A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

A central limit theorem for functions of stationary max-stable random fields on [formula omitted

Two kinds of estimation variance functions for estimating a local average (LA) of a stationary (homogeneous) random field (RF) are derived. One is local estimation (LE) and the other is general estimation (GE) of LA. The former is for estimating LA at the observation location, and the latter is for obtaining LA at any arbitrary location within the RF. The geotechnical implications of these two estimations are that LE is for estimating LA of geotechnical parameters at the spot where the investigations are made, whereas GE is for estimating LA at any arbitrary location within the same layer. The behavior of the two estimation variance functions differs greatly, controlled by the number of observations (i.e. sample size) and the normalized layer thickness (layer thickness divided by autocorrelation distance of RF). Based on the derived estimation variance functions, methods for determining reliable characteristic values of geotechnical parameters and necessary sample size are proposed. The methods are based on the same framework as that of the traditional statistical theory, i.e. confidence interval of estimated parameters. However, the assumption of independently and identically distributed (i.i.d.) samples in the traditional statistical theory is replaced by the assumption of correlated samples from a stationary RF. The results obtained from the proposed methods for LE and GE differ from each other as well as from the traditional results, which has significant implications for geotechnical parameter estimation in geotechnical engineering practice.

For perfect surface roughness description is not enough to know characteristics of surface profile. It is necessary to use topography methods, so called microtopography. Thereby, surface roughness in microtopographycal understanding must be described with three coordinates, whose in Cartesian coordinates system compose point under consideration height h, abscissa and ordinate, determines point position in the plane. Most efficient methods in irregular surface roughness research are random function theory methods. Therefore, microtopography, analogically to profile, may consider as random function, but two dimensional function, i.e. two variable x and y random field h(x,y). From analogy with random process, random field can be normal – ordinates are distributed by normal (Gaussian) distribution. Moreover, random field can be homogeneous and heterogeneous. Random field is deemed homogeneous if its mean value is discretionary and correlation function depends only from distance between surface points. Important characteristic of random field is correlation function, whose depends of two variables t1 and t2 – orthogonal Cartesian coordinates of vector t. Random field is homogeneous and isotropic when its characteristics are equivalent in any direction. There are three types of surface anisotropy: • General event of surface anisotropy. Characteristics of this event roughness parameters are depend of surface split direction. • Surface roughness with direct anisotropy. Those surfaces are with typical traces of tool and they proper two mutually perpendicular surface roughness directions. • Extended anisotropy area – special event of anisotropy roughness. Of analytical opinion, gainfully anisotropy roughness see as extended occasional isotropy area. This let easy cross from anisotropy surface to isotropy and contrariwise, thereby embrace amount class of surface roughness. Let’s formulate microtopography model of rough surface [1]. Surface roughness is described with homogeneous normal random field h(x,y) that has uninterrupted correlation function and uninterrupted deriviates. We may consider that E{h(x,y)}=0. The mean random field value is plane called mean plane. For describing random field we must know mathematical expectation and field correlation function, what in fact reduces on determining dispersion and rationed correlation function r(t1, t2). Homogeneous random field dispersion D{h} doesn’t depends of direction and can be founded in any surface split. Given model of rough surface let inspect surfaces produced by abrasive instruments and friction surfaces after wear-in period.

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

The problem of sequentially scanning and predicting data arranged in a multidimensional array is considered. We introduce the notion of a scandictor, which is any scheme for the sequential scanning and prediction of such multidimensional data. The scandictability of any finite (probabilistic) data array is defined as the best achievable expected performance on that array. The scandictability of any (spatially) stationary random field on /spl Zopf//sup m/ is defined as the limit of its scandictability on finite boxes (subsets of /spl Zopf//sup m/), as their edges become large. The limit is shown to exist for any stationary field, and essentially be independent of the ratios between the box dimensions. Fundamental limitations on scandiction performance in both the probabilistic and the deterministic settings are characterized for the family of difference loss functions. We find that any stochastic process or random field that can be generated autoregressively with a maximum-entropy innovation process is optimally scandicted the way it was generated. These results are specialized for cases of particular interest. The scandictability of any stationary Gaussian field under the squared-error loss function is given a single-letter expression in terms of its spectral measure and is shown to be attained by the raster scan. For a family of binary Markov random fields (MRFs), the scandictability under the Hamming distortion measure is fully characterized.

Because of digitalization, many organizations possess large datasets. Furthermore, measurement data are often not normally distributed. However, when samples are sufficiently large, the central limit theorem may be used for the sample means. In this article, we evaluate the use of the central limit theorem for various distributions and sample sizes, as well as its effects on the performance of a Shewhart control chart for these large non‐normally distributed datasets. To this end, we use the sample means as individual observations and a Shewhart control chart for individual observations to monitor processes. We study the unconditional performance, expressed as the expectation of the in‐control average run length (ARL), as well as the conditional performance, expressed as the probability that the control chart based on estimated parameters will have a lower in‐control ARL than a specified desired in‐control ARL. We use recently developed factors to correct the control limits to obtain a specified conditional or unconditional in‐control performance. The results in this paper indicate that the control chart should be applied with caution, even with large sample sizes.

