论文标题
嵌套卡林的占用方案中的小数
Small counts in nested Karlin's occupancy scheme generated by discrete Weibull-like distributions
论文作者
论文摘要
嵌套的卡林的占用方案是古典卡林的盒子中的球和加权分支过程的共生。为了定义它，请想象一个确定的加权分支过程，其中第一代人的权重由离散概率分布的要素给出。对于每个正整数$ j $，请确定$ j $ j $ th Generation Boxes的个人。球的集合是所有世代的一个或相同的，每个球都始于加权分支过程树的根部，并根据以下规则沿着树的根部移动：从母箱过渡到女儿盒子，以女儿和母重的比例给出了概率。 假设有$ n $ balls，而导致第一代的离散概率分布类似于Weibull。用$ \ Mathcal {k} _n^{（j）}（l）$和$ \ Mathcal {k} _n^{*（j）}（l）$分别包含$ j $ j $ l $ l $ l $ l $ balls的$ j $ thenoster盒的数量。我们证明了矩阵效率的过程$ \ big（\ Mathcal {k} _ {[{\ rm e}^{t+\ cdot}]}^{（j）}（j）}（l）\ big）_ { $ \ big（\ Mathcal {k} _ {[{\ rm e}^{t+\ cdot}]}}}^{*（j）}（l）\ big）_ {目前的FLT是Iksanov，Kabluchko和Kotelnikova（2022）证明的FLT的延伸，用于矢量值$ \ big（\ Mathcal {k} _ {[{{\ rm） e}^{t+\ cdot}]}}^{（j）}（1）\ big）_ {j \ in \ mathbb {n}} $。虽然每个极限矩阵值值过程的行是独立的且分布相同的，但每行的条目是固定的高斯过程，具有明确给定的协方差和跨交叉分配。我们为每一行提供一个不可或缺的表示。即使是卡林的占用计划，获得的结果也是新的。
A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's balls-in-boxes scheme and a weighted branching process. To define it, imagine a deterministic weighted branching process in which weights of the first generation individuals are given by the elements of a discrete probability distribution. For each positive integer $j$, identify the $j$th generation individuals with the $j$th generation boxes. The collection of balls is one and the same for all generations, and each ball starts at the root of the weighted branching process tree and moves along the tree according to the following rule: transition from a mother box to a daughter box occurs with probability given by the ratio of the daughter and mother weights. Assume that there are $n$ balls and that the discrete probability distribution responsible for the first generation is Weibull-like. Denote by $\mathcal{K}_n^{(j)}(l)$ and $\mathcal{K}_n^{*(j)}(l)$ the number of the $j$th generation boxes which contain at least $l$ balls and exactly $l$ balls, respectively. We prove functional limit theorems (FLTs) for the matrix-valued processes $\big(\mathcal{K}_{[{\rm e}^{T+\cdot}]}^{(j)}(l)\big)_{j,l\in\mathbb{N}}$ and $\big(\mathcal{ K}_{[{\rm e}^{T+\cdot}]}^{*(j)}(l)\big)_{j,l\in\mathbb{N}}$, properly normalized and centered, as $T\to \infty$. The present FLTs are an extension of a FLT proved by Iksanov, Kabluchko and Kotelnikova (2022) for the vector-valued process $\big(\mathcal{K}_{[{\rm e}^{T+\cdot}]}^{(j)}(1)\big)_{j\in\mathbb{N}}$. While the rows of each of the limit matrix-valued processes are independent and identically distributed, the entries within each row are stationary Gaussian processes with explicitly given covariances and cross-covariances. We provide an integral representation for each row. The results obtained are new even for Karlin's occupancy scheme.
