Two-point Distribution Research Articles

Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample autocovariance-based statistics.

Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.

A Wigner Quasiprobability Distribution of Work.

In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.

Intertemporal Price Discrimination via Randomized Promotions

Problem definition: The undesirable but inevitable consequence of running promotions is that consumers can be trained to time their purchases strategically. In this paper, we study randomized promotions, where the firm randomly offers discounts over time, as an alternative strategy of intertemporal price discrimination. Methodology/results: We consider a base model where a monopolist sells a single product to a market with a constant stream of two market segments. The segments are heterogeneous in both their product valuations and patience levels. The firm precommits to a price distribution, and in each period, a price is randomly drawn from the committed distribution. We characterize the optimal price distribution as a randomized promotion policy and show that it serves as an intertemporal price discrimination mechanism such that high-valuation customers would make a purchase immediately at a regular price upon arrival, and low-valuation customers would wait for a random promotion. Compared against the optimal cyclic pricing policy, which is optimal within the strategy space of all deterministic pricing policies, the optimal randomized pricing policy beats it if low-valuation customers are sufficiently patient and the absolute discrepancy between high and low customer valuations is large enough. We extend the model in three directions. First, we consider the case where a portion of customers are myopic and would never wait. We show that the existence of myopic customers is detrimental to the firm’s profitability, and the expected profit from an optimal randomized pricing policy decreases as the proportion of myopic customers in the population increases. Second, we consider Markovian pricing policies where prices are allowed to be intertemporally correlated in a Markovian fashion. This additional maneuver allows the firm to reap an even higher profit when low-valuation customers are sufficiently patient by avoiding consecutive promotions but, on average, running the promotion more frequently with a smaller discount size. Lastly, we consider a model with multiple customer segments and show that a two-point price distribution remains optimal, and our conclusion from the two-segment base model still holds under certain conditions that are adopted in the literature. Managerial implications: Our results imply that the firm may want to deliberately randomize promotions in the presence of forward-looking customers. Funding: This work was supported by the National Natural Science Foundation of China [Grant 72201124], the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2015-06757 and RGPIN-2021-04295], and the Youth Project of the Humanities and Social Science Foundation of the Ministry of Education of China [Grant 22YJC630006]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.1194 .

On the Conjecture of Berry Regarding a Bernoulli Two-Armed Bandit

In this paper, we study an independent Bernoulli two-armed bandit with unknown parameters ρ and λ, where ρ and λ have a pair of priori distributions such that dR(ρ)=CRρr0(1−ρ)r0′dμ(ρ),dL(λ)=CLλl0(1−λ)l0′dμ(λ) and μ is an arbitrary positive measure on [0,1]. Berry proposed the conjecture that, given a pair of priori distributions (R,L) of parameters ρ and λ, the arm with R is the current optimal choice if r0+r0′<l0+l0′ and the expectation of ρ is not less than that of λ. We give an easily verifiable equivalent form of Berry’s conjecture and use it to prove that Berry’s conjecture holds when R and L are two-point distributions as well as when R and L are beta distributions and the number of trials N≤r0r0′+1.

Two-Point Convergence of the Stochastic Six-Vertex Model to the Airy Process

In this paper we consider the stochastic six-vertex model in the quadrant started with step initial data. After a long time T, it is known that the one-point height function fluctuations are of order $$T^{1/3}$$ and governed by the Tracy–Widom distribution. We prove that the two-point distribution of the height function, rescaled horizontally by $$T^{2/3}$$ and vertically by $$T^{1/3}$$ , converges to the two-point distribution of the Airy process. The starting point of this result is a recent connection discovered by Borodin–Bufetov–Wheeler between the stochastic six-vertex model and the ascending Hall–Littlewood process (a certain measure on plane partitions). Using the Macdonald difference operators, we obtain formulas for two-point observables for the ascending Hall–Littlewood process, which for the six-vertex model give access to the joint cumulative distribution function for its height function. A careful asymptotic analysis of these observables gives the two-point convergence result under certain restrictions on the parameters of the model.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions.

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Statistical behavior of nonequilibrium and living biological systems subjected to active and thermal fluctuations.

