Title: Records in the Infinite Occupancy Scheme Show Chinese title

Title: 无限占用方案中的记录

Abstract: We consider the classic infinite occupancy scheme, where balls are thrown in boxes independently, with probability $p_j$ of hitting box $j$. Each time a box receives its first ball we speak of a record and, more generally, call an $r$-record every event when a box receives its $r$th ball. Assuming that the sequence $(p_j)$ is not decaying too fast, we show that after many balls have been thrown, the suitably scaled point process of $r$-record times is approximately Poisson. The joint convergence of $r$-record processes is argued under a condition of regular variation. Show Chinese abstract

Abstract: 我们考虑经典的无限装箱方案，其中球被独立地投掷到箱子里，击中箱$j$的概率为$p_j$。每次一个箱子第一次收到球时，我们称之为记录，更一般地，当一个箱子收到第$r$个球时，我们称其为$r$记录。假设序列$(p_j)$不是衰减得太快，我们证明在投掷了许多球之后，适当缩放的$r$记录时间点过程近似为泊松过程。在正则变化条件下，$r$记录过程的联合收敛性得到了论证。