Limit distributions for large P\'{o}lya urns

We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings, the composition vector has asymptotically a first deterministic term of order $n$ and a second random term of order $n^{m/S}$. The object of interest is the limit distribution of this random term. The method consists in embedding the discrete-time urn in continuous time, getting a two-type branching process. The dislocation equations associated with this process lead to a system of two differential equations satisfied by the Fourier transforms of the limit distributions. The resolution is carried out and it turns out that the Fourier transforms are explicitly related to Abelian integrals over the Fermat curve of degree $m$. The limit laws appear to constitute a new family of probability densities supported by the whole real line.Comment: Published in at http://dx.doi.org/10.1214/10-AAP696 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Support and density of the limit mm-ary search trees distribution

The space requirements of an $m$-ary search tree satisfies a well-known phase transition: when $m\leq 26$, the second order asymptotics is Gaussian. When $m\geq 27$, it is not Gaussian any longer and a limit $W$ of a complex-valued martingale arises. We show that the distribution of $W$ has a square integrable density on the complex plane, that its support is the whole complex plane, and that it has finite exponential moments. The proofs are based on the study of the distributional equation W\egalLoi\sum_{k=1}^mV_k^{\lambda}W_k, where $V_1, ..., V_m$ are the spacings of $(m-1)$ independent random variables uniformly distributed on $[0,1]$, $W_1, ..., W_m$ are independent copies of W which are also independent of $(V_1, ..., V_m)$ and $\lambda$ is a complex number

Smoothing equations for large P\'olya urns

Consider a balanced non triangular two-color P\'olya-Eggenberger urn process, assumed to be large which means that the ratio sigma of the replacement matrix eigenvalues satisfies 1/2<sigma <1. The composition vector of both discrete time and continuous time models admits a drift which is carried by the principal direction of the replacement matrix. In the second principal direction, this random vector admits also an almost sure asymptotics and a real-valued limit random variable arises, named WDT in discrete time and WCT in continous time. The paper deals with the distributions of both W. Appearing as martingale limits, known to be nonnormal, these laws remain up to now rather mysterious. Exploiting the underlying tree structure of the urn process, we show that WDT and WCT are the unique solutions of two distributional systems in some suitable spaces of integrable probability measures. These systems are natural extensions of distributional equations that already appeared in famous algorithmical problems like Quicksort analysis. Existence and unicity of the solutions of the systems are obtained by means of contracting smoothing transforms. Via the equation systems, we find upperbounds for the moments of WDT and WCT and we show that the laws of WDT and WCT are moment-determined. We also prove that WDT is supported by the whole real line and admits a continuous density (WCT was already known to have a density, infinitely differentiable on R\{0} and not bounded at the origin)

What Has 100 Billion Dollars Worth of Debt Relief Done for Low- Income Countries?

Between 1989 and 2003, low-income countries received $100 billion in debt relief. The stated objectives for much of this debt relief have been to reduce debt overhang and to free up recipient government resources for development spending that would otherwise have been used for debt service. In this paper we empirically assess the extent to which debt relief has been successful in meeting these objectives, using a newly-constructed database measuring the present value of debt relief for 62 low-income countries. We find little evidence that debt relief has affected the level and composition of public spending in recipient countries. We also do not find evidence that debt relief has raised growth, investment rates or the quality of policies and institutions among recipient countries. Although we cannot rule out the possibility that our failure to find evidence of positive impacts of debt relief is due to a variety of data and statistical problems, the evidence reported here does suggest that some skepticism is in order regarding the likely benefits of further large-scale debt relief.Debt Relief, HIPC, Low-Income Countries, Debt

Who gets debt relief ?

The authors use preliminary results from an ongoing effort to construct estimates of debt relief to study its allocation across a sample of 62 low-income countries. They find some evidence that debt relief, particularly from multilateral creditors, has been allocated to countries with better policies in recent years. Somewhat surprisingly, conditional on per capita incomes and policy, more indebted countries are not much more likely to receive debt relief. But countries that have large debts especially to multilateral creditors are more likely to receive debt relief. The authors do not find much evidence that debt relief responds to shocks to GDP growth. Finally, most of the persistence in debt relief is driven by slowly changing country characteristics, indicating that it may be difficult for countries to"exit"from cycles of repeated debt relief.External Debt,Banks&Banking Reform,Strategic Debt Management,Foreign Direct Investment,Economic Theory&Research

