Distribution Theorem Research Articles

Hidden Gaussian Markov model for distributed fault detection in wireless sensor networks

Wireless sensor networks are based on a large number of sensor nodes used to measure information like temperature, acceleration, displacement, or pressure. The measurements are used to estimate the state of the monitored system or area. However, the quality of the measurements must be guaranteed to ensure the reliability of the estimated state of the system. Actually, sensors can be used in a hostile environment such as, on a battle field in the presence of fires, floods, earthquakes. In these environments as well as in normal operation, sensors can fail. The failure of sensor nodes can also be caused by other factors like: the failure of a module (such as the sensing module) due to the fabrication process models, loss of battery power and so on. A wireless sensor network must be able to identify faulty nodes. Therefore, we propose a probabilistic approach based on the Hidden Markov Model to identify faulty sensor nodes. Our proposed approach predicts the future state of each node from its actual state, so the fault could be detected before it occurs. We use an aided judgment of neighbour sensor nodes in the network. The algorithm analyses the correlation of the sensors’ data with respect to its neighbourhood. A systematic approach to divide a network on cliques is proposed to fully draw the neighbourhood of each node in the network. After drawing the neighbourhood of each node (cliques), damaged cliques are identified using the Gaussian distribution theorem. Finally, we use the Hidden Markov Model to identify faulty nodes in the identified damaged cliques by calculating the probability of each node to stay in its normal state. Simulation results demonstrate our algorithm is efficient even for a huge wireless sensor network unlike previous approaches.

Statistics of Infima and Stopping Times of Entropy Production and Applications to Active Molecular Processes

We study the statistics of infima, stopping times and passage probabilities of entropy production in nonequilibrium steady states, and show that they are universal. We consider two examples of stopping times: first-passage times of entropy production and waiting times of stochastic processes, which are the times when a system reaches for the first time a given state. Our main results are: (i) the distribution of the global infimum of entropy production is exponential with mean equal to minus Boltzmann's constant; (ii) we find the exact expressions for the passage probabilities of entropy production to reach a given value; (iii) we derive a fluctuation theorem for stopping-time distributions of entropy production. These results have interesting implications for stochastic processes that can be discussed in simple colloidal systems and in active molecular processes. In particular, we show that the timing and statistics of discrete chemical transitions of molecular processes, such as, the steps of molecular motors, are governed by the statistics of entropy production. We also show that the extreme-value statistics of active molecular processes are governed by entropy production, for example, the infimum of entropy production of a motor can be related to the maximal excursion of a motor against the direction of an external force. Using this relation, we make predictions for the distribution of the maximum backtrack depth of RNA polymerases, which follows from our universal results for entropy-production infima.

Limit theory for the Gilbert graph

For a given homogeneous Poisson point process in Rd two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behavior of the resulting random graph, the Gilbert graph or random geometric graph, is investigated as the intensity of the Poisson point process is increased and the distance parameter goes to zero. The asymptotic expectation and covariance structure of a class of length-power functionals are computed. Distributional limit theorems are derived that have a Gaussian, a stable or a compound Poisson limiting distribution. Finally, concentration inequalities are provided using the convex distance.

The McLennan–Zubarev steady state distribution and fluctuation theorems

The McLennan–Zubarev steady state distribution is studied in the connection with fluctuation theorems. We derive the McLennan–Zubarev steady state distribution from the nonequilibrium detailed balance relation. Then, considering the cumulant function or cumulant functional, two fluctuation theorems for entropy and for currents are proved. Using the fluctuation theorem for currents, the current is expanded in terms of thermodynamic forces. In the lowest order of the thermodynamic force, we find that the transport coefficient satisfies the Onsager’s reciprocal relation. In the next order, we derived the correction term to the Green–Kubo formula.

Temporal Distributional Limit Theorems for Dynamical Systems

Suppose \(\{T^t\}\) is a Borel flow on a complete separable metric space X, \(f:X\rightarrow \mathbb R\) is Borel, and \(x\in X\). A temporal distributional limit theorem is a scaling limit for the distributions of the random variables \(X_T:=\int _0^t f(T^s x)ds\), where t is chosen randomly uniformly from [0, T], x is fixed, and \(T\rightarrow \infty \). We discuss such laws for irrational rotations, Anosov flows, and horocycle flows.

