Some remarks on the direct calculation of Probabilities in Urn Schemes

Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn

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 Pólya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three a…

无限尿方案中一类随机外测度的极限定理

Limit theorems for a class of random outer measures in infinite urn schemes

本文研究了无限尿方案中一类随机外测度的极限定理，通过函数中心极限定理分析了文本概率模型的应用。

Azuma-Hoeffding bounds for a class of urn models

We obtain Azuma bounds for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using Jordan vectors.

带有触发机制的尿布模型用于常见问题（FAQ）

Triggered urn models for frequently asked questions (FAQ)

本文提出了一种带有触发机制的尿布模型，用于分析FAQ的分布特性，通过渐近分析和模拟验证了该模型在预测和实际应用中的有效性。

The coupon collector urn model with unequal probabilities in ecology and evolution

The sequential sampling of populations with unequal probabilities and with replacement in a closed population is a recurrent problem in ecology and evolution. Many of these questions can be reformulated as urn problems, often as special cases of the coupon collector problem, most simply expressed as the number of coupons that must be collected to have a complete set. We aimed to apply the coupon collector model in a comprehensive manner to one example -hosts (balls) being searched (draws) and parasitized (ball color change) by parasitic wasps- to evaluate the influence of differences in sampli…

Limit theorems for forward and backward processes of numbers of non-empty urns in infinite urn schemes

We study the joint asymptotics of forward and backward processes of numbers of non-empty urns in an infinite urn scheme. The probabilities of balls hitting the urns are assumed to satisfy the conditions of regular decrease. We prove weak convergence to a two-dimensional Gaussian process. Its covariance function depends only on exponent of regular decrease of probabilities. We obtain parameter estimates that have a normal asymototics for its joint distribution together with forward and backward processes. We use these estimates to construct statistical tests for the homogeneity of the urn schem…