The Effects of Conflict on the Structure of the Economy

The presence of conflict affects peoples economic incentives. Some sectors of activity flourish, while others suffer. For understanding structural problems in developing countries and designing appropriate post-conflict reconstruction policies, it is essential to understand in what ways conflict affects the structure of the economy. We develop a simple model of conflict and multiple sectors of activity, where conflict efforts, the allocation of factor endowments and the production outputs are endogenous. We predict that for moderately destructive conflicts labor-intensive sectors are most affected by fighting, while for highly destructive conflicts capital-intensive sectors suffer most. In the latter case, under some conditions it is also possible that in the presence of endogenous conflict - an increase in the price of the capital-intensive commodity reduces the output of this same good. The model further predicts that export-sectors and sectors that require inter-temporal investments are particularly exposed to conflict activity. In the empirical part of the paper, we study the impact of various forms of conflict, separately and as an aggregate conflict index constructed with principal component analysis. We present some basic stylized facts about the effect of conflict on the productive structure of the economy. Conflict reduces the share of the manufacturing sector in the GDP, increases the exploitation of some simple natural resources (i.e. forestry) and reduces the production of crops. Using industrial level data for developing countries we study the channels through which conflict affects the manufacturing sector. As expected, we find that industries that are more institutional/transaction intensive are the ones that suffer most in conflictive societies. Laborintensive sectors are also negatively affected by conflict. It is also found that exporting industries and sectors requiring external financing suffer more during conflict. Our results are robust to sensitivity analysis. --Conflict,Production Structure,Resource Curse,Post-Conflict Reconstruction

Digital search trees and chaos game representation

In this paper, we consider a possible representation of a DNA sequence in a quaternary tree, in which on can visualize repetitions of subwords. The CGR-tree turns a sequence of letters into a digital search tree (DST), obtained from the suffixes of the reversed sequence. Several results are known concerning the height and the insertion depth for DST built from i.i.d. successive sequences. Here, the successive inserted wors are strongly dependent. We give the asymptotic behaviour of the insertion depth and of the length of branches for the CGR-tree obtained from the suffixes of reversed i.i.d. or Markovian sequence. This behaviour turns out to be at first order the same one as in the case of independent words. As a by-product, asymptotic results on the length of longest runs in a Markovian sequence are obtained

Is China the new land of football ?

In recent years, China has been in the spotlight for its determination to develop a relatively unexpected industry within its boundaries, football. The idea was introduced by the current President of the People’s Republic of China, Mr Xi Jinping, who intends to turn China into a top football nation. This is rather surprising considering that China has never been successful in football and has also never had a domestic football star. Thus, the Chinese government, the Chinese Football Association and local clubs have launched several considerable projects and concluded deals where large amounts of money were involved in order to help develop football in China. The Chinese large financial investments in football have made the headlines of many newspapers, notably The Financial Times and The Economist. These investments have led the football community to wonder whether the development of football in China is worthwhile. Many football professionals mainly criticised the lack of investments in grassroots of Chinese football and the surplus of money paid by Chinese clubs to attract foreign football players. Hence, this study aims to analyse the Chinese football business model and compare it with other nations’ football business models to see if Chinese financial investments in football are reasonable. It is shown later in this document that the Chinese financial investments in football are actually guided by a well thought-out strategy. Chinese are surely financially investing a lot but when compared to other nations, Chinese financial investments in football are far from excessive. In addition, the Chinese Football Association and some Chinese football clubs have succeeded to build a strong network with top football nations and clubs in order to improve their knowledge in the sport. The main challenge for the Chinese leaders of the national football project will be to create a football culture in China, which is essential if they want their country to succeed in this sport. The Chinese government, businesses, and national football association are already financing many projects to positively influence the growth of a football culture within the country. Nonetheless, a football culture cannot simply be bought and consequently results that are expected from this spending are not immediate. The main recommendation for the Chinese authorities involved in the national football project is to keep investing in football youth development while controlling overspending. In addition, they need to remain patient in order to see emerging generations of local football stars that will have benefited from the investments currently being made

Variable length Markov chains and dynamical sources

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the ``comb'' and the ``bamboo blossom'', we find a necessary and sufficient condition for the existence and the unicity of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences.Comment: 45 pages, 15 figure

Experimental control of Variable Cam Timing actuators

International audienceIn this paper, we propose a control scheme for the position of Variable Cam Timing (VCT) actuators. We take into account the observation by (Genç et al., 2001) that these actuators feature a significant nonlinearity, but we use a different approach. We perform a model reduction of mechanics, pressure, flows and balance equations. It serves to design an observer used in a motion planning strategy that is ultimately complemented by a tracking controller. Supportive vehicle experimental results are presented