The Central Limit Theorem (CLT) plays a foundational role in statistical inference, often serving as the rationale for assuming a normal approximation of the sample mean. Yet, the pace at which this assumption becomes valid is influenced by the shape of the parent distribution, especially its skewness. This research quantifies the minimum number of observations required for the mean of samples drawn from skewed, non-normal distributions specifically Gamma, Poisson, Binomial, and Beta to achieve a satisfactory normal approximation. We implemented a Monte Carlo simulation and applied both the Shapiro-Wilk and Kolmogorov-Smirnov tests to assess the adequacy of the normal approximation. Results indicate a nonlinear association between the degree of skewness and the sample size required for acceptable normal approximation. For distributions with mild asymmetry (|skewness| < 0.5), 20 samples often suffice, whereas more heavily skewed distributions (|skewness| ≥ 2.5) may necessitate sample sizes beyond 100. These findings call into question the blanket use of the "n ≥ 30" heuristic and suggest more tailored guidelines are necessary for accurate inference. A graphical overview summarizes these results across the examined distributional families, offering clear guidance for applied researchers working with non-normal data.

Every two-dimensional incompressible flow follows the level lines of some scalar function $\psi$ on $\mathbb{R}^2$; transport properties of the flow depend in part on whether all level lines are bounded. We study the structure of the level lines when $\psi$ is a stationary random field. We show that under mild hypotheses there is only one possible alternative to bounded level lines: the "treelike" random fields, which, for some interval of values of a, have a unique unbounded level line at each level a, with this line "winding through every region of the plane." If the random field has the FKG property, then only bounded level lines are possible. For stationary $C^2$ Gaussian random fields with covariance function decaying to 0 at $\infty$, the treelike property is the only alternative to bounded level lines provided the density of the absolutely continuous part of the spectral measure decays at $\infty$ "slower than exponentially," and only bounded level lines are possible if the covariance function is nonnegative.

We define multi-self-similar random fields, that is, random fields that are self-similar component-wise. We characterize them, relate them to stationary random fields using a Lamperti-type transformation and study these stationary fields. We also extend the notions of local stationarity and local stationarity reducibility to random fields. Our work is motivated by applications arising from climatological and environmental sciences. We illustrate these new concepts with the fractional Brownian sheet and the Lévy fractional Brownian random field.

Spectral representation of intrinsically stationary fields

The aim of this note is to discuss some statistical properties of the phase separation line in the 2D low-temperature Ising model. We prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation. The limiting Gaussian measure corresponds to a scaled Brownian bridge with direction dependent parameters. Up to the temperature factor, the variances of local increments of this limiting process are inversely proportional to the stiffness.

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.

Slope reliability analysis has always been a research focus for geotechnical engineers. Currently, most studies use stationary random fields to simulate the spatial variability of soil parameters. However, many field test data show that soil parameters such as cohesion and friction angle often exhibit obvious nonstationary characteristics along the burial depth direction. Moreover, the slope reliability analysis method based on Monte Carlo simulation requires a combination of the FEM and a large number of computational resources, resulting in extremely low computational efficiency. Therefore, this paper proposes a convolutional neural network (CNN)-based surrogate model for slope reliability analysis, emphasizing the nonstationary spatial variability of cohesion and friction angle. First, stationary and nonstationary random fields considering the spatial variability of shear strength parameters are generated, and sample data sets are collected using an automated computational program. Next, a CNN surrogate model is established to predict the factor of safety (FOS), which is trained using nine sample data sets containing stationary and nonstationary random fields. Finally, the model performance is validated using six slope models with different shear strength parameters. The results show that the CNN surrogate model can adapt to the data characteristics of different random fields and also shows good predictive performance in handling unseen sample data sets. The predicted FOS values show a high degree of consistency with the results calculated by the FEM, with the computation time being significantly reduced.

Phase diagram and tricritical behavior of an Ising metamagnet model in a trimodal random field

This paper introduces stationary and multi-self-similar random fields which account for stochastic volatility and have type G marginal law. The stationary random fields are constructed using volatility modulated mixed moving average (MA) fields and their probabilistic properties are discussed. Also, two methods for parameterizing the weight functions in the MA representation are presented: one method is based on Fourier techniques and aims at reproducing a given correlation structure, the other method is based on ideas from stochastic partial differential equations. Moreover, using a generalized Lamperti transform we construct volatility modulated multi-self-similar random fields which have type G distribution.