We present a path-integral formulation of the motion of a particle subjected to fluctuating active and thermal forces. This general framework predicts the statistical behavior associated with the stochastic trajectories of the particle, accounting for all possible realizations of Brownian and active forces, over an arbitrary potential landscape. Temporal correlations in the active forces result in non-Markovian statistics, necessitating the inclusion of a fixed active-force value at specified times within the statistical treatment. We specialize our theory to that of exponentially correlated active forces for a particle in a harmonic potential. We find the exact results for the statistical distributions for the initial position of the particle, accounting for the impact of the correlated active forces at all times prior to the initial time. Our theory is then used to find the two-point distribution for the active Brownian particle, which governs the joint probability that a particle begins and ends at specified locations. Analyses of the active Brownian statistics demonstrate that the impact of active forces can be interpreted through a time-dependent temperature whose influence depends on the competition of timescales of the active-force correlation and the relaxation time of the particle in the harmonic potential. The general results presented in this work are transferable to a broad range of nonequilibrium systems with active and Brownian motion, and the time-dependent temperature serves as a governing principle to describe the competition of timescales associated with active forces and internal relaxation processes.

Security Analysis of Continuous-Variable Measurement-Device-Independent Quantum Key Distribution Systems in Complex Communication Environments.

Continuous-variable measure-device-independent quantum key distribution (CV-MDI QKD) is proposed to remove all imperfections originating from detection. However, there are still some inevitable imperfections in a practical CV-MDI QKD system. For example, there is a fluctuating channel transmittance in the complex communication environments. Here we investigate the security of the system under the effects of the fluctuating channel transmittance, where the transmittance is regarded as a fixed value related to communication distance in theory. We first discuss the parameter estimation in fluctuating channel transmittance based on these establishing of channel models, which has an obvious deviation compared with the estimated parameters in the ideal case. Then, we show the evaluated results when the channel transmittance respectively obeys the two-point distribution and the uniform distribution. In particular, the two distributions can be easily realized under the manipulation of eavesdroppers. Finally, we analyze the secret key rate of the system when the channel transmittance obeys the above distributions. The simulation analysis indicates that a slight fluctuation of the channel transmittance may seriously reduce the performance of the system, especially in the extreme asymmetric case. Furthermore, the communication between Alice, Bob and Charlie may be immediately interrupted. Therefore, eavesdroppers can manipulate the channel transmittance to complete a denial-of-service attack in a practical CV-MDI QKD system. To resist this attack, the Gaussian post-selection method can be exploited to calibrate the parameter estimation to reduce the deterioration of performance of the system.

Møller operators and Hadamard states for Dirac fields with MIT boundary conditions

The aim of this paper is to prove the existence of Hadamard states for Dirac fields coupled with MIT boundary conditions on any globally hyperbolic manifold with timelike boundary once a suitable propagation of singularities theorem is assumed. To this avail, we consider particular pairs of weakly-hyperbolic symmetric systems coupled with admissible boundary conditions. We then prove the existence of an isomorphism between the solution spaces to the Cauchy problems associated with these operators – this isomorphism is in fact unitary between the spaces of L^2 -initial data. In particular, we show that for Dirac fields with MIT boundary conditions, this isomorphism can be lifted to a * -isomorphism between the algebras of Dirac fields and that any Hadamard state can be pulled back along this * -isomorphism preserving the singular structure of its two-point distribution.

Brownian bridge with random length and pinning point for modelling of financial information

In this paper, we introduce an extension of a Brownian bridge with a random length by including uncertainty also in the pinning level of the bridge. The main result of this work is that unlike for deterministic pinning point, the bridge process fails to be Markovian if the pining point distribution is absolutely continuous with respect to the Lebesgue measure. Further results include the derivation of formulae to calculate the conditional expectation of various functions of the random pinning time, the random pinning location, and the future value of the Brownian bridge, given an observation of the underlying process. For the specific case that the pining point has a two-point distribution, we state further properties of the Brownian bridge, e.g. the right continuity of its natural filtration and its semi-martingale decomposition. The newly introduced process can be used to model the flow of information about the behaviour of a gas storage contract holder; concerning whether to inject or withdraw gas at some random future time.

Optimal versioning strategy of enterprise software considering the customer cost-acceptance level

PurposeThe emergence of the Software-as-a-service (SaaS) licensing model dramatically changes how enterprise software is released. Especially, it is favored by small and medium enterprises (SMEs) because of the cost-friendly feature. In contrast, many large enterprises (LEs) own relatively abundant budgets and prefer the on-premise software to fulfill demands through customization. Considering the differentiated cost-acceptance level among customers, this study aims to address the versioning problem of the enterprise software faced by software firms.Design/methodology/approachA two-point distribution model is formulated to calculate the maximal profits software firm earned from both LEs and SMEs under three strategies (On-premise, SaaS and Hybrid). Then through profit comparison, this paper obtains the optimal versioning strategy and corresponding feasible conditions. Finally, the optimal solutions are derived concerning social welfare.FindingsA significant finding is that moving to SaaS becomes necessary for the software firms in product releases since the on-premise strategy will not be optimal. Based on this, this paper discovers that when LEs own a cost-acceptance level close to that of SMEs, the hybrid strategy is the only optimal choice. When LEs become less sensitive to costs, the hybrid strategy is suggested if the customization cost falls below the threshold. Otherwise, the SaaS strategy becomes the optimal option. The conclusions explain why some software vendors transit to “cloud companies” thoroughly and provide practical insights for software firms’ future decisions.Originality/valueTo the best of the authors’ knowledge, this paper is the first information economics study to consider consumer cost sensitivity in discussing enterprise software versioning. The differentiated cost-acceptance level is introduced to describe the customer utilities, and the results uncover the necessity of moving to SaaS under diversified customer composition. This work provides significant theoretical value and practical insights.

