arXiv subjects

F. Thomas Bruss

Publications and source records attributed to F. Thomas Bruss.

16 recordsLinked to original sources

arXiv2026

Win rates at first-passage times for biased simple random walks

We study the win rate $R_{N_d}/N_d$ of a biased simple random walk $S_n$ on $\mathbb{Z}$ at the first-passage time $N_d=\inf\{n\ge 0:S_n=d\}$, with $p=P[X_1=+1]\in[1/2,1)$. Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of $R_{N_d}/N_d$ along with monotonicity properties in the threshold $d$ and the bias $p$. We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of $π$ based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.

math.PR↗

arXiv2026

Algorithms for Robbins' Problem using Markov Decision Processes

In this paper, we consider Robbins' problem, which is a full information variant of the well-known secretary selection problem. In this version of the problem, the goal is to minimize the expected rank of the selected candidate among $n$ that are interviewed sequentially, and a decision to select or not the $m^{th}$ candidate needs to be taken right after the interview (so without seeing the last $n-m$ candidates and without recall). We first show how to model instances of Robbins' problem as infinite Markov Decision Processes (MDPs). Then we propose several finite-state abstractions of these MDPs that allow us to approximate the value of the problem for fixed $n$. While it is known that the full memory of past candidates' values is necessary for optimal expected rank minimization, making the analysis of the problem challenging, we highlight simple memory structures that are sufficient for obtaining near-optimal selection strategies. Additionally, we provide approximate values for Robbins' problem for numbers of candidates $n$ up to 100 for which no good approximations were previously known (the exact value is only known for instances where $n \leq 4$ and numerical approximations were for small values of $n$ not exceeding one digit), for all $n : 5 \leq n \leq 100$, we give better approximation than what was previously known.

cs.GT↗

arXiv2025

Interactions between resource dependent branching processes and equilibria

This paper is a supplement to the paper "Interactions between Human Populations and Related Problems of Optimal Transport" written by the same author in honour of Marc Hallin, Université Libre de Bruxelles, at the occasion of Hallin's $75$th birthday. It was announced in the main paper (Bruss (2024)) published in the Springer Festschrift entitled {\it Recent Advances in Econometrics and Statistics}. It contains the proofs which, given the space constraints required for the Festschrift, could not appear in the main paper. Moreover, we complement in the present supplement the main paper by brief comments on related problems which are likely to turn up in practice for problems of guiding human populations, namely problems of control and problems of optimal stopping.

math.PR↗

arXiv2023

Gambling under unknown probabilities as a proxy for real world decisions under uncertainty

We give elementary examples within a framework for studying decisions under uncertainty where probabilities are only roughly known. The framework, in gambling terms, is that the size of a bet is proportional to the gambler's perceived advantage based on their perceived probability, and their accuracy in estimating true probabilities is measured by mean squared-error. Within this framework one can study the cost of estimation errors, and seek to formalize the ``obvious" notion that in competitive interactions between agents whose actions depend on their perceived probabilities, those who are more accurate at estimating probabilities will generally be more successful than those who are less accurate.

math.PR↗

arXiv2023

Mathematical intuition, deep learning, and Robbins' problem

{\bf Abstract.} The present article is an essay about mathematical intuition and Artificial intelligence (A.I.), followed by a guided excursion to a well-known open problem. It has two objectives. The first is to reconcile the way of thinking of a computer program as a sequence of mathematically defined instructions with what we face nowadays with newer developments. The second and major goal is to guide interested readers through the probabilistic intuition behind Robbins' problem and to show why A.I., and in particular Deep Learning, may contribute an essential part in its solution. This article contains no new mathematical results, and no implementation of deep learning either. Nevertheless, we hope to find through its semi-historic narrative style, with well-known examples and an easily accessible terminology, the interest of mathematicians of different inclinations.

math.HO↗

arXiv2021

Galton-Watson processes and their role as building blocks for branching processes

This article is an essay, both expository and argumentative, on the Galton-Watson process as a tool in the domain of Branching Processes. It is at the same time the author's ways to honour two distinguished scientists in this domain, both from the Russian Academy of Science, and to congratulate them for their special birthdays coming up very soon. The thread of the article is the role, which the Galton-Watson process had played in the author's own research. We start with an article on a controlled Galton-Watson process. Then we pass to random absorbing processes, and also recall and discuss a problem in medicine. Further questions will bring us via the Borel-Cantelli Lemma to $φ$-branching processes and extensions. To gain more generality, we then look at bisexual Galton-Watson processes. Finally we briefly discuss relatively complicated resource dependent branching processes to show that, here again, using Galton-Watson reproduction schemes (whenever reasonable) can be a convincing approach to new processes which are then sufficiently tractable to obtain results of interest. Keywords: Controlled branching process; $φ$-branching process, Bisexual reproduction, Borel-Cantelli Lemma; Resource dependence; Society forms, Stopping times, Theorem of envelopment, BRS-inequality.

