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

9626236克拉斯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：具有固定求和数的二次(双线性)形式，具有随机求和数的形式，以及独立随机变量泊松和的改进的尾概率近似。在第一个这样的领域中，预计将获得识别固定数量的独立实值随机变量的双线性形式(以及广义U-统计量)的绝对值的函数的期望的数量级的结果，对于任何至多是多项式增长的非递减函数，而不需要进一步假设双线性形式、广义U-统计量、分布或独立变量的数量。第二个问题领域，在时间允许的情况下，将试图建立关于一个特定的双线性形式的独立且同分布的平均零随机变量的最大绝对值的期望的类似结果，最大限度地直到由变量本身确定的停止时间。第三，提出者和他的同事将尝试为满足某些约束的独立随机变量的泊松和产生改进的和最佳可能的尾概率近似。当数字很重要，而对情况的全面了解或控制变得不可能在人类身上实现时，概率问题就会出现。例如，一家连锁店(如西尔斯或梅西百货)的管理层必须不断做出决定，决定将多少资本分配给众多商品中的每一件。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，也不知道类似地区其他公司销售的可比或竞争产品的当前销售结果--管理层可能没有完全理解影响自己销售模式的贡献因素。为了做出好的财务决策，管理层需要一个相当全面并不断更新的财务状况数学模型。例如，它需要知道，在一家给定的门店以及整个公司，它有超过d件C或更高价格的未售出连衣裙的可能性有多大。为了评估其盈利能力对原材料和劳动力成本以及消费者需求变化的敏感性，该公司可能想要计算涉及数量的预期，数学家认为这些数量是二次型或广义U-统计量。因此，如果它们的效用得到适当的认识，这项建议中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.