On the accuracy of the poissonisation in the infinite occupancy scheme
Mikhail Chebunin
本文研究了无限占用方案中泊松化的渐近精度，针对具有正则变化分布的整值随机变量得出了相关结果。
我们获得了无限占用方案中泊松化的渐近精度。所有结果均针对具有正则变化分布的整值随机变量得出。
We obtain asymptotic accuracy of the poissonisation in the infinite occupancy scheme. All results are obtained for integer-valued random variables having a regularly varying distribution.
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We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.
Records in the Infinite Occupancy Scheme
本文研究经典无穷占位方案中的记录过程，在序列$(p_j)$衰减不太快时证明了缩放后的$r$-记录时间点过程近似为泊松过程，并在正则变化条件下论证了其联合收敛性。
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In this paper, we consider the occupancy distribution for an open network of infinite server queues with multivariate batch arrivals following a non-homogeneous Poisson process, and general service time distributions. We derive a probability generating function for the transient occupancy distribution of the network, and prove that it is necessary and sufficient for ergodicity that the expected occupancy time for each batch be finite. Further, we recover recurrence relations for the transient probability mass function formulated in terms of a distribution obtained by compounding the batch size…
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This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.
We investigate the randomized Karlin model with parameter $β\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $β/2\in(0,1/2)$. We show here that when the randomization is heavy-tailed with index $α\in(0,2)$, then the odd-occupancy process scales to a $(β/α)$-self-similar symmetric $α$-stable process with stationary increments.
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本文证明了独立指示变量无穷和的重对数律，根据期望与方差的相对增长确定归一化含迭代或单对数，并应用于Ginibre点过程与Karlin占位方案。
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An occupancy problem with an infinite number of bins and a random probability vector for the locations of the balls is considered. The respective sizes of bins are related to the split times of a Yule process. The asymptotic behavior of the landscape of first empty bins, i.e., the set of corresponding indices represented by point processes, is analyzed and convergences in distribution to mixed Poisson processes are established. Additionally, the influence of the random environment, the random probability vector, is analyzed. It is represented by two main components: an i.i.d. sequence and a fi…
We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labelled 1,2,... so that the urn $j$ at every draw gets a ball with probability $p_j$, $\sum_j p_j=1$. We prove functional central limit theorems for discrete time and the poissonised version for the urn occupancies process, for the odd-occupancy and for the missing mass processes extending the known non-functional central limit theorems.
On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking: the case of heavy tails
本文研究重尾情形下断棍生成的随机环境中嵌套占位方案的中间层，证明经归一化的中间层已占盒子数过程的有限维分布弱收敛于逆稳定从属子的Lebesgue-Stieltjes积分，拓展了此前有限矩情形的相关研究。
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Asymptotic bias of the plug-in Shannon entropy estimator under a regularly varying occupancy model
本文针对可数无限支撑离散分布的香农熵估计难题，在正则变化占用模型下推导了插件式熵估计量的渐近偏差关系，明确了幂律频率分布下偏差的渐近行为由基础分布尾特征决定。
Universal Exponent-Two Degree Laws in Range-Renewal Networks
该研究针对范围更新网络，推导其极限度分布的二次正则变化定律，结合概率估计与几何方法证明，经模拟验证结论，明确边方向与自环对极限度分布的影响。
Many-Server Asymptotics for Join-the-Shortest-Queue: Large Deviations and Rare Events
本文研究了处理器数量随到达率变化的Join-the-Shortest-Queue路由策略，在无限维路径空间中建立了占用过程的大偏差原理，并推导出长队列概率的显式指数衰减率。
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