This paper reviews P´olya urn models and their connection to random trees. Basic results are presented, together with proofs that underly the historical evolution of the accompanying thought process. Extensions and generalizations are given according to chronology: • P´olya-Eggenberger’s urn • Bernard Friedman’s urn • Generalized P´olya urns • Extended urn schemes • Invertible urn schemes ...

in this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. we also find the expectation of the number of nodes of a given outdegree in a branch of such trees.

We investigate the distribution, mean value, variance and some limiting properties of an urn model of white and red balls under random multiple drawing (either with or without replacement) when the number of white and red balls added follows a schedule that depends on the number of white balls chosen in each drawing.

Motivated by mathematical tissue growth modelling, we consider the problem of approximating dynamics multicolor P\'olya urn processes that start with large numbers balls different colors and run for a long time. Using strong approximation theorems empirical quantile processes, establish Gaussian process approximations processes. The are sums multivariate Brownian motion an independent linear dr...

A way of making Bayesian inference for concave distribution functions is introduced. This is done by uniquely transforming a mixture of Dirichlet processes on the space of distribution functions to the space of concave distribution functions. The approach also gives a way of making Bayesian analysis of mul-tiplicatively censored data. We give a method for sampling from the posterior distributio...

We de ne a reinforced urn process (RUP) to be a reinforced random walk on a state space of urns and we show its partial exchangeability. When it is recurrent, a RUP is a mixture of Markov chains and we characterize its mixing distribution on the space of stochastic matrices. Many Bayesian nonparametric priors, like P olya trees, the beta-Stacy process and, in general, neutral to the right proce...

A new O(n) algorithm is given for evaluating univariate polynomials of degree n in the P olya basis. Since the Lagrange, Bernstein, and monomial bases are all special instances of the P olya basis, this technique leads to e cient evaluation algorithms for these special bases. For the monomial basis, this algorithm is shown to be equivalent to Horner's rule. 1 P olya basis functions Let nk(t) be...

Ostreolysin A (OlyA) is an ∼15-kDa protein that has been shown to bind selectively to membranes rich in cholesterol and sphingomyelin. In this study, we investigated whether OlyA fluorescently tagged at the C-terminal with mCherry (OlyA-mCherry) labels cholesterol/sphingomyelin domains in artificial membrane systems and in membranes of Madin-Darby canine kidney (MDCK) epithelial cells. OlyA-mCh...

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights unbounded number possible colors. Under fairly mild assumptions on the expected reinforcement, we show that predictive empirical distributions converge almost surely (a.s.) in total variation to same probability measure $\tilde{P}$; moreover...