Problem In Probability Theory Research Articles

Evolutionary dynamics of incubation periods.

The incubation period for typhoid, polio, measles, leukemia and many other diseases follows a right-skewed, approximately lognormal distribution. Although this pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here, we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease.

Propagation of input uncertainties in particle transport and the distribution of the sum of n independent stochastic variables a generalization of the Irwin–Hall distribution

We show, under wide hypothesis, that the problem of forward propagation of uncertainties in a physical model reduces to a well known problem in probability theory: the determination of the weighted sum of N independent stochastic variables obeying arbitrary distributions supported on bounded intervals. We determine various exact expressions for the probability distribution of the sum, possibly weighted, of N independent stochastic variables obeying arbitrary polynomial distributions on different intervals. These exact results provide the mathematical basis for the treatment of the given problem. We discuss the relevance of these findings for parameter uncertainty quantification in the context of particle transport computations. As a byproduct an explicit formula for the convolution of an arbitrary number of polynomials supported on bounded intervals is obtained.

Optimal transport and Skorokhod embedding

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.

Phase transition on the convergence rate of parameter estimation under an Ornstein-Uhlenbeck diffusion on a tree.

Diffusion processes on trees are commonly used in evolutionary biology to model the joint distribution of continuous traits, such as body mass, across species. Estimating the parameters of such processes from tip values presents challenges because of the intrinsic correlation between the observations produced by the shared evolutionary history, thus violating the standard independence assumption of large-sample theory. For instance (Ho and Ané, Ann Stat 41:957-981, 2013) recently proved that the mean (also known in this context as selection optimum) of an Ornstein-Uhlenbeck process on a tree cannot be estimated consistently from an increasing number of tip observations if the tree height is bounded. Here, using a fruitful connection to the so-called reconstruction problem in probability theory, we study the convergence rate of parameter estimation in the unbounded height case. For the mean of the process, we provide a necessary and sufficient condition for the consistency of the maximum likelihood estimator (MLE) and establish a phase transition on its convergence rate in terms of the growth of the tree. In particular we show that a loss of [Formula: see text]-consistency (i.e., the variance of the MLE becomes [Formula: see text], where n is the number of tips) occurs when the tree growth is larger than a threshold related to the phase transition of the reconstruction problem. For the covariance parameters, we give a novel, efficient estimation method which achieves [Formula: see text]-consistency under natural assumptions on the tree. Our theoretical results provide practical suggestions for the design of comparative data collection.

A solution to the reversible embedding problem for finite Markov chains

The embedding problem for Markov chains is a famous problem in probability theory and only partial results are available up till now. In this paper, we propose a variant of the embedding problem called the reversible embedding problem which has a deep physical and biochemical background and provide a complete solution to this new problem. We prove that the reversible embedding of a stochastic matrix, if it exists, must be unique. Moreover, we obtain the sufficient and necessary conditions for the existence of the reversible embedding and provide an effective method to compute the reversible embedding. Some examples are also given to illustrate the main results of this paper.

Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd

Real Gibbs measures arise in many problems of probability theory and statistical mechanics. This measure, related to the Boltzmann distribution, generalizes the notion of canonical ensemble. In addition, Gibbs measure is unique measure that maximizes the entropy of the expected energy. But non-archimedean (p-adic) analogue of Gibbs measures have been little studied. It is known that in the case of real numbers concepts of Gibbs measure and Markov random field are identical. But in the p-adic case, the class of p-adic Markov random fields is wider than the class of p-adic Gibbs measures [1]. One of the main problems of physics is to study the set of all p-adic Gibbs measures (see e.g. [1, 2]). Let us present some main definitions from the theory of p-adic numbers (see [3–5]). Let p be a prime number. Every rational number x = 0 can be represented in the form x = p n m , where r, n ∈ Z, m is a positive number, (n,m) = 1, where m and n are not divisible by p. A p-adic norm of rational number x = p n m is defined as follows

On Hoeffding and Bernstein type inequalities for sums of random variables in non-additive measure spaces and complete convergence

Working with real phenomena, one often faces situations where additivity assumption is unavailable. Non-additive measures and Choquet integral are attracting much attention from scientists in many different areas such as financial economics, economic modelling, probability theory and statistics. Hoeffding’s and Bernstein’s inequalities are two powerful tools that can be applied in many studies of the asymptotic behaviour of inference problems in probability theory, model selection, stochastic processes and economic modelling. One thing that seems missing is the developments of Hoeffding’s and Bernstein’s inequalities for sums of random variables in non-additive cases. The purposes of this paper are to extend Hoeffding’s and Bernstein’s inequalities for sums of random variables from probability measure space to non-additive measure space, and then establish two complete convergence theorems for more general form.

