Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.

It is known [8] that a certain class of bond-decorated graphs exhibits multiple AB percolation phase transitions. Sufficient conditions are given under which the corresponding AB percolation critical probabilities may be identified as points of intersection of the graph of a certain polynomial with the boundary of the percolative region of an associated two-parameter bond-site percolation model on the underlying undecorated graph. The main result of the article is used to prove that the graphs in [8] exhibit multiple AB percolation critical probabilities. The possibility of identifying AB percolation critical exponents with corresponding limits for the bond-site model is discussed.

A sequence of objects with independent, identically distributed qualities is presented to a selector who must choose two on-line, i.e. without anticipation or recall. The selector's aim is to obtain a satisfactory pair as quickly as possible. Two versions of the problem are considered, and optimal selection rules are derived and compared. An investigation is also made of a heuristic rule suitable for a selector who has no prior knowledge of the nature of the object sequence.

Aven and Jensen (1999) proposed an approximation to the downtime distribution for arbitrary coherent systems with binary and independent repairable components, where each component follows an alternating renewal process. The approximation is based on a mixture of the duration distributions of the cut sets in the system. We extend this approach to coherent systems composed by multistate independent components, where every component follows a semi-Markov process and where we introduce the concept of a minimal cut set in the multistate setting. We test our proposal on a simple power demand-generation system, by comparing the analytic approximation with simulation results, and we find it is accurate when the system is highly available.

A transformation is introduced to stabilize the variance of the largest value in a sample from a normal distribution. The transformed distribution is found to approximate closely to the limiting Gumbel form for all sample sizes. The practical implications of the result are discussed.

This paper considers several models for biological processes in which animate individuals live and die as members of groups which can split to form smaller groups. Resulting distributions of individuals over groups are compared and contrasted. In particular, two qualitatively different types of distributions are identified. It is clear that distinguishing between models giving rise to the same distribution types is difficult. Implications for more complex models are discussed and avenues for further research are outlined.

Consider a device which begins operating at time t = 0 and then operates continuously in time. The operation of the device is satisfactory so long as a physical parameter X(t) of the device remains less than a critical value c > 0 (X(0) = 0). The device does not operate satisfactorily (i.e., it is “failed”) at a time t if X(t) ≧ c. Suppose that X(t) is a random variable with mean EX(t) = ta (a> 0), i.e., the device parameter X drifts with a linear rate a.

We introduce variance-optimal semi-static hedging strategies for a given contingent claim. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy we use a Fourier approach in a multidimensional factor model. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in the Heston model, which is affine, in the 3/2 model, which is not, and in a market model including jumps.

We analyze here the optimal premium rates for an insurance company for the case where the claims process is a stochastic process with unknown parameters. The beliefs of the insurance company about these parameters are represented by some prior distribution function, and the premium rates are determined on the basis of this distribution. The insurer learns about the unknown parameters from the past behavior of the claims process, and revises the prior distribution whenever new information is obtained (i.e., a claims stream is observed).

The learning process introduces multi-period dynamic considerations to the problem of the rate determination, since the rates determined at any period affect the claims process, and thus the information available for future decisions. We show how the optimal rates can be obtained by using some dynamic programming techniques. We also prove that the optimal rates obtained in our case are lower than those determined by an insurance company that does not revise its distribution.

In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X 1,...,X n are independent random variables with known continuous distributions and V n (X 1,...,Xn ):=supτℙ(X τ=M n ), where M n ≔max{X 1,...,X n } and the supremum is over all stopping times adapted to X 1,...,X n then V n (X 1,...,X n )≥(1-1/n)n-1, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each n, we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.

In his well-known textbook Probability with Martingales, David Williams (1991) introduces the Mabinogion sheep problem in which there is a magical flock of sheep, some black, some white. At each stage n=1,2,..., a sheep (chosen randomly from the entire flock, independently of previous events) bleats; if this bleating sheep is white, one black sheep (if any remain) instantly becomes white; if the bleating sheep is black, one white sheep (if any remain) instantly becomes black. No births or deaths occur. Suppose that one may remove any number of white sheep from the flock at (the end of) each stage n=0,1,.... The object is to maximize the expected final number of black sheep. By applying the martingale optimality principle, Williams showed that the problem is solvable and admits a simple nice solution. In this paper we consider a generalization of the Mabinogion sheep problem with two parameters 0≤p, q≤1, denoted M(p,q), in which at each stage, when the bleating sheep is white (black, respectively), a black (white, respectively) sheep (if any remain) instantly becomes white (black, respectively), with probability p (q, respectively) and nothing changes with probability 1-p (1-q, respectively). Note that the original problem corresponds to (p,q)=(1,1). Following Williams' approach, we solve the two cases (p,q)=(1,1/2) and (1/2,1) which admit simple solutions.

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

It is apparent that much of the congestion in road traffic is due to interference between vehicles at intersections, and this has led to the appearance of several mathematical discussions of the problems of delay at intersections of various types (Cohen and Stam (1963), Cohen and de Lange (1965), Hawkes (1965), (1966), Tanner (1962), Weiss and Maradudin (1962)). Comparatively little work has appeared on the problem of delay caused by vehicles turning right* across an opposing stream of vehicles, although Hawkes (1966) and Newell (1959) discuss this problem explicitly, and the models considered by all of these authors may be regarded as having some application to the right-turning problem.

In this article we investigate the minimal entropy martingale measure for continuous-time Markov chains. The conditions for absence of arbitrage and existence of the minimal entropy martingale measure are discussed. Under this measure, expressions for the transition intensities are obtained. Differential equations for the arbitrage-free price are derived.

An iterated sequence of Bernoulli trials is carried out and the success probability estimated at each point on the sequence by the current success ratio. We find the probability P1 that this estimate always lies above some pre-selected rational fraction p′, and its complement P2, the probability that it will reach p′ or below at least once. In the region p′ ≧ p, P1 = 0. In the region p′ < p, P1 ≠ 0 and is furthermore a discontinuous function of p′ at every rational p′.

Let Xi,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. We transform (‘prune’) the sequence {X1,…,Xn},n∈ ℕ, of discrete random samples into a sequence {0,1,2,…,Yn}, n∈ ℕ, of contiguous random sets by replacing Xn+1 with Yn +1 if Xn+1 >Yn. We consider the asymptotic behaviour of Yn as n→∞. Applications include path growth in digital search trees and the number of tables in Pitman's Chinese restaurant process if the latter is conditioned on its limit value.

For given functions, g(x), h(x), F(x) and a constant, a, f(x) is said to be a solution of the optimal inventory equation if where f(y - z) = f(0) when z > y. (The same convention holds in similar equations below.) A method of deriving a solution to (1) is to define inductively a sequence of approximating functions, fn(x), by:

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

For every hth member of a two-state Markov chain the value of a random variable Y is observed where the distribution of Y is conditional on the state of the corresponding member of the chain. A recursive set of equations is derived giving the posterior probabilities for both the observed and unobserved members. The use of this recursive solution to investigate the optimality of certain simple classification rules is discussed, and a “classification by runs” is also presented.