Further Study of Random Sums, Bilinear Forms, Multilinear Forms, Stopping Times, Expectations, Tail Probabilities & Limit Theorems

摘要首席研究员和他的同事研究随机和的行为。提出了三个方面的工作：具有固定数目和的二次（双线性）形式，具有随机数目和的二次（双线性）形式，以及独立随机变量泊松化和的改进尾部概率近似。在第一个这样的领域，预计将获得的结果将确定一个函数的绝对值的期望的数量级的双线性形式（以及广义u统计量）的一个独立的实值随机变量的固定数量，对于任何非递减函数的多项式增长，而不进一步假设双线性形式，广义u统计量，分布，或自变量的数量。第二个问题领域，在时间允许的情况下，将尝试建立关于独立和同分布的平均零随机变量的特定双线性形式的最大绝对值的期望的类似结果，最大化到由变量本身决定的停止时间。第三，对于满足一定约束的独立随机变量泊松化和，提议者及其同事将尝试产生改进的和最好的尾概率近似。当数字很重要时，概率问题就会出现，而对情况的全面了解或控制是人类不可能做到的。例如，连锁商店（如西尔斯或梅西百货）的管理层必须不断地决定将多少资金分配给众多项目中的每一项。然而，它并不真正了解整个销售情况。管理层不仅不知道未来的趋势，也不知道其他公司在类似地区销售的可比或竞争性产品的当前销售结果如何——管理层可能不完全了解影响其销售模式的因素。为了做出好的财务决策，管理层需要一个相当全面和不断更新的财务状况数学模型。例如，它需要知道在给定的商店以及整个公司中，超过d件价格在c或c以上的未售出连衣裙的可能性有多大。为了评估其盈利能力对材料和劳动力成本以及消费者需求变化的敏感性，公司可能需要计算涉及数学家认为是二次型或广义u统计量的数量的期望。因此，如果适当地认识到它们的效用，本建议将获得的结果很可能具有重大的工业利益。

9626236 Klass ABSTRACT The Principal Investigator and his colleagues study the behavior of random sums. Work in three areas is proposed: quadratic (bilinear) forms having a fixed number of summands, ones having a random number of summands, and improved tail probability approximations for Poissonized sums of independent random variables. In the first such area it is anticipated that results will be obtained which identify the order of magnitude of the expectation of a function of the absolute value of a bilinear form (and also of a generalized U-statistic) of a fixed number of independent real-valued random variables, for any non-decreasing function of at most polynomial growth without further assumptions on the bilinear form, the generalized U-statistics, the distributions, or the number of independent variates. The second problem area, worked on if time permits, will attempt to establish similar results regarding the expectation of the maximum absolute value of one specific family of bilinear forms of independent and identically distributed mean zero random variates, maximized up to a stopping time determined by the variates themselves. Thirdly, the proposer and colleagues will attempt to produce improved and best possible tail probability approximations for Poissonized sums of independent random variables satisfying certain constraints. Probabilistic issues crop up whenever numbers are important and total knowledge or command of the situation is rendered humanly impossible. For example, the management of a chain store (e.g. Sears or Macy's) must continually make decisions concerning how much of its capital to allocate to each of a multitude of items. Yet it does not really know the entire sales picture. Not only does management not know future trends, it doesn't know what the current sales results are for comparable or competitive items sold by other companies in similar locations -- and management may not have full comprehension of the contributory factors influencing its own sales pattern. To make good financial decisions, management needs a fairly comprehensive and continually updated mathematical model of its fiscal situation. For instance, it needs to know what is the chance that it will have more than d unsold dresses of cost c or more in a given store, as well as throughout the company. To assess the sensitivity of its profitability to variations in materials and labor costs as well as consumer demand the company may want to compute expectations involving quantities which mathematicians recognize as quadratic forms or generalized U-statistics. Thus, if their utility were properly recognized, the results to be obtained in this proposal might well be of substantial industrial interest.