AbstractA series of independent trials is considered in which one ofk≥ 2 mutually exclusive and exhaustive outcomes occurs at each trial. The series terminates when m outcomes of any one type have occurred. The limiting distribution (as m → ∞) of the number of trials performed until termination is found with particular attention to the situation where a Dirichlet distribution is assigned to thekvector of probabilities for each outcome. Applications to series of races involvingkrunners and to spares problems in reliability modeling are discussed. The problem of selecting a stopping rule so that the probability of the series terminating on outcomeiisk−1(i.e., a “fair” competition) is also studied. Two generalizations of the original asymptotic problem ar