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

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

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

Appendix to "Approximating perpetuities"

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation $Y=_d UY + U(1-U)$ is constructed, where $Y$ and $U$ are independent and $U$ is uniformly distributed on $[0,1]$. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code

Methodol Comput Appl Probab (2008) 10:507–529 DOI 10.1007/s11009-007-9059-x Approximating Perpetuities

Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well

A note on the approximation of perpetuities

We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be well approximated

Pólya urns via the contraction method

We propose an approach to analyze the asymptotic behavior of Pólya urns 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. A decomposition of the 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 Pólya urns that lead to limit laws with normal limit distributions, with non-normal limit distributions and with asymptotic periodic distributional behavior