math.PR↗

arXiv2020

Answer to an open question concerning the $1/e$-strategy for best choice under no information

This paper answers a long-standing open question concerning the $1/e$-strategy for the problem of best choice. $N$ candidates for a job arrive at times independently uniformly distributed in $[0,1]$. The interviewer knows how each candidate ranks relative to all others seen so far, and must immediately appoint or reject each candidate as they arrive. The aim is to choose the best overall. The $1/e$ strategy is to follow the rule: `Do nothing until time $1/e$, then appoint the first candidate thereafter who is best so far (if any).' The question, first discussed with Larry Shepp in 1983, was to know whether the $1/e$-strategy is optimal if one has `no information about the total number of options'. Quite what this might mean is open to various interpretations, but we shall take the proportional-increment process formulation of \cite{BY}. Such processes are shown to have a very rigid structure, being time-changed {\em pure birth processes}, and this allows some precise distributional calculations, from which we deduce that the $1/e$-strategy is in fact not optimal.

math.PR↗

arXiv2020

On the $1/e$-strategy for the best-choice problem under no information

The main purpose of this paper is to correct an error in the previously submitted version [*] := arXiv:2004.13749v1. [*] had been already accepted for publication in a scientific journal, but withdrawn by the author after the discovery of the error. For the withdrawal from arXiv we follow their preference to maintain what remains of interest. The background of the open problem, and the brief survey which comes with it, stay relevant. These keep their place in the present corrected version. The same is true for two new modified odds-theorems proved in [*] since they are applicable for several different stopping problems. Then, and in particular, we show where exactly the error occurred in [*], why it invalidates its main theorem and title,and what the conclusions are. The final discussion of optimal strategies without value in Section 4 is believed to be of general independent interest.

math.PR↗

arXiv2020

The BRS-inequality and its Applications A Survey

This article is a survey of results concerning an inequality, which may be seen as a versatile tool to solve problems in the domain of Applied Probability. The inequality, which we call BRS-inequality, gives a convenient upper bound for the expected maximum number of non-negative random variables one can sum up without exceeding a given upper bound $s>0.$ One fine property of the BRS-inequality is that it is valid without any hypothesis aboutindependence of the random variables. Another welcome feature is that, once one sees that one can use it in a given problem, its application is often straightforward or, not very involved. This survey is focussed, and we hope that it is pleasant and inspiring to read. The focus is easy to achieve, given that BRS-inequality and its most useful versions can be displayed in three Theorems, one Corollary, and their proofs. We try to do this in an appealing way. The objective to be inspiring is harder, and the best we can think of is offering a variety of applications. Our examples include comparisons between sums of iid versus non-identically distributed and/or dependent random variables, problems of condensing point processes, subsequence problems, knapsack problems, online algorithms, tiling policies, Borel-Cantelli type problems, up to applications in the newer theory of resource dependent branching processes.

math.PR↗

arXiv2020

Numerical Aspects of Computing Possible Equilibria for Resource Dependent Branching Processes with Immigration

This article studies the stability of solutions of equilibrium equations arising in so-called resource dependent branching processes. We argue that these new models, building on the model already presented by Bruss (1984 a), refined and elaborated in Bruss and Duerinckx (2015) and now extended to allow immigration, are suitable to cope with specific properties of human populations. Our main interest is here to understand under which conditions immigration may lead to an equilibrium. At the same time, we would like to advertize resource dependent branching processes as possibly the best models to study such questions. The equilibrium equations for the new models we obtain are clear and informative for several important stability questions. The goal of the study of the specific examples we provide is to see where the impact of immigration is most visible, and in how far increased efforts of integration can cope with dangers of instability. Moreover we discuss the advantages and a weaker point of our model, and also include a brief look at continuous-state, continuous-time branching processes as an alternative.

math.PR↗

arXiv2020

The rencontre problem

Let $\left\{X^{1}_k\right\}_{k=1}^{\infty}, \left\{X^{2}_k\right\}_{k=1}^{\infty}, \cdots, \left\{X^{d}_k\right\}_{k=1}^{\infty}$ be $d$ independent sequences of Bernoulli random variables with success-parameters $p_1, p_2, \cdots, p_d$ respectively, where $d \geq 2$ is a positive integer, and $ 0<p_j<1$ for all $j=1,2,\cdots,d.$ Let \begin{equation*} S^{j}(n) = \sum_{i=1}^{n} X^{j}_{i} = X^{j}_{1} + X^{j}_{2} + \cdots + X^{j}_{n}, \quad n =1,2 , \cdots. \end{equation*} We declare a "rencontre" at time $n$, or, equivalently, say that $n$ is a "rencontre-time," if \begin{equation*} S^{1}(n) = S^{2}(n) = \cdots = S^{d}(n). \end{equation*} We motivate and study the distribution of the first (provided it is finite) rencontre time.

