Let I1,I2,…,In be independent indicator functions on some probability space We suppose that these indicators can be observed sequentially. Furthermore, let T be the set of stopping times on (Ik), k=1,…,n, adapted to the increasing filtration where The odds algorithm solves the problem of finding a stopping time τ ∈ T which maximises the probability of stopping on the last Ik=1, if any. To apply the algorithm, we only need the odds for the events {Ik=1}, that is, rk=pk/(1-pk), where The goal of this paper is to offer tractable solutions for the case where the pk are unknown and must be sequentially estimated. The motivation is that this case is important for many real-world applications of optimal stopping. We study several approaches to incorporate sequential information. Our main result is a new version of the odds algorithm based on online observation and sequential updating. Questions of speed and performance of the different approaches are studied in detail, and the conclusiveness of the comparisons allows us to propose always using this algorithm to tackle selection problems of this kind.

Let X1, X2, …, Xn be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w(t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w(t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.

Consider N players, respectively owning x1, x2, …, xN monetary units, who play a sequence of games, winning from and losing to each other integer amounts according to fixed rules. The sequence stops as soon as (at least) one player is ruined. We are interested in the ruin process of these N players, i.e. in the probability that a given player is ruined first, and also in the expected ruin time. This problem is called the N-player ruin problem. In this paper, the problem is set up as a multivariate absorbing Markov chain with an absorbing state corresponding to the ruin of each player. This is then discussed in the context of phase-type distributions where each phase is represented by a vector of size N and the distribution has as many absorbing points as there are ruin events. We use this modified phase-type distribution to obtain an explicit solution to the N-player problem. We define a partition of the set of transient states into different levels, and on it give an extension of the folding algorithm (see Ye and Li (1994)). This provides an efficient computational procedure for calculating some of the key measures.

Let X1, X2,…, Xn be independent, identically distributed random variables, uniform on [0,1]. We observe the Xk sequentially and must stop on exactly one of them. No recollection of the preceding observations is permitted. What stopping rule τ minimizes the expected rank of the selected observation? This full-information expected-rank problem is known as Robbins' problem. The general solution is still unknown, and only some bounds are known for the limiting value as n tends to infinity. After a short discussion of the history and background of this problem, we summarize what is known. We then try to present, in an easily accessible form, what the author believes should be seen as the essence of the more difficult remaining questions. The aim of this article is to evoke interest in this problem and so, simply by viewing it from what are possibly new angles, to increase the probability that a reader may see what seems to evade probabilistic intuition.

Consider N towers each made up of a number of counters. At each step a tower is chosen at random, a counter removed which is then added to another tower also chosen at random. The probability distribution for the time needed to empty one of the towers is obtained in the case N = 3. Arguments are set forward as to why no simple formulae can be expected for N > 3. An asymptotic expression for the mean time before one of the towers becomes empty is derived in the case of four towers when they all initially contain a comparably large number of counters. We then study related problems, in particular the ruin problem for three players. Here we use simple martingale methodology as well as a solution proposed by T. S. Ferguson for a slightly modified problem. Throughout the paper it is our main objective to shed light on the reasons why the case N > 3 is so substantially different from the case N ≤ 3.

Strings are generated by sequences of independent draws from a given alphabet. For a given pattern H of length l and an integer n ≥ l, our goal is to maximize the probability of stopping on the last appearance of the pattern H in a string of length n (if any), given that, if we choose to stop on an occurrence of H, we must do so right away. This contrasts with the goals of several other investigations on patterns in strings such as computing the expected occurrence time and the probability of finding exactly r patterns in a string of fixed length. Several motivations are given for our problem ranging from relatively simple best choice problems to more difficult stopping problems allowing for a variety of interesting applications. We solve this problem completely in the homogeneous case for uncorrelated patterns. However, several of these results extend immediately to the inhomogeneous case. In the homogeneous case, optimal success probabilities are shown to vary, depending on characteristics of the pattern, essentially between the value 1/e and a new asymptotic constant 0.619…. These results demonstrate a considerable difference between the true optimal success probability compared with what a first approximation by heuristic arguments would yield. It is interesting to see that the so-called odds algorithm which gives the optimal rule for l = 1 yields an excellent approximation of the optimal rule for any l. This is important for applications because the odds algorithm is very convenient. We finally give a detailed asymptotic analysis for the homogeneous case.

Let I1,I2,…,In be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if Ik = 1. The sequence I1,I2,…,In is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

We consider a generalization of the house-selling problem to selling k houses. Let the offers, X1, X2, · ··, be independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneously k stopping rules, N1, · ··, Nk, one for each component. The payoff is the sum over j of the jth component of minus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are non-decreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908.

Let X1, X2, · ··, Xn be positive i.i.d. random variables with known distribution function having a finite mean. For a given s ≥0 we define Nn = N(n, s) to be the largest number k such that the sum of the smallest k Xs does not exceed s, and Mn = M(n, s) to be the largest number k such that the sum of the largest k X's does not exceed s. This paper studies the precise and asymptotic behaviour of E(Nn), E(Mn), Nn, Mn, and the corresponding ‘stopped' order statistics and as n →∞, both for fixed s, and where s =sn is an increasing function of n.

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

Cowan and Zabczyk (1978) have studied a continuous-time generalization of the so-called secretary problem, where options arise according to a homogeneous Poisson processes of known intensity λ. They gave the complete strategy maximizing the probability of accepting the best option under the usual no-recall condition. In this paper, the solution is extended to the case where the intensity λ is unknown, and also to the case of an inhomogeneous Poisson process with intensity function λ (t), which is either supposed to be known or known up to a multiplicative constant.

This note deals with extinction criteria for bisexual Galton–Watson processes with arbitrary mating functions in terms of the averaged reproduction mean per mating unit. It gives a satisfactory answer to a question put forward by Hull (1982).

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ, P), the divergence of ΣnP(An) is necessary for P(An i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the An are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by for some . We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ϕ -branching processes will show the tractability of the method.

A discrete Galton–Watson process is modified by an absorbing process, which, within each generation, eliminates a subset of the living particles without leaving offspring. The absorbing process will be only roughly specified by its expected efficiency on the associated process. We give a sharp sufficient condition for the joint process to be extinguished with probability one and give, after an easy generalization, an example for possible biomedical applications.