论文标题
关于杆折的随机环境中嵌套占用方案的中间水平。
On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking I
论文作者
论文摘要
考虑通过单位长度敲打（又称剩余分配模型）获得的间隔长度产生的加权分支过程，并与每个权重签名为“盒子”。鉴于重量“球”被独立投入到第一代的盒子中，可能会击中等于其重量的盒子。每个球的$ J $ th Generation（与其他人都独立）击中了一个女儿盒子（J+1）$第三代的盒子，概率等于女儿体重和母重的比例。这就是我们在随机环境中所说的嵌套占用方案。将注意力限制在特定一代人中，在随机环境中获得了经典的Karlin占用计划。 Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_n u\rfloor$ for $u>0$, where $j_n$ diverges to infinity at a sublogarithmic rate as $n$ becomes large.用$ k_n（j）$表示$ j $ th Generation中被占领的盒子的数量。结果表明，该过程$（k_n（\ lfloor j_n u \ rfloor）的有限维分布）_ {u> 0} $，适当地归一化和居中，弱收敛到布朗运动的整体功能的趋势。还分析了更普遍的破坏棍子的情况。
Consider a weighted branching process generated by the lengths of intervals obtained by stick-breaking of unit length (a.k.a. the residual allocation model) and associate with each weight a `box'. Given the weights `balls' are thrown independently into the boxes of the first generation with probability of hitting a box being equal to its weight. Each ball located in a box of the $j$th generation, independently of the others, hits a daughter box in the $(j+1)$th generation with probability being equal the ratio of the daughter weight and the mother weight. This is what we call nested occupancy scheme in random environment. Restricting attention to a particular generation one obtains the classical Karlin occupancy scheme in random environment. Assuming that the stick-breaking factor has a uniform distribution on $[0,1]$ and that the number of balls is $n$ we investigate occupancy of intermediate generations, that is, those with indices $\lfloor j_n u\rfloor$ for $u>0$, where $j_n$ diverges to infinity at a sublogarithmic rate as $n$ becomes large. Denote by $K_n(j)$ the number of occupied (ever hit) boxes in the $j$th generation. It is shown that the finite-dimensional distributions of the process $(K_n(\lfloor j_n u\rfloor))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a Brownian motion. The case of a more general stick-breaking is also analyzed.
