für eine spezielle Klasse von Fixpunktgleichungen werden universelle Abschätzungen angegeben, im Allgemeinen muss eine solche Abschätzung jedoch für den Einzelfall hergeleitet werden. Dies wird exemplarisch an einem Beispiel aus der probabilistischen Analyse von Algorithmen durchgeführt, für das auch der Algorithmus implementiert und eine Tafel der Verteilungsfunktion generiert wird. Um die Qualität der erhaltenen Fehlerschranken und die praktische Verwendbarkeit des Algorithmus zu beurteilen, werden abschließend einige Beispiele untersucht, in denen die Dichten oder zumindest gewisse Eigenschaften bereits bekannt sind. Hierbei zeigt sich, dass die theoretischen Fehlerschranken stets deutlich unterschritten werden und die Approximation in praktikabler Laufzeit bereits sehr gute Ergebnisse liefert. Der verwendete Algorithmus beruht auf einem bekannten Verfahren, das jedoch für eine andere Klasse von Fixpunktgleichungen entwickelt wurde. Bei der Anpassung an den hier betrachteten Fall konnte eine wesentliche Verbesserung erreicht werden, die sich auch auf den ursprünglichen Algorithmus übertragen lässt

Pólya urns via the contraction method

In this thesis, the asymptotic behaviour of Pólya urn models is analyzed, using an approach based on the contraction method. For this, a combinatorial discrete time embedding of the evolution of the composition of the urn into random rooted trees is used. The recursive structure of the trees is used to study the asymptotic behavior using ideas from the contraction method. The approach is applied to a couple of concrete Pólya urns that lead to limit laws with normal distributions, with non-normal limit distributions, or with asymptotic periodic distributional behavior. Finally, an approach more in the spirit of earlier applications of the contraction method is discussed for one of the examples. A general transfer theorem of the contraction method is extended to cover this example, leading to conditions on the coefficients of the recursion that are not only weaker but also in general easier to check

Local Government and Affordable Housing Tools

The issue of affordable housing is a complex and multi-faceted one. It often runs counter to the market principles, and is viewed with the same scrutiny and judgment as welfare programs at large. Affordable housing programs are funded primarily at the federal level, administered at the state level, and finally implemented through local communities. Given this relationship, it may seem that local communities have very little say in the way that affordable housing develops. This paper explores how empowering localities to tackle this issue through planning initiatives can have a significant impact on quality of life for low-income individuals. In this paper, the challenges for local governments are reviewed and evaluated: involved are the ability and willingness to use innovative planning and zoning techniques and strategic partnerships to achieve affordable housing goals tailored to the needs of the locality. Some localities focus more on place based strategies while others are people-based. Affordable housing policies change frequently, leading to lives of continuous upheaval for those in need of this safety net. The cases of two Michigan communities, Grand Rapids and Ann Arbor are reviewed. Both communities face high needs for affordable housing. Each has either embraced innovation at the planning level or has remained unable to fully adopt innovations. Grand Rapids has worked with many nonprofits to provide affordable housing, in combination with a revamping of traditional zoning codes to serve their low income populations more effectively than has Ann Arbor, which also has a high student population in need of affordable housing. Recommendations for local government officials seeking to meet the needs of lower-income constituents are presented

Public and Private Firms in Natural Resource Industry: Comparing the Development of the Lithium Industry in South America

Literature on public and private firms has traditionally focused on the efficiency effects of differing firm ownership, although the literature has been largely dormant since the 1980s. State-led development models in Asia and Latin America warrant continued analysis of this fundamental issue in International Political Economy. This paper attempts to frame debate on natural resource governance by identifying the factors that affect the mix of public and private firms in extractive industry. Historical institutionalism is used as a framework for analyzing policy change, and political economy theory of governance institutions is compared to the traditional economic theory of the firm. Analyzing the lithium industry in Chile and Bolivia as a case study, this paper suggests that temporal factors like historical stability of economic modeling and the sustained protection of property rights are critical in determining the optimum mix of public and private firms. This paper synthesizes existing scholarly contributions with original fieldwork gathered in the region