Limiting Probability Distribution Research Articles

A discrete probabilistic model yielding multidimensional q − Gaussians in the thermodynamic limit for specific parameter values

We show that the N → ∞ limiting probability distributions (with N being the number of random variables to be summed) of a particular case belonging to a family of d − dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1) − dimensional hyperpyramids (introduced in Rodríguez and Tsallis, J. Math. Phys 2012) are given, after an appropriate change o variables, by d − dimensional q − Gaussian distributions ( fq,β(x)∝eq−βxTΣ−1x , with x taking values in a subset of Rd , q and β real parameters, Σ a positive definite matrix, and eqx=[1+1(1−q)x]+11−q , with [u]+=max{u,0} ), for the particular parameter values q = 3 and β < 0. Some features of these distributions, which arise in the context of Nonextensive Statistical Mechanics, as well as a connection between Dirichlet distributions and q − Gaussian distributions, which generalizes the one given for dimension 1 by Rodríguez et al, are also exposed.

Queuing models of links carrying streaming and elastic services

We consider an access link carrying data generated by streaming and elastic services requested by fixed or mobile end users, and subjected to an admission control (AC) algorithm. For the performance analysis of such link we develop a new queuing model and we show that, with the considered AC, the queuing model admits a product form expression for the joint limiting probability distribution of the numbers of active services of the different types. In addition, we prove that, when mobility can be neglected, i.e., in the case of either fixed access or slow mobility, the queuing model is insensitive to the distribution of the amount of data to be transferred for the fulfillment of the different service requests. Numerical results show unexpected oscillating behaviors for several performance metrics, and provide interesting insight into the link performance.

A generalization of Bernstein–Vazirani algorithm with multiple secret keys and a probabilistic oracle

A probabilistic version of the Bernstein-Vazirani problem (which is a generalization of the original Bernstein-Vazirani problem) and a quantum algorithm to solve it are proposed. The problem involves finding one or more secret keys from a set of multiple secret keys (encoded in binary form) using a quantum oracle. From a set of multiple unknown keys, the proposed quantum algorithm is capable of (a) obtaining any key (with certainty) using a single query to the probabilistic oracle and (b) finding all keys with a high probability (approaching 1 in the limiting case). In contrast, a classical algorithm will be unable to find even a single bit of a secret key with certainty (in the general case). Owing to the probabilistic nature of the oracle, a classical algorithm can only be useful in obtaining limiting probability distributions of $ 0 $ and $ 1 $ for each bit-position of secret keys (based on multiple oracle calls) and this information can further be used to infer some estimates on the distribution of secret keys based on combinatorial considerations. For comparison, it is worth noting that a classical algorithm can be used to exactly solve the original Bernstein-Vazirani problem (involving a deterministic oracle and a single hidden key containing $n$ bits) with a query complexity of $\mathcal{O}(n)$. An interesting class of problems similar to the probabilistic version of the Bernstein-Vazirani problem can be construed, where quantum algorithms can provide efficient solutions with certainty or with a high degree of confidence and classical algorithms would fail to do so.

Moments of the 2D Directed Polymer in the Subcritical Regime and a Generalisation of the Erdös–Taylor Theorem

We compute the limit of the moments of the partition function Z_{N}^{beta _N} of the directed polymer in dimension d=2 in the subcritical regime, i.e. when the inverse temperature is scaled as beta _N sim hat{beta } sqrt{tfrac{pi }{log N}} for hat{beta } in (0,1). In particular, we establish that for every h in {mathbb {R}}, lim _{N rightarrow infty } {{mathbb {E}}} big [big (Z_{N}^{beta _N}big )^h big ]=big (frac{1}{1-hat{beta }^2}big )^{frac{h(h-1)}{2}}. We also identify the limit of the moments of the averaged field tfrac{sqrt{log N}}{N} sum _{x in {mathbb {Z}}^2} varphi (tfrac{x}{sqrt{N}})big (Z_{N}^{beta _N} (x)-1 big ), for varphi in C_c({mathbb {R}}^2), as those of a gaussian free field. As a byproduct, we identify the limiting probability distribution of the total pairwise collisions between h independent, two dimensional random walks starting at the origin. In particular, we derive that πlogN∑1≤i<j≤hLN(i,j)→N→∞(d)Γ(h(h-1)2,1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\pi }{\\log N}\\sum _{1 \\le i<j\\le h} {\ extsf{L}}_N^{(i,j)}\\xrightarrow [N \\rightarrow \\infty ]{(d)} \\Gamma \\big ( \ frac{h(h-1)}{2},1\\big ) , \\end{aligned}$$\\end{document}where {{textsf{L}}}^{(i,j)}_N denotes the collision local time by time N between copies i, j and Gamma denotes the Gamma distribution. This generalises a classical result of Erdös and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960).