Cosmic bandits: Exploration versus exploitation in CMB B-mode experiments

A preferred method to detect the curl-component, or B-mode, signature of inflationary gravitational waves (IGWs) in the cosmic microwave background (CMB) polarization, in the absence of foregrounds and lensing, is a prolonged integration over a single patch of sky of a few square degrees. In practice, however, foregrounds abound and the sensitivity to B modes can be improved considerably by finding the region of sky cleanest of foregrounds. The best strategy to detect B modes thus involves a tradeoff between exploration (to find lower-foreground patches) and exploitation (through prolonged integration). This problem is akin to the multi-armed bandit (MAB) problem in probability theory, wherein a gambler faces a series of slot machines with unknown winning odds and must develop a strategy to maximize his/her winnings with some finite number of pulls. While the optimal MAB strategy remains to be determined, a number of algorithms have been developed in an effort to maximize the winnings. Here, based on this resemblance, we tackle the search for IGW B modes with single frequency experiments in the presence of spatially varying foregrounds by developing adaptive survey strategies to optimize the sensitivity to IGW B modes. We demonstrate, using realistic foreground models and taking lensing-induced B modes into account, that adaptive experiments can substantially improve the upper bound on the tensor-to-scalar ratio (by factors of 2 and 3 in single frequency experiments, and possibly even more). Similar techniques can be applied to other surveys, including 21-cm measurements of signatures of the epoch of reionization, searches for a stochastic primordial gravitational wave background, deep-field imaging by the James Webb Space Telescope or various radio interferometers, and transient follow-up searches.

English

The Humbert-Bessel are multi-index functions with various applications in electromagnetism. New families of functions sharing some similarities with Bessel functions are often introduced in the mathematical literature, but at a closer analysis they are not new, in the strict sense of the word, and are shown to be expressible in terms of already discussed forms. This is indeed the case of the remodified Bessel functions, whose properties have been analyzed within the context of coincidence problems in probability theory. In this paper we show that these functions are particular cases of the Humbert-Bessel ones.

Методика обучения решению задач теории вероятностей в военном вузе

Методика обучения решению задач теории вероятностей в военном вузе

Unconditional Convergence of Functional Series in Problems of Probability Theory

We study the unconditional convergence of series in Banach spaces. We consider series of special type (Hadamard series), obtain the condition of their unconditional convergence, and discuss some of their applications. Further, we examine the almost sure unconditional convergence of random series in Banach spaces and, in the case of Gaussian series, we establish the relationship between the almost sure unconditional convergence and the geometry of Banach spaces. We also consider some probabilistic problems related to the convergence of series.

Komlós–Major–Tusnády approximation under dependence

The celebrated results of Komlós, Major and Tusnády [Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58] give optimal Wiener approximation for the partial sums of i.i.d. random variables and provide a powerful tool in probability and statistics. In this paper we extend KMT approximation for a large class of dependent stationary processes, solving a long standing open problem in probability theory. Under the framework of stationary causal processes and functional dependence measures of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150–14154], we show that, under natural moment conditions, the partial sum processes can be approximated by Wiener process with an optimal rate. Our dependence conditions are mild and easily verifiable. The results are applied to ergodic sums, as well as to nonlinear time series and Volterra processes, an important class of nonlinear processes.

Phase transitions in number theory: From the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed.

The Birthday Problem of People with a Given Distribution

The problem of birthday is an ancient and interesting problem in probability theory. The traditional birthday problem has a conclusion that the probability of the same birthday among people will reach 50% when the number of people reached 23. This paper studies the issue of people with a given distribution and discusses the probability that people have the same birthday among a given crowd.

