Abstract A man has a house with n doors. Initially he places k pairs of walking shoes at each door. For each walk, he chooses one door at random, and puts on a pair of shoes, returns after the walk to a randomly chosen door and takes off the shoes at the door. Let Tn be the first time a door is chosen to walk out but with no shoes available. We show that as Tn has the same asymptotic distribution and moments as the number of choices required to choose among n equally likely alternatives repeatedly until any one of the alternatives has appeared k + 1 times. To derive these results, we need to consider a more general setting where the numbers of pairs of shoes initially placed at the doors (initial configuration) are not necessarily equal. We show that Tn increases in the sense of stochastic ordering if the initial configuration is more evenly distributed in the sense of majorization.

中文翻译：

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当门的数量很大时，步行次数的渐近性直到没有鞋子

摘要 一个人有一栋有 n 扇门的房子。最初，他在每扇门上放了 k 双步行鞋。每次步行，他随机选择一个门，穿上一双鞋，步行后返回随机选择的门，在门口脱鞋。让 Tn 成为第一次选择一扇门走出但没有鞋子可用。我们表明，由于 Tn 具有相同的渐近分布和矩作为在 n 个同样可能的备选方案中反复选择所需的选择数量，直到任何一个备选方案出现 k + 1 次。为了得出这些结果，我们需要考虑一个更一般的设置，其中最初放置在门上的鞋子的数量（初始配置）不一定相等。