Vertical and slanted sound propagation in the near-ground atmosphere: Coherence and distributions.

Atmospheric turbulence causes acoustic signals to fluctuate and diminishes their coherence. These phenomena are important in applications such as source localization and sonic boom propagation. This article provides formulations for the spatial, cross-frequency, and temporal coherences of narrowband acoustic signals propagating over vertical and slanted paths in the atmosphere. Formulations for single- and two-point distributions of acoustic signals are also overviewed. The theoretical formulations are compared with data from a comprehensive sound propagation experiment carried out in 2018 at the National Wind Technology Center (Boulder, CO). The theories for sound propagation in a turbulent atmosphere, when combined with turbulence models incorporating shear and buoyancy instabilities, correctly predict the measured spatial coherence, which is primarily affected by small-scale isotropic turbulence. For relatively small coherence times, this approach also correctly predicts the temporal coherence. However, the approach underpredicts the cross-frequency coherence and temporal coherence for relatively large coherence times, which are affected by large-scale anisotropic buoyancy-driven velocity fluctuations. For different regimes ranging from unsaturated to fully saturated scattering, the measured distributions agree well with the theoretical predictions.

Static Symmetric Solutions of the Semi-Classical Einstein\u2013Klein\u2013Gordon System

We consider solutions of the semi-classical Einstein–Klein–Gordon system with a cosmological constant Lambda in mathbb {R}, where the spacetime is given by Einstein’s static metric on mathbb {R}times mathbb {S}^3 with a round sphere of radius a>0 and the state of the scalar quantum field has a two-point distribution omega _2 that respects all the symmetries of the metric. We assume that the mass mge 0 and scalar curvature coupling xi in mathbb {R} of the field satisfy m^2+xi R>0, which entails the existence of a ground state. We do not require states to be Hadamard or quasi-free, but the quasi-free solutions are characterised in full detail. The set of solutions of the semi-classical Einstein–Klein–Gordon system depends on the choice of the parameters (a,Lambda ,m,xi ) and on the renormalisation constants in the renormalised stress tensor of the scalar field. We show that the set of solutions is either (i) the empty set, or (ii) the singleton set containing only the ground state, or (iii) a set with infinitely many elements. We characterise the ranges of the parameters and renormalisation constants where each of these alternatives occur. We also show that all quasi-free solutions are given by density matrices in the ground state representation and we show that in cases (ii) and (iii) there is a unique quasi-free solution which minimises the von Neumann entropy. When m=0 this unique state is a beta -KMS state. We argue that all these conclusions remain valid in the reduced order formulation of the semi-classical Einstein equation.

Target-based distributionally robust optimization for single machine scheduling

In the traditional single-machine scheduling problem with uncertain processing time, distributionally robust optimization models are established in situations where the distribution information is not known exactly. One well-studied robust optimization method used in scheduling is the distributionally β-robust optimization technique. The goal of this technique is to maximize the probability that the total flow time of the job sequence does not exceed the predetermined target level T. However, it is possible that the excess value of the total flow time beyond the target T may be incredibly large, and therefore it is necessary to consider the optimized excess value. To address this concern, two new target-based risk measures are introduced. First, target-oriented CVaR, the worst-case conditional value-at-risk (CVaR) is used to minimize the excess value beyond T, which accounts for both the excess probability and the excess value. Second, the essential riskiness index (ERI) is extended to a multi-piecewise linear function for excess value control in single-machine scheduling problems. To obtain a tractable closed-form solutions of the risk measures, the CVaR-based measure is decomposed into an assignment problem and a second-order cone programming problem, and the ERI is converted to its equivalent semi-definite programming and second-order cone counterparts. To efficiently solve the risk measures for single-machine scheduling, a bisection search algorithm and a Benders decomposition algorithm are designed. In the numerical analysis, the excess value and the probability are investigated with respect to two distributions, namely the two-point distribution and the truncated normal distribution. The experiment design also considers different ranges of expectation and variance, tolerance levels, and problem scales. It is discovered that the CVaR and ERI-based methods are always better than the distributionally β-robust scheduling approach in terms of excess value.

