Related papers: Polya urns via the contraction method
We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous' theorem giving the…
This is a technical note which extends the results of Kosygina, Mountford and Peterson (Ann. Probab., 51(5):1684-1728, 2023, Section 4) about generalized P\'olya's urns from a specific weight function $w(n) = (n+1)^{-\alpha}$ to a general…
We study linear recurrences of Eulerian type of the form \[ P_n(v) = (\alpha(v)n+\gamma(v))P_{n-1}(v) +\beta(v)(1-v)P_{n-1}'(v)\qquad(n\ge1), \] with $P_0(v)$ given, where $\alpha(v), \beta(v)$ and $\gamma(v)$ are in most cases polynomials…
Urn models play an important role to express various basic ideas in probability theory. Here we extend this urn model with tubes. An urn contains coloured balls, which can be drawn with probabilities proportional to the numbers of balls of…
In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily…
In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random…
We exploit the multiplicative structure of P\'olya Tree priors to establish novel consistency results on $p$-dimensional trees, conditions to obtain Kullback-Leibler minimax contraction rates for univariate density estimation and a…
We consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number…
We give an intuitive method--using local, cyclic replica symmetry--to isolate exponential tree decay in truncated (connected) correlations. We give an expansion and use the symmetry to show that all terms vanish, except those displaying…
This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated in Bai and Hu (1999,2005) and Bai, Hu and Shen (2002). We reformulate the dynamics of both the urn composition and…
We define the notions of disjoint unions and products for generalised P\'olya urns, proving that this turns the set of isomorphism classes of urns into a commutative semiring. The set of square matrices up to similarity by a permutation…
For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law P\'olya's urn" of Hammond and Sheffield we prove functional convergence…
We consider a stochastic process in which independent identically distributed random matrices are multiplied and where the Lyapunov exponent of the product is positive. We continue multiplying the random matrices as long as the norm,…
Boltzmann samplers and the recursive method are prominent algorithmic frameworks for the approximate-size and exact-size random generation of large combinatorial structures, such as maps, tilings, RNA sequences or various tree-like…
We study random trees which are invariant in law under the operation of contracting each edge independently with probability $p\in(0,1)$. We show that all such trees can be constructed through Poissonian sampling from a certain class of…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
We investigate the profile of random P\'olya trees of size $n$ when only nodes of degree $d$ are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process…
We discuss a complementary asymptotic analysis of the so called minimal random walk. More precisely, we present a version of the almost sure central limit theorem as well as a generalization of the recently proposed quadratic strong laws.…
The election is a classical problem in distributed algorithmic. It aims to design and to analyze a distributed algorithm choosing a node in a graph, here, in a tree. In this paper, a class of randomized algorithms for the election is…
Drawing (a multiset of) coloured balls from an urn is one of the most basic models in discrete probability theory. Three modes of drawing are commonly distinguished: multinomial (draw-replace), hypergeometric (draw-delete), and Polya…