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Register a new userRandom probability measures via Polya sequences: revisiting the Blackwell-MacQueen urn scheme
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Sufficient conditions are developed for a class of generalized Polya urn schemes ensuring exchangeability. The extended class includes the Blackwell-MacQueen Polya urn and the urn schemes for the two-parameter Poisson-Dirichlet process and finite dimensional Dirichlet priors among others.
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In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls depending on two discrete random variables X and Y with finite means and variances. Via the stochastic approximation algorithm, we give limit theorems describing the asymptotic behavior of white balls.
An extended Polya urn Model with two colors, black and white, is studied with some SLLN and CLT on the proportion of white balls.
We introduce a class of birth-and-death Polya urns, which allow for both sampling and removal of observations governed by an auxiliary inhomogeneous Bernoulli process, and investigate the asymptotic behaviour of the induced allelic partitions. By exploiting some embedded models, we show that the asymptotic regimes exhibit a phase transition from partitions with almost surely infinitely many blocks and independent counts, to stationary partitions with a random number of blocks. The first regime corresponds to limits of Ewens-type partitions and includes a result of Arratia, Barbour and Tavare (1992) as a special case. We identify the invariant and reversible measure in the second regime, which preserves asymptotically the dependence between counts, and is shown to be a mixture of Ewens sampling formulas, with a tilted Negative Binomial mixing distribution on the sample size.
We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A functional analog of this geometric inequality is derived, giving large and small deviation inequalities from a median, in terms of a modulus of regularity. Our methods are of independent interest, we find that log-affine sequences are the extreme points of the set of log-concave sequences belonging to a half-space slice of the simplex. This amounts to a discrete analog of the localization lemma of Lovasz and Simonovits. Further applications of this lemma are used to produce a discrete version of the Prekopa-Leindler inequality, large deviation inequalities for log-concave measures about their mean, and provide insight on the stability of generalized log-concavity under convolution.
An infinite sequence of real random variables $(xi_1, xi_2, dots)$ is said to be rotatable if every finite subsequence $(xi_1, dots, xi_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(xi_1, xi_2, dots)$ is rotatable if and only if $xi_j = tau eta_j$ for all $j$, where $(eta_1, eta_2, dots)$ is a sequence of independent standard Gaussian random variables and $tau$ is an independent nonnegative random variable. Freedmans theorem is equivalent to a classical result of Schoenberg which says that a continuous function $phi : mathbb{R}_+ to mathbb{C}$ with $phi(0) = 1$ is completely monotone if and only if $phi_n: mathbb{R}^n to mathbb{R}$ given by $phi_n(x_1, ldots, x_n) = phi(x_1^2 + cdots + x_n^2)$ is nonnegative definite for all $n in mathbb{N}$. We establish the analogue of Freedmans theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenbergs result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincare, and Borel which says that if $(zeta_1, ldots, zeta_n)$ is uniformly distributed on the sphere of radius $sqrt{n}$ in $mathbb{R}^n$, then, for fixed $k in mathbb{N}$, the distribution of $(zeta_1, ldots, zeta_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n to infty$.
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