The Interplay of Population Size and Mutation Probability in the ( $$1+\lambda $$ 1 + λ ) EA on OneMax

The ($$1+\lambda $$1+ź) EA with mutation probability c / n, where $$c>0$$c>0 is an arbitrary constant, is studied for the classical OneMax function. Its expected optimization time is analyzed exactly (up to lower order terms) as a function of c and $$\lambda $$ź. It turns out that 1 / n is the only optimal mutation probability if $$\lambda =o(\ln n\ln \ln n/\ln \ln \ln n)$$ź=o(lnnlnlnn/lnlnlnn), which is the cut-off point for linear speed-up. However, if $$\lambda $$ź is above this cut-off point then the standard mutation probability 1 / n is no longer the only optimal choice. Instead, the expected number of generations is (up to lower order terms) independent of c, irrespectively of it being less than 1 or greater. The theoretical results are obtained by a careful study of order statistics of the binomial distribution and variable drift theorems for upper and lower bounds. Experimental supplements shed light on the optimal mutation probability for small problem sizes.

Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

Consider branching random walks on the integer lattice Zd, where the branching mechanism is governed by a supercritical Galton–Watson process and the particles perform a symmetric nearest-neighbor random walk whose increments equal to zero with probability r∈[0,1). We derive exact convergence rate in the local limit theorem for distributions of particles. When r=0, our results correct and improve the existing results on the convergence speed conjectured by Révész (1994) and proved by Chen (2001). As a byproduct, we obtain exact convergence rate in the local limit theorem for some symmetric nearest-neighbor random walks, which is of independent interest.

Markov Chain Monte Carlo confidence intervals

For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ is finite and positive. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of $\sigma_P^2(h)$. We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.

A Frobenius theorem for corank-1 continuous distributions in dimensions two and three

We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical Frobenius theorem, which says that an involutive [Formula: see text] distribution is uniquely integrable.

Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory

In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For $n \geq 3$, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in $\mathbb{R}^n$.

How to identify housing bubbles? A decision support model

PurposeThe purpose of this paper is to provide a decision support model for the early diagnosis of housing bubbles in the UK during the maturity process of the phenomenon.Design/methodology/approachThe development process of the model is divided into four stages. These stages are driven by the normal distribution theorem coupled with the case study approach. The application of normal distribution theory is allowed through the usage of several parametric tools. The case studies tested in this research include the last two UK housing bubbles, 1986 to 1989 and 2001/2002 to 2007. The central hypothesis of the model is that during housing bubbles, all speculative activities of market participants follow an approximate synchronisation, and therefore, an irrational, synchronous and periodic increase on a wide range of relevant variables must occur to anticipate the bubble component. An empirical application of the model is conducted on UK housing market data over the period of 1983-2011.FindingsThe new approach successfully identifies the well-known UK historical bubble episodes over the period of 1983-2011. The study further determines that for uncovering housing bubbles in the UK, house price changes have the same weight with the debt–burden ratio when their velocity is positive. Finally, the application of this model has led us to conclude that the model’s outputs fluctuate approximately in line with phases of the UK real estate cycle.Originality/valueThis paper proposes a new measure for studying the presence of housing bubbles. This measure is not simply an ex post detection technique but dating algorithms that use data only up to the point of analysis for an on-going bubble assessment, giving an early warning diagnostic that can assist market participants and regulators in market monitoring.

Spectral analysis of the Moore–Penrose inverse of a large dimensional sample covariance matrix

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix Σn let S̃n denote the usual sample covariance (centered by the mean) and Sn the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p>n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore–Penrose inverse of Sn and S̃n. We consider the large dimensional asymptotics when the number of variables p→∞ and the sample size n→∞ such that p/n→c∈(1,+∞). We present a Marchenko–Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore–Penrose inverse of S̃n differs in the mean from that of Sn.

Nonequilibrium study of the intrinsic free-energy profile across a liquid-vapour interface.

We calculate an atomistically detailed free-energy profile across a heterogeneous system using a nonequilibrium approach. The path-integral formulation of Crooks fluctuation theorem is used in conjunction with the intrinsic sampling method to calculate the free-energy profile for the liquid-vapour interface of the Lennard-Jones fluid. Free-energy barriers are found corresponding to the atomic layering in the liquid phase as well as a barrier associated with the presence of an adsorbed layer as revealed by the intrinsic density profile. Our findings are in agreement with profiles calculated using Widom's potential distribution theorem applied to both the average and the intrinsic profiles as well as the literature values for the excess chemical potential.

