This research is to study the problem of obtaining an accurate description of the probabilistic behavior of sums of dependent random variables. Essential tools for the proposed work will be recent results in the area of decoupling inequalities. A decoupling inequality transforms problems on sums of dependent random variables to problems on sums of conditionally independent random variables. In the proposed work, particular attention will be focused on developing tail probability approximations for randomly stopped sums, quadratic forms of independent random variables, and the general problem of sums of arbitrarily dependent random variables. This research is to study some important problems in probability theory. In particular, the investigator will look at the behavior of sums of dependent observations. These sums will be studied by comparing them to appropriate sums of independent observations.
