The asymptotic values of the variances and covariances for the distribution of the transition numbers AA, AB, BA and BB for a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for the k-state chain under the restriction that the first and the last observations belong to the rth and the sth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the order nr (p11 – p21)n−r where r is the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for the k-state also. This is done by differentiating directly the determinantal characteristic equation.

Cumulants of some distributions arising from a two-state Markoff chain

Asymptotic Cumulants for Distributions of k -State Markov Chains

Probability distributions arising from points on a line

This paper considers the maximization of the expected maximum value of a portfolio of random variables subject to a budget constraint. We refer to this as the optimal college application problem. When each variable’s cost, or each college’s application fee, is identical, we show that the optimal portfolios are nested in the budget constraint, yielding an exact polynomial-time algorithm. When colleges diﬀer in their application fees, we show that the problem is NP-complete. We provide four algorithms for this more general setup: a branch-and-bound routine, a dynamic program that produces an exact solution in pseudopolynomial time, a diﬀerent dynamic program that yields a fully polynomial-time approximation scheme, and a simulated-annealing heuristic. Numerical experiments demonstrate the algorithms’ accuracy and eﬃciency.

M. N. Kapur

Papers

Probability distributions arising from points on a line

The College Application Problem