A Markovian Incremental Stochastic Subgradient Algorithm

In this article, a stochastic incremental subgradient algorithm for the minimization of a sum of convex functions is introduced. The method sequentially uses partial subgradient information, and the sequence of partial subgradients is determined by a general Markov chain. This makes it suitable to be used in networks, where the path of information flow is stochastically selected. We prove convergence of the algorithm to a weighted objective function, where the weights are given by the Cesàro limiting probability distribution of the Markov chain. Unlike previous works in the literature, the Cesàro limiting distribution is general (not necessarily uniform), allowing for general weighted objective functions and flexibility in the method.

A non-parametric evaluation method in staircase test for improving fatigue limit assessment

A new evaluation method based on Kernel Density Estimation (KDE) was proposed to estimate the fatigue limit distribution from staircase tests without prior knowledge. The proposed method notably reduced the standard deviation bias of the fatigue limit using small sample tests. A simulation-based investigation was performed to compare the effectiveness of the proposed method with other evaluation methods. The obtained results reveal that the proposed technique offers a better estimation of the median, the standard deviation, and the probability distribution of the fatigue limit. Moreover, this technique is insensitive to the test parameters of the staircase method.

The runsort permuton

Suppose we choose a permutation π uniformly at random from Sn. Let runsort(π) be the permutation obtained by sorting the ascending runs of π into lexicographic order. Alexandersson and Nabawanda recently asked if the plot of runsort(π), when scaled to the unit square [0,1]2, converges to a limit shape as n→∞. We answer their question by showing that the measures corresponding to the scaled plots of these permutations runsort(π) converge with probability 1 to a permuton (limiting probability distribution) that we describe explicitly. In particular, the support of this permuton is {(x,y)∈[0,1]2:x≤ye1−y}.

Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules

&#x0D; &#x0D; &#x0D; We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.&#x0D; &#x0D; &#x0D;

Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.

Higher-dimensional open quantum walk in environment of quantum Bernoulli noises

Open quantum walks (OQWs) (also known as open quantum random walks) are quantum analogs of classical Markov chains in probability theory, and have potential application in quantum information and quantum computation. Quantum Bernoulli noises (QBNs) are annihilation and creation operators acting on Bernoulli functionals, and can be used as the environment of an open quantum system. In this paper, by using QBNs as the environment, we introduce an OQW on a general higher-dimensional integer lattice. We obtain a quantum channel representation of the walk, which shows that the walk is indeed an OQW. We prove that all the states of the walk are separable provided its initial state is separable. We also prove that, for some initial states, the walk has a limit probability distribution of higher-dimensional Gauss type. Finally, we show links between the walk and a unitary quantum walk recently introduced in terms of QBNs.

Asymptotic Analysis of Finite-Source M/GI/1 Retrial Queueing Systems with Collisions and Server Subject to Breakdowns and Repairs

This paper deals with a retrial queuing system with a finite number of sources and collision of the customers, where the server is subject to random breakdowns and repairs depending on whether it is idle or busy. A significant difference of this system from the previous ones is that the service time is assumed to follow a general distribution while the server’s lifetime and repair time is supposed to be exponentially distributed. The considered system is investigated by the method of asymptotic analysis under the condition of an unlimited growing number of sources. As a result, it is proved that the limiting probability distribution of the number of customers in the system follows a Gaussian distribution with given parameters. The Gaussian approximation and the estimations obtained by stochastic simulations of the prelimit probability distribution are compared to each other and measured by the Kolmogorov distance. Several examples are treated and figures show the accuracy and area of applicability of the proposed asymptotic method.

Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein–Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies (Camassa et al., 2019) numerically demonstrated that the decaying passive scalar’s long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed (McLaughlin and Majda, 1996). Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the N-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein–Uhlenbeck case to the white noise case in the limit γ→∞ of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of γ, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the N-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of Bronski et al. (2007). The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations.

Quasi-stationarity of scalar turbulent mixing statistics in a non-symmetric case

The existence of an asymptotic shape at large times of the probability density function (PDF) of a non-reacting scalar, mixed by a solenoidal turbulent velocity field and molecular diffusive transport, was investigated by Sinai and Yakhot [Y. G. Sinai and V. Yakhot, “Limiting probability distributions of a passive scalar in a random velocity field,” Phys. Rev. Lett. 63, 1962 (1989)]. The quasi-stationarity of the mixing statistics along the time evolution by Valiño et al. [“Quasistationary probability density functions in the turbulent mixing of a scalar field,” Phys. Rev. Lett. 72, 3518 (1994)] was an extension to symmetric scalar pdfs; analytic solutions for the scalar fluctuation dissipation rates, conditional upon the scalar value, and the pdf were obtained. This manuscript examines the generalization of the latter results to asymmetric scalar pdfs, further scrutinizes underlying mechanisms for a quasi-stationary statistics, and shows a Monte Carlo implementation which allows non-Gaussian relaxations to prescribed values of skweness and kurtosis.

