Distributional Properties of CUSUM Stopping Times

Let S n be the partial sum of i.i.d. random variables X 1,X 2,…, and let N be the usual CUSUM stopping time based on S n . Under suitable conditions we determine ψ(α,β) = E exp(α S N − β N), where β > 0 and α is a suitable number. The given formula can be used to study the distributional properties of N, S N , and S N − h. Because the CUSUM based on maxima is reducible to N, the formula can be used to obtain the distributional properties of the maximal process as well. Several examples are discussed, and certain applications are shown in the so called trading securities. The formulas can also be used to study the distributional properties of a symmetric two-sided CUSUM.