Asymptotic Probability Density Research Articles

On the Horizontal Divergence Asymmetry in the Gulf of Mexico

Due to the geostrophic balance, horizontal divergence-free is often assumed when analyzing large-scale oceanic flows. However, the geostrophic balance is a leading-order approximation. We investigate the statistical feature of weak horizontal compressibility in the Gulf of Mexico by analyzing drifter data (the Grand LAgrangian Deployment (GLAD) experiment and the LAgrangian Submesoscale ExpeRiment (LASER)) based on the asymptotic probability density function of the angle between velocity and acceleration difference vectors in a strain-dominant model. The results reveal a notable divergence at scales between 10 km and 300 km, which is stronger in winter (LASER) than in summer (GLAD). We conjecture that the divergence is induced by wind stress with its curl parallel to the Earth’s rotation.

Trajectories of square lattice staircase polygons

The distribution of monomers in a coating of grafted and adsorbing polymers is modelled using a grafted staircase polygon in the square lattice. The adsorbing staircase polygon consists of a bottom and a top lattice path (branches) and the asymptotic probability density that vertices in the lattice are occupied by these paths is determined. This is a model consisting of two linear polymers grafted to a hard wall in a coating. The probability that either the bottom path, or the top path, of a staircase polygon passes through a lattice site with coordinates , for 0 < ϵ < 1 and δ ≥ 0, is determined asymptotically as n → ∞ . This gives the probability density of vertices in the staircase polygon in the scaling limit (as n → ∞ ): = The densities of other cases, grafted and adsorbed, and at the adsorption critical point (or special point), are also determined. In these cases the most likely and mean trajectories of the paths are determined. The results show that low density regions in the distribution induce entropic forces on test particles which confine them next to the hard wall, or between branches of the staircase polygon. These results give qualitative mechanisms for the stabilisation of a drug particle confined to a polymer coating in a drug delivery system such as a drug-eluding stent covered by a grafted polymer.

Vibration Analysis of Driving-Point System with Uncertainties Using Polynomial Chaos Expansion

A vibration transfer analysis method based on polynomial chaos expansion (PCE) is proposed in this study and is used to analyze the stochastic dynamic compliance of uncertain systems with the Gaussian distribution. The random dynamic compliance is established by utilizing mode superposition on the system as the parameters of system uncertainties are regarded as input variables. Considering the asymptotic probability density function of mode shape, the dynamic compliance is decomposed into the mean of mode shape and the subsystem represented as an orthogonal polynomial expansion. Following this, the vibration transmission analysis approach is proposed for the random vibration. Results of a numerical simulation carried out employing the PCE approach show that broad-band spectrum analysis is more effective than narrow-band spectrum analysis because the former jump of the dynamic compliance amplitude is weakened. This proposed approach is valid and feasible, but since broad-band spectrum analysis loses some important information about the random vibration, both the aforementioned processes need to simultaneously be applied to analyze the random vibration transmission of low-medium frequency systems.

Transient quantum beats, Rabi oscillations, and delay time of modulated matter waves

Transient phenomena of phase modulated cut-off wavepackets are explored by deriving an exact general solution to Schr\"odinger's equation for finite range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual \textit{self-induced two-level system} with energies $E_{+}$, and $E_{-}$, due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi-oscillations characterized by a frequency $\Omega=(E_{+}-E_{-})/\hbar$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a \textit{quantum beat} effect with a beating frequency, $\Omega$. We also find a regime characterized by a \textit{time-diffraction} phenomenon that allows to measure unambiguously the delay-time, which can be described by an exact analytical formula. It is found that the delay-time agrees with the phase-time only for the case of strictly monochromatic waves.

