Extensions of the Generalized Pólya Process

Methodology and Computing in Applied Probability - A new self-exciting counting process is here considered, which extends the generalized Pólya process introduced by Cha (Adv Appl Probab...     

The continuous-time triangular Pólya process

We study poissonized triangular (reducible) urns on two colors, which we take to be white and blue. We analyze the number of white and blue balls after a certain period of time has elapsed. We show that for balanced processes in this class, a different scaling is needed for each color to produce nontrivial limits, contrary to the distributions in the usual irreducible urns which only require the same scaling for both colors. The limit distributions (of the scaled variables) underlying triangular urns are Gamma. The technique we use couples partial differential equations with the method of moments applied in a bootstrapped manner to produce exact and asymptotic moments. For the dominant color, we get exact moments, while relaxing the balance condition. The exact moments include alternating signs and Stirling numbers of the second kind.     

On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.     

Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as . We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.     

A generalization of the Pólya–Vinogradov inequality

In this paper we consider an approach of Dobrowolski and Williams which leads to a generalization of the Pólya–Vinogradov inequality. We show how the Dobrowolski–Williams approach is related to the classical proof of Pólya–Vinogradov using Fourier analysis. Our results improve upon the earlier work of Bachman and Rachakonda (Ramanujan J. 5:65–71, 2001). In passing, we also obtain sharper explicit versions of the Pólya–Vinogradov inequality.     

Computational Aspects of Optional Pólya Tree

Optional Pólya tree (OPT) is a flexible nonparametric Bayesian prior for density estimation. Despite its merits, the computation for OPT inference is challenging. In this article, we present time complexity analysis for OPT inference and propose two algorithmic improvements. The first improvement, named limited-lookahead optional Pólya tree (LL-OPT), aims at accelerating the computation for OPT inference. The second improvement modifies the output of OPT or LL-OPT and produces a continuous piecewise linear density estimate. We demonstrate the performance of these two improvements using simulated and real date examples.     

Functional Limit Theorems for the Pólya Urn

Journal of Theoretical Probability - For the plain Pólya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls, assuming that the...     

The Pólya sum process: a Cox representation

In [1], Zessin constructed the so-called Pólya sum process via partial integration. Here we use the technique of integration by parts to the Pólya sum process to derive representations of the Pólya sum process as an infinitely divisible point process and a Cox process directed by an infinitely divisible random measure. This result is related to the question of the infinite divisibilty of a Cox process and the infinite divisibility of its directing measure. Finally we consider a scaling limit of the Pólya sum process and show that the limit satisfies an integration by parts formula, which we use to determine basic properties of this limit.     

A basic analog of a theorem of Pólya

We derive a q-analog of a theorem of George Pólya (1918), , concerning the zeros of basic cosine and sine transforms. The results are established without further restrictions on q.      

On a theorem of G. Pólya

f(z), :f(n)=0 (n=0, ±1, ±2, ...). ((n)} L _p ,p>1, .     

Martingale characterization of Pólya processes and sequences

It is proved that a mixed Poisson process ξ_t is a Pólya process if and only if there exists a nondegenerate linear transform ξ_t → η_t = a(t)ξ_t + b(t) such that η_t is a martingale. A similar result is valid for Pólya sequences.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.     

On the Mixed Pólya–Szeg? Principle

In this paper,the mixed Pólya-Szeg? principle is established.By the mixed Pólya-Szeg? principle,the mixed Morrey-Sobolev inequality and some new analytic inequalities are obtained.     

The Pólya–Chebotarev problem and inverse polynomial images

Consider the problem, usually called the Pólya–Chebotarev problem, of finding a continuum in the complex plane including some given points such that the logarithmic capacity of this continuum is minimal. We prove that each connected inverse image ${\mathcal {T}}_{n}^{-1} ([-1,1])$ of a polynomial  ${\mathcal {T}}_{n}$ is always the solution of a certain Pólya–Chebotarev problem. By solving a nonlinear system of equations for the zeros of ${\mathcal {T}}_{n}^{2}-1$ , we are able to construct polynomials ${\mathcal {T}}_{n}$ with a connected inverse image.     

Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques.In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.