Series Of Random Variables Research Articles

ЧАСОВА ДИНАМІКА, РОЗПОДІЛ ТА СУЧАСНІ ЗМІНИ ПАРАМЕТРІВ ТА ХАРАКТЕРИСТИК ВЕСНЯНОГО ВОДОПІЛЛЯ НА Р. ДЕСНА – М. ЧЕРНІГІВ

The article analyzes the long-term variability of the main parameters and characteristics of spring irrigation on the Desna River – the city of Chernihiv. The main methods used are the methods of mathematical statistics, systematization, generalization and analysis of initial data and obtained results. To carry out the research, a bank of average daily water consumption of the Desna River – Chernihiv city for the observation period 1895–2020 (126 years) was created. The analysis of the initial information was carried out and the following series of random variables were formed over a multi-year period: a series of dates of the start of spring irrigation (in the number of days from January 1); range of maximum water consumption of spring irrigation (in m3/c); a number of dates of maxima (in the number of days from January 1); a number of dates of the end of spring watering (in the number of days from January 1); range of duration of spring irrigation (in number of days). The method of comparison is used to identify modern changes. To do this, all the above-mentioned series of parameters and characteristics of spring irrigation are divided into two periods equal in number of years – 1895-1957 (63 years) and 1958-2020 (63 years). The multi-year average values (norms), coefficients of variation and asymmetry of spring irrigation start dates, maximum water flows during spring irrigation and their onset dates, end dates and duration of spring irrigation on the Desna River – Chernihiv city were determined for all studied periods. Constructed histograms of relative frequencies in the form of partial intervals of all studied indicators and characteristics of the spring irrigation of the Desna River – the city of Chernihiv demonstrated the peculiarities of their distribution during the observation period 1895–2020.

Another objection to the homogeneously weighted moving average control chart

AbstractIt was recently demonstrated that homogeneously weighted moving average (HWMA) control charts are improper devices for an online detection of changes within a series of random variables. However, another attempt was made to bail out the concept underlying these charts. Here we want to re‐emphasize that HWMA charts are dispensable by rebutting the arguments of the latter contribution.

Точность и статистическое качество решения обратных задач фотограмметрии

The issues of evaluating the results of solving inverse photogrammetry matters are considered. The necessary definitions are given and the key concepts of mathematical statistics theory important for equation calculations are given. The typical set of photogrammetric measurements was determined. The importance of the correct division of surveys into series of random variables within the weighted least squares method is noted. The indicators of the equalization accuracy by the remanent residuals of indirect measurements and by the elements of the covariance matrix of the equalized parameters are considered. It is shown that the accuracy estimate is insufficient for an objective representation of the found parameters proximity to the corresponding values that actually existed during the survey. It is indicated that the reliability of the parameters found can be assessed by an indicator of the statistical quality of the equalization. A brief conclusion of the analytical expression for this indicator is given. A modification of the statistical quality of equalization indicator adopted in geodesy and analytical photogrammetry, which provides for the use of the mean square error of the weight unit, is considered. The main factors influencing the improvement of equalization statistical quality are considered and brief recommendations aimed at achieving this goal are given. The different affecting the indicator of the mentioned statistical quality of direct and indirect measurements is theoretically justified.

Robust Design Optimization of Car-Door Structures with Spatially Varied Material Uncertainties

This paper presents an effective approach for robust design optimization of car-door structures with spatially varied material properties. This spatially varied material property causes structural response quantities; for example, the natural frequency and the lateral stiffness coefficient become random variables. In this regard, the Karhunen-Loève expansion is first used to represent the elastic modulus and the mass density random fields as a series of random variables. Then, a stochastic finite-element model is formulated for uncertainty quantification of the car-door structure. Combined with a polynomial-based response surface model to mimic the true performance indicator, this allows one to efficiently evaluate probability constraints for the robust design optimization of the uncertain car-door structure. In numerical simulations, design variables of the uncertain car-door structure are defined as thickness values of the tailor rolled blank structure at various regions, whereas multiple design objectives are formulated via the structural weight, the first-order natural frequency, and the lateral stiffness coefficient. Results have shown that the mean value of performance indicators can be generally improved, whereas the response variance is further minimized to archive the robust design objective. The probability-based constraint is significant to relate the Pareto optimum set to the targeted structural safety level. The proposed approach is simple, suggesting an attractive tool for the robust design optimization of car-door structures with spatially varied material uncertainties.