A Low-Complexity AEPDF-assisted Precoding Scheme for Massive MIMO Systems with Transmit Antenna Correlation

Multi-user massive multiple-input multiple-output (MU M-MIMO) system employs array of hundred (or thousand) antennas and precoding/equalization capability at the base station (BS) has been adopted in recent 5G new radio (5G NR) standards. By taking advantage of the multiplexing, diversity and antenna gain inherited from this versatile system configuration, unprecedented enhancement in capacity, spectral efficiency and signal integrity can be realized. While many MU M-MIMO system related issues such as pilot contamination, channel estimation and precoding/detection algorithms have been extensively examined by research communities, the investigation of a low-complexity, effective precoding scheme aims to address performance degradation due to spatial correlation (i.e., packing vast number of antennas in a limited physical space) remained scarce. In this paper, an iterative precoding scheme based on Chebyshev acceleration is proposed for one-sided Kronecker model (i.e., Rayleigh flat fading channel with transmit (Tx) antenna correlation), often encountered in uniform linear array (ULA) configuration. From the asymptotic eigenvalue probability density function (AEPDF) constructed for the non-commutative Kronecker channel, near-optimal recurrence coefficients, initial and final estimates of the proposed Chebyshev iterative method can be obtained free from expensive eigenvalue computation. Simulation shows that the proposed algorithm achieves outstanding BER in all correlation scenarios (i.e., from low to high) while exhibiting much lower complexity compared to other state-of-the-art precoding schemes.

Estimation of reliability of multicomponent stress-strength inverted exponentiated Rayleigh model

ABSTRACTIn this paper, we consider the estimation of the multicomponent reliability by assuming the inverted exponentiated Rayleigh distribution. Both stress and strength are assumed to have an inverted exponentiated Rayleigh distribution with common scale parameter. The random variable representing the stress experienced by the system and representing the strength of system available to overcome the stress. The system works flawlessly only if at least out of strength variables exceed the random stress. The multicomponent reliability of the system is given by . We estimate by using frequentist and Bayesian approaches. Bayes estimates of have been obtained by using Markov Chain Monte Carlo methods since joint posteriors of the parameters does not have the explicit forms. We also construct asymptotic and highest probability density credible intervals for . The behavior of the proposed estimators is studied on the basis of estimated risks through Monte Carlo simulations. Finally, a data set is analyzed for illustrative purposes.Abbreviations: PDF: Probability Density Function; CDF: Cumulative Density Function; IER: Inverted Exponentiated Rayleigh; MLE: Maximum likelihood estimators; HPD: Highest Posterior Density; UBT: Upside down Bathtub; HNC: Head and Neck Cancer data; MCMC: Monte-Carlo Markov Chains; CP: Coverage Probability; KS: Kolmogorov - Smirnov.

A Note on “Limit Distributions of Self-Normalized Sums” Using Cauchy-Generated Samples

In this case study, we would like to illustrate the utility of characteristic functions, using an example of a sample statistic defined for samples from Cauchy distribution. The derivation of the corresponding asymptotic probability density function is based on [1], elaborating and expanding the individual steps of their presentation, and including a small extension; our reason for such a plagiarism is to make the technique, its mathematical tools and ingenious arguments available to the widest possible audience.

Algorithm of micro-seismic source localization based on asymptotic probability distribution of phase difference between two random stationary Gaussian processes

The article is devoted to the problem of estimating the coordinates of micro-seismic sources (localizing the sources) using multichannel data recorded by a surface seismic array. A new statistical algorithm is proposed for the source localization, which is mainly based on the phases of the Discrete Finite Fourier Transforms of the array sensor seismograms. This algorithm was constructed using the Maximum Likelihood concept under the following constraints: (a) noise components of the array seismograms are statistically independent stationary Gaussian processes with different power spectral densities; (b) the signal-to-noise ratios in the array seismograms are small, but the duration of signals generated by a micro-seismic source in the sensors is quite large; (c) the time function of a micro-seismic source can be approximated by a stationary Gaussian random process. The asymptotic probability density function was obtained in the paper for the phase differences of two Gaussian stationary random time series. This function provided a theoretical basis for constructing the new statistical phase algorithm. The algorithm requires evaluation of the coherence functions for all pairs of the sensor seismograms. For this reason, it inquires more calculations for the source localization than the known phase algorithms. But Monte Carlo simulation has shown that the new phase algorithm provides a more accurate estimation of micro-seismic source coordinates compared to the most popular phase algorithm.