A queueing inventory system with threshold based priority service and a flexible server

In this paper, we analyze two-server queueing - inventory system which consists of two-parallel queues and two category of customers i.e., high and low priority customers. The high priority customer generates demand for an item whereas the low priority customers arrive only for repair (not demanding items). Server 1 offers his service to first queue which involves only high priority customers and server 2 is a flexible server who offers his service to both the queues. When there is no availability of service in both the queues, server 1 will be at idle state whereas server 2 goes on vacation. We find a priority service rule to minimize the loss probability of low priority customers. The low priority customers will be served on threshold-based service such that a threshold L which is placed in the second queue and the service starts only if the number customers in the second queue exceeds the threshold level. The limiting probability distribution of the four random variables is computed in steady state. Also, some significant change in measures of the system are derived. Finally, the suitable cost function is defined and significant numerical examples are projected.

The explicit expression of non-trivial stationary distribution of SDEs under regime switching

This paper mainly focuses on the stationary probability distribution of stochastic differential equations (SDEs) under regime switching. We give the explicit expression of stationary distribution according to the limit probability distribution of SDEs under some conditions. As a special case, we obtain the stationary distribution of SDEs with Markovian switching as the weighted mean of the stationary distribution of its subsystems. In the end, an example and some conclusions are given to illustrate our theoretical results of this paper.

Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises.

As a discrete-time quantum walk model on the one-dimensional integer lattice , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

A New Limit Theorem for Quantum Walk in Terms of Quantum Bernoulli Noises.

In this paper, we consider limit probability distributions of the quantum walk recently introduced by Wang and Ye (C.S. Wang and X.J. Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Inf. Process. 15 (2016), no. 5, 1897–1908). We first establish several technical theorems, which themselves are also interesting. Then, by using these theorems, we prove that, for a wide range of choices of the initial state, the above-mentioned quantum walk has a limit probability distribution of standard Gauss type, which actually gives a new limit theorem for the walk.

A half-normal distribution scheme for generating functions

We present a general theorem on the structure of bivariate generating functions which gives sufficient conditions such that the limiting probability distribution is a half-normal distribution. If X is a normally distributed random variable with zero mean, then |X| obeys a half-normal distribution. In the second part, we apply our result to prove three natural appearances in the domain of lattice paths: the number of returns to zero, the height, and the sign changes are under zero drift distributed according to a half-normal distribution. This extends known results to a general step set. Finally, our result also gives a new proof of Banach’s matchbox problem.

Neighborhood effect within polycrystalline materials: Elastoplastic micromechanical analysis by cellular automaton and finite element

Polycrystals heterogeneities generate stress concentration which can be responsible for premature failure of mechanical parts and thereby need to be considered in mechanical models. A grain’s close neighborhood has shown to significantly impact its mechanical behavior. The more the crystal is anisotropic, the more this impact can be, which in some specific configuration can double a grain’s stress level than it would normally have in an “average” random environment. The impact of such stress concentration on the elastoplastic behavior of polycrystalline material has been studied by means of a cellular automaton (CA) and finite element (FE) models. The studied aggregates were single-phase with grains of identical size and spherical shape. The grains’ anisotropy and crystallographic orientations were the only sources of heterogeneities studied. The CA model, originally developed for the study of elastic loadings, was adapted for the study of elastoplastic loadings in high cycle fatigue regime and compared to the FE model. Using the CA model, a statistical study was carried out to determine the true elastic limit probability distribution due to the random character of the neighborhood effect. The obtained probability distribution could be linked to the fatigue test scatter observed experimentally.

Stationarity and Convergence of the Metropolis-Hastings Algorithm: Insights into Theoretical Aspects

Markov chain Monte Carlo (MCMC) is a versatile sampling approach that is useful in a wide range of estimation and simulation applications. Fundamentally, MCMC is a powerful general means of generating random samples from probability distributions from which it is otherwise difficult to draw samples. The MCMC method is named for its reliance on the construction of a Markovian (dependent) sequence of random variables. Under modest conditions, the sequence has a limiting probability distribution that corresponds to the distribution of interest, often called the target distribution. The sequence, as will be seen, is easily constructed, and the target distribution may be almost any distribution of interest. These two features of MCMC make it a popular choice for Monte Carlo simulation. The primary forms of MCMC are the Metropolis-Hastings (M-H) algorithm and Gibbs sampling. Although both forms are useful, we focus on M-H, which is more flexible and easier to implement. A general discussion of many aspects of MCMC, including several examples, is given in [1]. As discussed in Summary, the purpose of this article is to provide some of the theoretical support for M-H that was not given in [1], focusing specifically on the stationarity and convergence of the underlying Markov process.