Service-oriented performance of inventory models with partial information on unimodal demand lead-time distributions

Facing uncertainty in demand, companies try to avoid stock-outs by holding safety inventories, depending on a pre-set customer service level. The knowledge of the demand distribution during lead-time serves to determine the safety inventory level. Many times the distribution is not fully known, except maybe for its range, mean or variance. However literature shows that the performance of holding safety stock strongly depends on the characteristics of the distribution. One option is to protect against the worst case distribution given some information like range or moments. But this worst case is a two-point distribution, bringing unbelief to managers that such an occurrence would ever appear. Mostly they share the opinion that the demand distribution is unimodal. This research develops a technique to derive the safety stock for unimodal demand distributions of which the mode either is known or can be estimated. In this way, the managers obtain solutions to the decision problem including a higher belief that the related type of distribution might appear in practice.

Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems

This paper studies tight upper bounds for the mean and higher moments of the steady-state waiting time in the GI/GI/1 queue given the first two moments of the interarrival-time and service-time distributions. We apply the theory of Tchebycheff systems to obtain sufficient conditions for classical two-point distributions to yield the extreme values. These distributions are determined by having one mass at 0 or at the upper limit of support.

SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE QUEUE GIVEN PARTIAL INFORMATION

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

Mechanism Design for Managing Hidden Rebates and Inflated Quotes of a Procurement Service Provider

Problem definition : When sourcing through a procurement service provider (PSP), the PSP often collects rebates from unethical manufacturers in developing countries (as referral fees) that are “hidden” from the retailers. Recognizing that a PSP has a strong incentive to solicit quotes from unethical manufacturers, we examine a situation in which the retailer insists on soliciting a quote from a manufacturer designated by the retailer and a separate quote from an unethical manufacturer selected by the PSP. However, when the designated manufacturer is ethical, the PSP has an incentive to inflate the quote from this ethical manufacturer in order to help the unethical manufacturer to win. Facing this situation, is there a mechanism for the retailer to control hidden rebates? Academic/practical relevance : The issue of hidden rebates is a “known secret” in global supply chain practice. Also, hidden rebates increase the customs duty for U.S. importers because of the first sales rule for customs valuation of U.S. imports. Therefore, there is a need to understand the implications of hidden rebates and to control this unethical practice. Methodology : To circumvent the issue of hidden rebates and quote inflations, we develop a deterministic, incentive-compatible mechanism that is based on a simple selection rule (for selecting a manufacturer) and a contingent service fee (as a reward for the service provided by the PSP). Results : Our optimal mechanism creates incentives to (1) deter the PSP from inflating the quote submitted from the ethical manufacturer, (2) reduce the incidence of hidden rebates, and (3) reduce the retailer’s procurement cost and the corresponding import tax significantly. More importantly, relative to the “lowest quote wins” selection rule, the optimal mechanism is Pareto-improving for the retailer and the service provider when the hidden rebate is below a certain threshold. Furthermore, we extend our analysis to the case in which (1) the retailer is not sure whether the designated manufacturer is ethical or not, (2) the retailer does not know the exact value of hidden rebate (but it follows a two-point distribution), and (3) the retailer may verify the quote with its designated manufacturer before a formal contract. We also explore the stochastic incentive-compatible mechanism for the cases in which the penalty is unenforceable or enforceable. Managerial implications : When law enforcement is inconsistent in developing countries, retailers should beware of the existence and implications of hidden rebates. We provide a simple mechanism that a retailer can consider as a practical way to deter the PSP from inflating certain quotes and put hidden rebates under control.

Correlation function inadequacy in random-sequence entropy measures.

Considering symbolic and numerical random sequences in the framework of the additive Markov chain approach, we establish a relation between their correlation functions and conditional entropies. We express the entropy by means of the two-point probability distribution functions and then evaluate the entropy for the numerical random chain in terms of the correlation function. We show that such approximation gives a satisfactory result only for special types of random sequences. In general case the conditional entropy of numerical sequences obtained in the two-point distribution function approach is lower. We derive the conditional entropy of the additive Markov chain as a sum of the Kullback-Leibler mutual information and give an example of random sequence with the exactly zero correlation function and the nonzero correlations.

The regularized Hadamard expansion

A local expansion is proposed for two-point distributions involving an ultraviolet regularization in a four-dimensional globally hyperbolic space-time. The regularization is described by an infinite number of functions which can be computed iteratively by solving transport equations along null geodesics. We show that the Cauchy evolution preserves the regularized Hadamard structure. The resulting regularized Hadamard expansion gives detailed and explicit information on the global dynamics of the regularization effects.