On the joint distribution of stopping times and stopped sums in multistate exchangeable trials

Abstract

Let T be a stopping time associated with a sequence of independent and identically distributed or exchangeable random variables taking values in {0, 1, 2, . . . , m}, and let S_T,i be the stopped sum denoting the number of appearances of outcome 'i' in X₁, . . . , X_T, 0 ≤ i ≤ m. In this paper we present results revealing that, if the distribution of T is known, then we can also derive the joint distribution of (T, S_T,0, S_T,1, . . . , S_T,m). Two applications, which have independent interest, are offered to illustrate the applicability and the usefulness of the main results.