Stochastic Dynamic Model of Sulfate Corrosion Reactions in Concrete Materials considering the Effects of Colored Gaussian Noises

The corrosion reactions in concrete materials subjected to external environment attack can lead to the deterioration of concrete. However, the effects of internal fluctuations on the corrosion reaction process have not been reported in current studies on damage of concrete materials. To comprehensively describe the effects of internal fluctuations, the stochastic dynamic model of corrosion reactions in concrete materials subjected to sulfate attack is established based on the law of mass conservation and random process theory, in which internal fluctuations and the parameters of the chemical system are, respectively, regarded as colored Gaussian noises and a series of random variables. An experiment of sulfate corrosion reactions in concrete material is carried out to verify the effectiveness of the proposed method. Furthermore, the effects of variations of the initial reactant concentrations on the concentration evolution processes of the corrosion products are investigated. Results show that the stochastic dynamical responses of the corrosion reactions in concrete can be comprehensively investigated by the proposed stochastic mathematical model; the probabilistic information of the corrosion products can also be obtained conveniently. The concentration evolution process of sulfate corrosion products is a random process. The experimental data are only some samples of the random process. Concentrations of the corrosion products in concrete materials significantly fluctuate with the variations of the initial reactant concentrations.

Statistical evaluation of performance impact of manufacturing variability by an adjoint method

Statistical evaluations of the aerodynamic performance changes due to the manufacturing variability for a turbine blade using the first order (FO) and second order (SO) sensitivities are presented in the paper. On the premise that the geometric variation at each scanning point on the blade meets a standard normal distribution, a modeling method taking into account the spatial correlations is used to produce the contour of geometric variations, the basis modes of which are extracted by principal component analysis. Then the geometric variations of any manufactured blades can be described by a series of random variables regarding as the weights of the basis modes. Calculations of the FO and SO sensitivities by using a continuous adjoint method are firstly introduced in detail. The sensitivities of mass flow rate and adiabatic efficiency of the turbine blade to a finite number of primary basis modes are then calculated and sensitivity validations are presented, revealing the significant improvements on performance evaluations by the SO sensitivities. Finally, the statistics of performance changes due to the manufacturing variability are evaluated by Monte Carlo simulations with different probability density functions given for the weights of basis modes. The results demonstrate that by the FO sensitivities, the performance exhibits almost linear changes versus the manufacturing tolerance, while by the SO sensitivities the nonlinear dependence of performance on the geometric variations can be accurately evaluated.

On Absolute Convergence of Series of Random Variables Almost Surely

New conditions are found for the absolute convergence of series of random variables almost surely. The results contain no independence assumptions. A generalization in analytical terms is obtained. Bibliography: 5 titles.

A time-variant reliability analysis method for structural systems based on stochastic process discretization

In this paper, we propose a new method for analyzing time-variant system reliability based on stochastic process discretization, which provides an effective tool for reliability design of many relatively complex structures considering the whole lifecycle. Within a design lifetime, the stochastic process is discretized into a series of random variables, and meanwhile, we can derive a time-invariant limit-state function in each time interval; the discretized random variables from the stochastic processes and the original random variables are transformed to the independent normal space, and a conventional time-invariant system reliability problem is derived through the linearization to each discretized limit-state functions; by solving this time-invariant system reliability problem, we can obtain the structural reliability or failure probability within the design lifetime. Finally, in this paper, we provide four numerical examples to verify the effectiveness of the method.

