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.
