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

9626236 Klass摘要首席研究员和他的同事研究随机和的行为。 工作在三个领域提出：二次（双线性）形式具有固定数量的被加数，具有随机数量的被加数，和改进的尾概率近似Poissonized和独立的随机变量。 在第一个这样的领域，预计将获得确定双线性形式的绝对值的函数的期望的数量级的结果（以及广义U-统计量）的固定数量的独立实值随机变量，对于任何至多多项式增长的非减函数，而无需进一步假设双线性形式，广义U-统计量，分布，或者独立变量的数量。 第二个问题领域，如果时间允许的话，将试图建立类似的结果，关于一个特定的家庭的双线性形式的独立和同分布的平均零随机变量的最大绝对值的期望，最大化到一个停止时间由变量本身确定。 第三，提议者和同事将尝试产生改进的和最好的可能的尾概率近似的泊松化和的独立随机变量满足一定的约束。 每当数字很重要，而人类不可能完全了解或掌握情况时，就可能出现问题。 例如，一家连锁店（如西尔斯或梅西百货）的管理层必须不断地决定将多少资本分配给众多商品中的每一种。 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产品的当前销售结果，而且管理层可能对影响其销售模式的促成因素没有充分的理解。 为了做出好的财务决策，管理层需要一个相当全面的、不断更新的财务状况数学模型。 例如，它需要知道在一个给定的商店以及整个公司中有超过d件成本为c或更高的未售出服装的机会是多少。 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，公司可能希望计算数学家认为是二次形式或广义U统计量的期望值。 因此，如果它们的效用得到适当的承认，本提案所取得的结果很可能具有重大的工业意义。

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.