Beyond uniqueness: the birthday paradox, source attribution and individualization in forensic science testimony

For many decades, forensic science identification experts have insisted that they can ‘individualize’ traces such as fingerprints and toolmarks to the one and only one object that produced them. They have relied on a theory of global uniqueness of patterns as the basis for such individualization. Although forensic practitioners and theorists are moving toward a more probabilistic understanding of pattern matching, textbooks and reference works continue to assert that uniqueness justifies individualization and that experience demonstrates discernible uniqueness. One response to the last claim applies a famous problem in probability theory — the Birthday Problem — to the forensic realm to show that even an extensive record of uniqueness does little to prove that all such patterns are unique. This essay describes the probabilistic reasoning and its limits. It argues that the logic of the Birthday Paradox does indeed undercut the theory of global, general uniqueness, but that the reasoning is logically compatible with opinion testimony that a specific object is nearly certain to be the source of a pattern or trace. It also notes some alternatives to categorical claims of individualization, whether those claims are based on the theory of global, general uniqueness or instead on some less sweeping and more defensible theory.

Phase Transition of Message Propagation Speed in Delay-Tolerant Vehicular Networks

Delay-tolerant network (DTN) architectures have recently been proposed as a means to enable efficient routing of messages in vehicular area networks (VANETs), which are characterized by alternating periods of connectivity and disconnection. Under such architectures, when multihop connectivity is available, messages propagate at the speed of radio over connected vehicles. On the other hand, when vehicles are disconnected, messages are carried by vehicles and propagate at vehicle speed. Our goal in this paper is to analytically determine what gains are achieved by DTN architectures and under which conditions, using the average message propagation speed as the primary metric of interest. We develop an analytical model for a bidirectional linear network of vehicles, as found on highways. We derive both upper and lower bounds on the average message propagation speed by exploiting a connection with the classical pattern-matching problem in probability theory. The bounds reveal an interesting phase transition behavior. Specifically, we find out that, below a certain critical threshold, which is a function of the traffic density in each direction, the average message speed is the same as the average vehicle speed, i.e., DTN architectures provide no gain. On the other hand, we determine another threshold above which the average message speed quickly increases as a function of traffic density and approaches radio speed. Based on the bounds, we also develop an approximation model for the average message propagation speed that we validate through numerical simulations.

Registrar: a complete-memory operator to enhance performance of genetic algorithms

We examine the concept of storing all evaluated chromosomes and directly reuse them in Genetic Algorithms (GAs). This is achieved by a fully encapsulated operator, called Registrar, which is effortlessly placed between the GA and the objective function. The Registrar does not approximate the objective function. Instead, it replaces the chromosomes requested by the GA with similar ones taken from the registry, bypassing the function evaluation. Unlike other methods that use external memory to increase genetic diversity, our simple implementation encourages revisits in order to avoid evaluations in an aggressive manner. Significant increase in performance is observed which is present even at the early stages of evolution, in accordance with the Birthday Problem of probability theory. Implementation with Standard GA shows great promise, while the encapsulation of the code facilitates implementation with other Evolutionary Algorithms.

On a non-homogeneous difference equation from probability theory

Abstract The so-called gambler’s ruin problem in probability theory is considered for a Markov chain having transition probabilities depending on the current state. This problem leads to a non-homogeneous difference equation with non-constant coefficients for the expected duration of the game. This mathematical expectation is computed explicitly.

AVERAGE-CASE "MESSY" BROADCASTING

Current studies of "messy" broadcasting have so far concentrated on finding worst-case times. However, such worst-case scenarios are extremely unlikely to occur in general. Hence, determining average-case times or tight upper bounds for completing "messy" broadcasting in various network topologies is both necessary and meaningful in practice. In this paper, we focus on seeking the average-case "messy" broadcast times of stars, paths, cycles, and d-ary trees, and finding good upper bounds for hypercubes. Finally, we derive a recursive formula to express the average-case time for a specific "messy" broadcast model on a complete graph using a classical occupancy problem in probability theory, and provide a nice simulation result which indicates that this model behaves like classical broadcasting.

Strong convergence of pairwise NQD random sequences

Strong limit theory is one of the most important problems in probability theory. Some results on the convergence of pairwise NQD random sequences have been presented. This paper further analyzes the strong convergence of pairwise NQD sequences and generalizes partial results of Wu [Q.Y. Wu, Convergence properties of pairwise NQD random sequences, Acta Math. Sinica 45 (3) (2002) 617–624 (in Chinese)]. Since no general moment inequalities are given as so far, we avoid this problem and obtain a class of strong limit theorem for NQD sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem, which are the supplements to the previous fruits.