Distributional Properties Of Cusum Stopping Times

摘要

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(αS_N - βN), where β > 0 and a 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.