F. Thomas Bruss

A Note on Extinction Criteria for Bisexual Galton-Watson Processes

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).

A Counterpart of the Borel-Cantelli Lemma

The general part of the Borel-Cantelli lemma says that for any sequence of events (A,) defined on a probability space (fl, 1, P), the divergence of Y, P(A, ) is necessary for P(A, i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the A, are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by A,,, A,+ C A+2 for some t E N. We will see that this property of (A,) 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 ,-branching processes will show the tractability of the method. BOREL-CANTELLI LEMMA; RANDOM WALK; O-BRANCHING PROCESSES

On the maximum and its uniqueness for geometric random samples

Given n independent, identically distributed random variables, let pn 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 pns in the case of geometric random variables. We find a function Q such that (pn - D(n)) -- 0 as n - 0o. In particular, we show that pn does not converge as n -- oo. 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

Minimizing the expected rank with full information

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 nondecreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908. §

Embedding optimal selection problems in a Poisson process

We consider optimal selection problems, where the number N1 of candidates for the job is random, and the times of arrival of the candidates are uniformly distributed in [0, 1]. Such best choice problems are generally harder than the fixed-N counterparts, because there is a learning process going on as one observes the times of arrivals, giving information about the likely values of N1. In certain special cases, notably when N1 is geometrically distributed, it had been proved earlier that the optimal policy was of a very simple form; this paper will explain why these cases are so simple by embedding the process in a planar Poisson process from which all the requisite distributional results can be read off by inspection. Routine stochastic calculus methods are then used to prove the conjectured optimal policy.

Multiple buying or selling with vector offers

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 XN, 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.

Invariant record processes and applications to best choice modelling

Let X1, X2,...be identically distributed random variables from an unknown continuous distribution. Further let Ir(1), Ir(2),...be a sequence of indicator functions defined on X1, X2,...by Ir(k) = 0 if k

On the Perception of Time

In this article, we review scientific work and present new results on the perception of time, that is, on the feeling of time as perceived by individuals. The phenomenon of time being felt passing faster with growing age is well known, and there are numerous interesting studies to shed light on the question why this is so. Many of these are based on studies in psychology and social sciences. Others range from symptoms of the ageing process to related symptoms of decreasing memory capacities. Again other explanations, quite different in nature from the preceding ones, involve event intensities in the life of individuals. The relative decrease of interesting new events as one grows older is seen as an important factor contributing to the feeling that time is thinned out. The last type of possible explanations can be made more explicit in a mathematical model. Quantitative conclusions about the rate of decrease of the feeling of time can be drawn, and, interestingly, without restrictive assumptions. It is shown that under this model the feeling of time is thinned out at least logarithmically. Numerical constants will depend on specific hypotheses which we discuss but the lower-bound logarithmic character of the thinning-out phenomenon does not depend much on these. The presented model can be generalized in several ways. In particular we prove that there are, a priori, no logical incompatibilities in a model leading to the very same distribution of time perception for individuals with completely different pace and style of life. Our model is built to explain long-time perception. No claim is made that the feeling of time being thinned out is omnipresent for very individual. However, this is typically the case and we explain why.

Half-Prophets and Robbins’ Problem of Minimizing the Expected Rank

Let X 1, X 2,…X n be i.i.d. random variables with a known continuous distribution function. Robbins’ problem is to find a sequential stopping rule without recall which minimizes the expected rank of the selected observation. An upper bound (obtained by memoryless threshold rules) and a procedure to obtain lower bounds of the value are known, but the difficulty is that the optimal strategy depends for all n > 2 in an intractable way on the whole history of preceding observations. The goal of this article is to understand better the structure of both optimal memoryless threshold rules and the (overall) optimal rule. We prove that the optimal rule is a “stepwise” monotone increasing threshold-function rule and then study its property of, what we call, full history-dependence. For each n, we describe a tractable statistic of preceding observations which is sufficient for optimal decisions of decision makers with half-prophetical abilities who can do generally better than we. It is shown that their advice can always be used to improve strictly on memoryless rules, and we determine such an improved rule for all n sufficiently large. We do not know, however, whether one can construct, as n → ∞ asymptotically relevant improvements.

The odds algorithm based on sequential updating and its performance

Let I 1,I 2,…,I n 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 (I k ), 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 I k =1, if any. To apply the algorithm, we only need the odds for the events {I k =1}, that is, r k =p k /(1-p k ), where The goal of this paper is to offer tractable solutions for the case where the p k 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.