Optimal filtering for uncertain systems with stochastic nonlinearities, correlated noises and missing measurements

The globally optimal recursive filtering problem is studied for a class of systems with random parameter matrices, stochastic nonlinearities, correlated noises and missing measurements. The stochastic nonlinearities are presented in the system model to reflect multiplicative random disturbances, and the additive noises, process noise and measurement noise, are assumed to be one-step autocorrelated as well as two-step cross-correlated. A series of random variables is introduced as the missing rates governing the intermittent measurement losses caused by unfavorable network conditions. The aim of the addressed filtering problem is to design an optimal recursive filter for the uncertain systems based on an innovation approach such that the filtering error is globally minimized at each sampling time. A numerical simulation example is provided to illustrate the effectiveness and applicability of the proposed algorithm.

On Almost Sure Convergence of Series of Random Variables Irrespective of Their Joint Distributions

In this article, necessary and sufficient conditions are presented for a series of random variables to converge absolutely almost surely irrespective of their joint distributions. The summands are not assumed to be integrable. Illustrative examples and counterexamples are presented.

Spectrum Harvesting and Sharing in Multi-Hop CRNs Under Uncertain Spectrum Supply

The essential impediment to apply cognitive radio (CR) technology for efficient spectrum utilization lies in the uncertainty of licensed spectrum supply. In this paper, we propose a novel architecture for spectrum harvesting and sharing, and investigate the joint routing and frequency scheduling problem in multi-hop cognitive radio networks (CRNs) under uncertain spectrum supply. We introduce a new service provider, Secondary Service Provider (SSP), to facilitate the accessing for secondary users (SUs). We model the vacancy of available bands with a series of random variables, and mathematically describe the corresponding frequency scheduling and flow routing constraints. From the SSP's point of view, we characterize the CRN performance with a pair of parameters (α, β), and present an optimization problem to minimize the required network-wide spectrum resource at the (α,β) level. Given that (α, β) level is specified, we obtain a lower bound for the optimization problem and develop a threshold based coarse-grained fixing algorithm for a feasible solution. Simulation results show that (i) for any (α,β) level, the proposed algorithm provides a near-optimal solution to the formulated NP-hard problem, and (ii) the (α,β) based solution is better than the expected bandwidth based one in terms of blocking ratio and spectrum utilization in multi-hop CRNs.

Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-lipschitz conditions

In the paper, stochastic differential equations with random impulses and Markovian switching are brought forward, where the so-called random impulse means that impulse ranges are driven by a series of random variables and impulse times are a random sequence, so these equations extend stochastic differential equations with jumps and Markovian switching. Then the existence and uniqueness of solutions to such equations are investigated by employing the Bihari inequality under non-Lipschtiz conditions.

Seismic Reliability of Long-Span Bridges Based on the First Order Reliability Method

In this paper, first order reliability method (FORM) is generalized to the seismic reliability analysis of long-span bridges and the first excursion probability of a single tower cable-stayed bridge is studied. The seismic motivation is firstly dispersed as a series of random variables. Then the cross velocities on the dispersed time points can be obtained by solving the motivation of the design checking points with FORM. The upper bound of first excursion probability is also obtained by integrating the cross velocities at different time. The results show that the seismic reliability analysis method based on the FORM is feasible and effective to solve the first excursion probability problem. The single tower cable-stayed bridge studied in this paper has a higher reliability under strong seismic.

The Sequential Occupancy Problem through Group Throwing of Indistinguishable Balls

The occupancy problem is generalized to the case where instead of throwing one ball at a time, a fixed size group of indistinguishable balls are distributed sequentially into cells. Bose-Einstein statistics is used for analyzing the distribution of the waiting time until each cell is occupied by at least one ball. Each trial is classified according to its jump size, i.e. the number of newly occupied cells. We propose an approach to decompose the occupancy and filling processes in terms of the jumps sizes using a multi-dimensional representation. A set of recursive equations is built in order to obtain the joint generating probability function of a series of random variables, each of which denotes the number of trials for a given jump size that occurred during the filling process. As a special case, the joint probability function of these random variables is obtained.

