An algebraic approach to Polya processes

P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large)

The influence of urban form on travel patterns. An application to the metropolitan area of Bordeaux

The influence of urban form on travel patterns is of growing interest among researchers. It has been notably argued that high density, mixed land use settlements reduce automobile use and distances travelled, hence energy consumption per capita. A precise characterization of urban form calls analysis at an infra-urban level. We have questioned the interaction between land use and mobility in the metropolitan area of Bordeaux, France, by using OLS regressions for usual transportation variables and a multinomial logit model for modal shares. The results confirm a strong influence of both residential and firm density on mobility patterns. Mixed land use patterns doesn’t seem to influence mobility. Some economic and demographic characteristics have an influence on travel patterns. Thus it is unavoidable to take them in account. But sometimes it leads to a logical indecision, and it is difficult to determine the direction of the causal relationship. Keywords : urban sprawl, density, compact city, land use – mobility interaction JEL classification : R12, R14, R41

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

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)

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

The motives of mobility: an infra-urban level analysis. The case of Bordeaux, France.

The influence of urban form on travel patterns is of growing interest among researchers. It has been notably argued that high density, mixed land use settlements reduce automobile use and distances travelled, hence energy consumption per capita. A precise characterization of urban form calls analysis at an infra-urban level. We have questioned the interaction between land use and mobility in the metropolitan area of Bordeaux, France, by using OLS regressions for usual transportation variables and a multinomial logit model for modal shares. The results confirm a strong influence of both residential and firm density on mobility patterns. Mixed land use patterns doesn’t seem to influence mobility. Some economic and demographic characteristics have an influence on travel patterns. Thus it is unavoidable to take them in account. The strong interdependence between variables, and the difficulty to determine the direction of the causal relationships shows a strong degree of complexity of the problem. We’re led to conclude by proposing an explanatory framework for a better understanding of the factors of infra-urban mobility patterns.urban sprawl urban form density compact city mobility

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

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