On the distribution of the maximum value of the characteristic polynomial of GUE random matrices

Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of as and . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of DN (x).

On the Statistical Properties of Capacity Outage Intervals in OSTBC-MIMO Rayleigh Fading Channels

This paper deals mainly with the study of the asymptotic probability density functions (PDFs) of the outage durations of the instantaneous capacity of orthogonal space-time block code (OSTBC) multiple-input multiple-output (MIMO) systems over Rayleigh channels. Drawing upon known statistical properties for the asymptotic behavior of chi-squared processes at low levels, we provide approximate solutions for the PDF, the cumulative distribution function (CDF), and the $k$ th-order moments of the outage intervals of the underlying capacity processes. Then, as an application of the derived PDF, the performance assessment of capacity simulators is reported. Following this, we introduce the newly defined average-to-trough capacity ratio (ATCR) metric, and investigate its PDF and CDF. The validity of the derived results is confirmed by specializing them to already known ones, although they have been obtained by using a rather different approach. The accuracy and applicability of the asymptotic analysis to practical values of capacity thresholds have been examined by means of computer simulations. The theoretical results presented contribute to the statistical characterization of outage events, the study of which has mainly been limited to the analysis of the probability, the occurrence rate, and the average duration of outages.

Reduced Mean-Square Error Quadratic Inverse Spectrum Estimator

A new spectrum estimator is introduced. The new estimator exploits quadratic-inverse theory to attain improved mean-square error performance over the standard multitaper spectrum estimators. The standard, non-adaptive, eigenvalue weighted multitaper estimator is obtained by averaging a high-resolution inconsistent spectrum estimator over the estimator bandwidth. The improved performance of the proposed estimator results from replacing this average with a weighted average computed in the space spanned by the quadratic-inverse basis. The weighting, determined analytically, is chosen such that the resulting estimator minimizes the sum of the variance and the square of the in-band bias; neglecting bias due to spectral leakage and potential bias due to the possible incompleteness of the quadratic-inverse basis. For a white spectrum the neglected bias is found to be as small as that of a standard, non-adaptive multitaper spectrum estimator. The relative reduction of the mean-square error of the proposed spectrum estimator is validated by simulation for an ARMA(4,2) process, and results in a typical mean-square error reduction of 5% for large time-bandwidth parameters and 20% for a time-bandwidth parameter of four, when compared to the non-adaptive, non-eigenvalue weighted multitaper estimator. When compared to the adaptive multitaper spectrum estimator, larger mean-square error improvements are attainable. An expression for the theoretical probability density function for the proposed estimator is given. It is found to be as accurate as the asymptotic probability density function for the standard multitaper estimator.

Normalized structure factor analysis with theCentroMKprogram

Statistical analysis of the normalized structure factorEis important during space-group determination. Several approaches to solve this problem have been described in the literature. In this paper, the most popular approach, the ideal asymptotic probability density function developed by Wilson, is compared with the more accurate exact probability density functions described by Shmueli and co-workers. Furthermore, a new computer program,CentroMK, for normalized structure factor analysis, is presented. The program is capable of plotting histograms of the normalized structure factors and exact probability density functions. Moreover, the program calculates five estimators helpful during the space-group determination: 〈|E|〉, 〈|E2 − 1|〉, %E&gt; 2, %E&lt; 0.25 and the discrepancyRfunction. The two approaches and the error rates of the five listed estimators are compared for nearly 30 600 crystal structures obtained fromActa Crystallographica Section E.It is shown that within a space group the means 〈|E|〉 and 〈|E2 − 1|〉 of real crystal structures show high variability. The comparison shows that decisions based on the exact probability density function are more accurate, the computing time is reasonable, and estimators 〈|E|〉, %E&lt; 0.25 andRare the most accurate and should be preferred during space-group determination.