math.PR↗

arXiv2019

Odds-Theorem and Monotonicity

Given a finite sequence of events and a well-defined notion of events being interesting, the Odds-theorem (Bruss (2000)) gives an online strategy to stop on the last interesting event. It is optimal for independent events. Here we study questions in how far optimal win probabilities mirror monotonicity properties of the underlying sequence of probabilities of events. We make these questions precise, motivate them, and then give complete answers. This note, concentrating on the original Odds-theorem, is elementary, and the answers are hoped to be of interest. We include several applications.

math.PR↗

arXiv2018

Equilibrium Equations for Human Populations with Immigration

The objective of this article is to create a framework to study asymptotic equilibria in human populations with a special focus on immigration. We present a new model, based on Resource Dependent Branching Processes, which is now broad enough to cope with the goal of finding equilibrium criteria under reasonable hypotheses. Our equations are expressed in terms of natality rates, mean productivity and mean consumption of the home-population and the immigrant population as well as policies of the Society to distribute resources among individuals. We also study the impact of integration of one sub-population into the other one, and in a third model, the additional influence of a continuous stream of new immigrants. Proofs of the results are based on classical limit theorems, on Borel-Cantelli type arguments, on the Theorem of envelopment of Bruss and Duerinckx (2015), on a maximum inequality of Bruss and Robertson (1991) and, in particular, on an extension of J.M. Steele (2016) of the latter. Conditions for the existence of an equilibrium often prove to be severe, and sometimes surprisingly sensitive. This underlines how demanding the real world of immigration can be for politicians trying to make sound decisions. Our main objective is to provide help through insights from an adequate theory. Another objective of the present study is to learn which of the possible control measures are best for combining feasibility and efficiency to reach an equilibrium, and to recognise the corresponding steps one has to take towards controls. We also make preliminary suggestions to envisage ways to optimal control. As far as the author is aware, all results are new.

math.PR↗

arXiv2018

A Mathematical Approach to Comply with Ethical Constraints in Compassionate Use Treatments

Patients who are seriously ill may ask doctors to treat them with unapproved medication, about which not much is known, or else with known medication in a high dosage. Apart from strict legal constraints such cases may involve difficult ethical questions as e.g. how long a series of treatments of different patients should be continued. Similar questions also arise in less serious situations. A physician trusts that a certain combination of freely available drugs are efficient against a specific disease and tries to help patients and to follow at the same time the primum-non-nocere principle. The objective of this paper is to contribute to the research on such questions in the form of mathematical models. Arguing in a step-to-step approach, we will show that certain sequential optimisation problems comply in a natural way with the true spirit of major ethical principles in medicine. We then suggest protocols and associate algorithms to find optimal, or approximately optimal, treatment strategies. Although the contribution may sometimes be difficult to apply in medical practice, the author thinks that the rational behind the approach offers a valuable alternative for finding decision support and should attract attention.

stat.OT↗

arXiv2015

Resource dependent branching processes and the envelope of societies

Since its early beginnings, mankind has put to test many different society forms, and this fact raises a complex of interesting questions. The objective of this paper is to present a general population model which takes essential features of any society into account and which gives interesting answers on the basis of only two natural hypotheses. One is that societies want to survive, the second, that individuals in a society would, in general, like to increase their standard of living. We start by presenting a mathematical model, which may be seen as a particular type of a controlled branching process. All conditions of the model are justified and interpreted. After several preliminary results about societies in general we can show that two society forms should attract particular attention, both from a qualitative and a quantitative point of view. These are the so-called weakest-first society and the strongest-first society. In particular we prove then that these two societies stand out since they form an envelope of all possible societies in a sense we will make precise. This result (the envelopment theorem) is seen as significant because it is paralleled with precise survival criteria for the enveloping societies. Moreover, given that one of the "limiting" societies can be seen as an extreme form of communism, and the other one as being close to an extreme version of capitalism, we conclude that, remarkably, humanity is close to having already tested the limits.

math.PR↗

arXiv2013

Last-Hitting Times and Williams' Decomposition of the Bessel Process of Dimension 3 at its Ultimate Minimum

In this note we shortly recall the importance of last-hitting times in theory and applications of optimal stopping. As a small contribution to this domain we then propose a concise proof of David Williams' decomposition of the Bessel Process of dimension 3 (BES(3)), starting from r > 0 at its ultimate minimum. This discussion is strongly motivated by our interest in properties of last hitting times in general, and here specifically, directly linked with the forthcoming reading guide of Nikeghbali and Platen on this subject.

math.PR↗