7 - Miscellaneous Applications

Abstract

This chapter illustrates miscellaneous applications of infinite sequences, which are consequent on limiting behavior. It also presents the uses of finite exchangeable sequences, particularly in conjunction with combinational arguments for random walk and ballot problem. Finite exchangeable sets cannot necessarily be extended to infinite exchangeable sequences. The variables in an infinite exchangeable sequence are necessarily non-negatively correlated. A random variable is n-symmetric if it is a function of (X1, X2 …), which is unchanged under any permutation of the first n variables. The chapter also focuses on stochastic approximation. In certain statistical applications, such as bioassay, sensitivity testing, or fatigue trials, problems are obtained that can be conveniently attacked using stochastic approximation methods with a minimum of distributional assumptions. At each observation time, the unknown parameters of the system are estimated and a control policy, which is optimal in a certain sense, is computed as if the estimates were the exact values of the parameters. This is used to control the system in the next time period.