The Asymptotic Distribution of Maxima of Independent and Identically Distributed Sums of Correlated or Non-Identical Gamma Random Variables and its Applications

In this paper, we show that the asymptotic probability density function (pdf) of the maxima of n independent and identically distributed (i.i.d.) sums of independent non-identically (i.n.i.d.) distributed gamma random variables (RVs) is a Gumbel pdf using Extreme Value Theory (EVT). We will also show that the asymptotic pdf of the maxima of n i.i.d. sums of correlated gamma RVs is a Gumbel pdf. Some applications in wireless communication are discussed where the maxima of n i.i.d. sums of correlated gamma RVs and maxima of n i.i.d. sums of i.n.i.d. gamma RVs arise. We discuss the utility of our results in the context of these applications.

Asymptotic and Finite User PER Analysis of Successive Interference Cancellation for DS-CDMA

An expression is derived for the average Packet Error Rate (PER) of a Successive Interference Canceller (SIC) for DS-CDMA when the number of users asymptotically tends to infinity. The asymptotic probability density function of the interference power is governed by a Fokker-Planck differential equation with drift and (asymptotically vanishing) diffusion depending on the PER function of the adopted forward error correcting code (FEC). In addition to the asymptotic solution for the PER, a particle-based algorithm is also developed for computing efficiently the PER in the finite user case.

Lévy flights in confining potentials

We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump-type processes are considered: those driven by Langevin equation with Lévy noise and those, named topological Lévy processes (occurring in systems with topological complexity such as folded polymers or complex networks), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary probability density, even if their detailed dynamical behavior look different. This near-equilibrium observation seems to be generic to a broad class of Lévy noise-driven processes, such as e.g., superdiffusion on folded polymers, geophysical flows, and even climatic changes.

On asymptotic posterior normality for controlled branching processes

For a controlled branching process (CBP) with offspring distribution belonging to the power series family, the asymptotic normality of the posterior distribution of the basic parameter and the offspring mean is proved. As practical applications, we calculate asymptotic high probability density credibility sets for the offspring mean and we provide a rule to make inference about the value of this parameter. Moreover, the asymptotic posterior normality of the respective parameters of two classical branching models, namely the standard Galton–Watson process and the Galton–Watson process with immigration, is derived as particular cases of the CBP.

STOKES' DRIFT AND HYPERSENSITIVE RESPONSE WITH DICHOTOMOUS MARKOV NOISE

Stochastic Stokes' drift and hypersensitive transport driven by dichotomous noise are theoretically investigated. Explicit mathematical expressions for the asymptotic probability density and drift velocity are derived including the situation in which particles cross unstable fixed points. The results are confirmed by numerical simulations.

Stochastic resonance in biological nonlinear evolution models.

We investigate stochastic resonance in the nonlinear, one-dimensional Fisher-Eigen model (FEM), which represents an archetypal model for biological evolution based on a global coupling scheme. In doing so we consider different periodically driven fitness functions which govern the evolution of a biological phenotype population. For the case of a simple harmonic fitness function we are able to derive the exact analytic solution for the asymptotic probability density. A distinct feature of this solution is a phase lag between the driving signal and the linear response of the system. Furthermore, for more complex systems a general perturbation theory (linear response approximation) is put forward. Using the latter approach, we investigate stochastic resonance in terms of the spectral amplification measure for a quadratic, a quartic single-peaked, and for a bistable fitness function. Our analytical results are also compared with those of detailed numerical simulations. Our findings vindicate that stochastic resonance does occur in these nonlinear, globally coupled biological systems.

Drift by dichotomous Markov noise.

We derive explicit results for the asymptotic probability density and drift velocity in systems driven by dichotomous Markov noise, including the situation in which the asymptotic dynamics crosses unstable fixed points. The results are illustrated on the problem of the rocking ratchet.

Asymptotic probability density of nonlinear phase noise.

The asymptotic probability density of nonlinear phase noise, often called the Gordon-Mollenauer effect, is derived analytically when the number of fiber spans is large. Nonlinear phase noise is the summation of infinitely many independently distributed noncentral chi2 random variables with two degrees of freedom. The mean and the standard deviation of those random variables are both proportional to the square of the reciprocal of all odd natural numbers. Nonlinear phase noise can also be accurately modeled as the summation of a noncentral chi2 random variable with two degrees of freedom and a Gaussian random variable.