Simulation of Random Crack Generation in Concrete Members with Uniform Stress Fields

Abstract The randomness of strength and deformation of concrete material is serious and should be considered both in theoretical analyses such as Finite Element Methods and engineering practice, specially for those structural members with a uniform stress field, where stresses or strains are approximately the same under loading. A mathematical approach of producing a series of random variables of the ultimate tensile strain in concrete is proposed to describe the randomness ofconcrete deformation. With reinforced concrete finite elements a real model calculation method is found for the randomness of initial cracks determined by a minimum tension strain within the uniform stress fields of concrete members. The proposed methods in our paper have as aim to improve the existing method used by FEM and other relative approaches, which normally pay less attention to randomness with consequences that may possibly differ from testing or practice. The method and sample computation as indicated is meaningful and comply with testing and engineering practice.

Drought Analysis Based on a Marked Cluster Poisson Model

Abstract This paper presents an operational definition of drought events based on an “excess over threshold” approach applied on rainfall series and develops a stochastic model for describing droughts defined in that way. The model consists of a Poisson cluster process to represent drought occurrence and a marked process composed of three series of random variables (duration, deficit, and maximum intensity) to describe drought severity; it is theoretically justified, and adequate procedures to check its validity are suggested and applied on five Spanish rainfall series. Useful parameters for the design and planning of water resource systems, together with their confidence intervals, are estimated from the model.

Density dependence tests, are they?

A large number of time series of abundances of insects and birds from a variety of data sets were submitted to a new density dependence test. The results varied enormously between data sets, but the relation between the frequency of statistically significant density dependence (SSDD) and the length of the series was similar to that of the power curve of the test, making the results consistent with the hypothesis of the density-dependent model being universally applicable throughout the data used. Pest and non-pest species did not differ in the incidence of SSDD. The more sampling error present in the data, the higher the percentages of SSDD. This was expected given that the power of the test increases with increasing sampling error in the data. Many of the data used here, as well as in the literature, clearly violate the basic assumption of the test that the organism concerned should be univoltine and semelparous. Yet the incidence of SSDD was the same in univoltine as in bi/polyvoltine species and the same in semelparous organisms as in birds that are reproductively active in more than one year. The seasonal migrant Autographa gamma in Britain and Czechoslovakia and even rainfall data were found to have SSDD. Statistical significance, however, does not automatically lead to the conclusion of density-dependent regulation. Any series of random variables which are in a stochastic equilibrium, such as a series of independent, identically distributed, random variables, is typically described better by the alternative (density-dependent) model than by the null (density-independent) model. Significant test results were often obtained with sloppy data, with data that clearly violate the basic assumptions of the test and with other data where an interpretation of the results in terms of densitydependent regulation was absurd. Given the fact that other explanations have to be found for significant test results for all these cases, mechanisms other than regulation may very well be applicable too where the data are entirely appropriate for the test. The test is simply a data-based choice between a model without and one with a stochastic equilibrium. A time series as such does not contain any information about the causes of the fluctuation pattern, so that one cannot expect statistics to produce such information from that time series. A significant test result using suitable data is entirely consistent with the hypothesis of density-dependent regulation, but also with any other suitable hypotheses. Because the test results were generally consistent with the hypothesis of a universal applicability of the density-dependence model, a negative test result may only mean that the time series was not long enough for the density dependence that was present to become statistically significant. Positive results are difficult to interpret, but so are negative results. A final decision needs to be based not so much on the test result as on much detailed information about the population concerned. Because the "density-dependence test" does not test for the presence of the mechanism of density-dependent regulation and because of the loaded, multiple meanings of the term "density-dependence", calling the test a "test of statistical density dependence" may be preferable.

Mathematical Fun with the Compound Binomial Process

AbstractThe compound binomial model is a discrete time analogue (or approximation) of the compound Poisson model of classical risk theory. In this paper, several results are derived for the probability of ruin as well as for the joint distribution of the surpluses immediately before and at ruin. The starting point of the probabilistic arguments are two series of random variables with a surprisingly simple expectation (Theorem 1) and a more classical result of the theory of random walks (Theorem 2) that is best proved by a martingale argument.

Random symmetric polynomials

Let {Q(n)(x1,...,xn)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {Xn1,...,Xnn}, n⩾ k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Yn=Q(n)(Xn1, ... Xnn), n⩾ k One obtains limit theorems for the sequence Yn, under very general assumptions.

Rate of convergence of series of random variables

Rate of convergence of series of random variables