A general parametric Stein characterization

We present a general characterization theorem for parametric probability distributions in terms of a differential operator akin to the so-called Stein operators from the literature on Stein’s method.

Training Sequence Design for Feedback Assisted Hybrid Beamforming in Massive MIMO Systems

Massive multiple-input multiple-output (MIMO) communication is an emerging technology that uses an excess of transmit antennas to realize high spectral efficiency. Achieving potential gains with large-scale antenna arrays hinges on sufficient channel estimation accuracy. Much prior work focuses on time-division duplex (TDD)-based networks, relying on reciprocity between the uplink and downlink channels. However, most currently deployed commercial wireless systems are frequency-division duplex (FDD)-based, making it difficult to exploit channel reciprocity. In massive MIMO FDD systems, the problem of channel estimation becomes even more challenging due to the substantial training resources and feedback requirements which scale with the number of antennas. In this paper, we consider the problem of training sequence design and the mapping of training signals to training periods. We focus on reduced-dimension training sequence designs, along with transmit precoder designs, aimed at reducing both hardware complexity and power consumption. The resulting designs are extended to hybrid analog-digital beamforming systems, which employ a limited number of active RF chains for transmit precoding, by applying the Toeplitz distribution theorem to large-scale linear antenna systems. A practical guideline for training sequence parameter selection is presented along with performance analysis.

A Bode-Like Integral for Discrete Linear Time-Periodic Systems

We present a generalization of the Bode integral formula for discrete-time linear periodic systems. It is shown that similar to the classical Bode integral formula, the sensitivity integral for a discrete-time linear periodic system depends only on the open-loop dynamics of the system; in particular, on the open-loop characteristic multipliers outside the unit circle in the complex plane. The integral is derived by using an asymptotic eigenvalue distribution theorem for block Toeplitz matrices, which does not require the open loop system to be stable or the disturbance to be Gaussian. The result is demonstrated through application to a class of multi-rate sampled data systems with commensurate rates.

A generalized Peano Kernel Theorem for distributions of exponential decay

The Peano Kernel Theorem (PKT) is a classical representation theorem in numerical integration. The idea is that if T is a quadrature rule that exactly integrates polynomials up to degree n−1 on a bounded interval [a,b], then there exists a kernel K, depending only on T, such thatE(f)=T(f)−∫abf(t)dt=∫abK(t)f(n)(t)dt, whenever f∈Cn([a,b]). In this work, we generalize the PKT from the class of linear functionals on Cn([a,b]) to the class of Laplace transformable tempered distributions of exponential decay. In particular, it is not necessary that the functionals being approximated have compact support. The generalized result is proven using an approach that provides a formula for computing the kernel K in the Fourier domain, which can be more computationally tractable and efficient in many cases. We conclude with examples of how the generalized Peano Kernel Theorem can be used for error analysis in signal processing.

Distributional Behavior of Time Averages of Non- $$L^1$$ L 1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

Observed Measures and Fluctuations in Dissipative Infinite Ergodic Systems: Randomization Theory for the Infinite-Modal Maps with Ant-Lion Property

Universal aspects of a certain class of infinite ergodic systems are studied by using the infinite-modal maps with a special interest to their observed measures. It is shown that the ant-lion (AL) property is an important nature to realize the infinite ergodicity in the dissipative dynamics. The AL-property, which seems to be a little bit paradoxical one, is characterized by the monotonical relaxation of mean orbits into a singular stable point, but it causes the divergence of the Lyapunov exponent as well as the emergence of a number of absolutely-continuous invariant measures. Our main concern is to characterize the unique observed measure in those many admissible ergodic measures. To this end, firstly the randomization formulae are developed on the basis of the uniform distribution theorem by Weyl, to derive the stochastic aspects of the AL-property. Actually, it is shown that the statistical natures of the infinite-modal maps are well explained by the randomization formulae. Furthermore, it is shown that the observed measure derived from the randomization formulae is universal, and that the asymptotic form obeys the power law with the exponent −1, in agreement with numerical simulations.

Richter’s local limit theorem and Black–Scholes type formulas

We prove a Black–Scholes type formula when the geometric Brownian motion originates from approximations by multinomial distributions. It is shown that the variance appearing in the Black–Scholes formula for option pricing can be structured according to occurrences of different types of events at each time instance using a local limit theorem for multinomial distributions in Richter (1956). The general approach has first been developed in Kan (2005).