Sequence Of Variables Research Articles

On the (p,q)$(p,q)$‐type strong law of large numbers for sequences of independent random variables

AbstractLi, Qi, and Rosalsky (Trans. Amer. Math. Soc., 368 (2016), no. 1, 539–561) introduced a refinement of the Marcinkiewicz–Zygmund strong law of large numbers (SLLN), the so‐called ‐type SLLN, where and . They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: , , and . Results for the case where and remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space , but the results are new even when is the real line. Furthermore, the conditions for a sequence of random variables satisfying the ‐type SLLN are shown to provide an exact characterization of stable type p Banach spaces.

An almost sure invariance principle for some classes of non-stationary mixing sequences

In this note we (in particular) prove an almost sure invariance principle (ASIP) for non-stationary and uniformly bounded sequences of random variables which are exponentially fast ϕ-mixing. The obtained rate is of order o(Vn14+δ) for an arbitrary δ>0, where Vn is the variance of the underlying partial sums Sn. For certain classes of inhomogeneous Markov chains we also prove a vector-valued ASIP with similar rates.

The Role of Cellulose in Microbial Diversity Changes in the Soil Contaminated with Cadmium

Cadmium is an essential element for plant growth and development. Its accumulation in soil is more hazardous to human and animal health than to plants and microorganisms. A pot greenhouse experiment was conducted to determine the usability of Sinapis alba L. and Avena sativa L. for the phytoremediation of soil contaminated with cadmium and to verify cellulose viability in the remediation of soil under cadmium pressure in doses from 4 to 16 mg Cd2+ kg−1 soil d.m. (dry matter) The effect of cadmium on soil microbiome was investigated with the culture method and the variable region sequencing method. Sinapis alba L. and Avena sativa L. were found viable in the phytoremediation of soil contaminated with Cd2+. Avena sativa L. was more potent to accumulate Cd2+ in roots than Sinapis alba L. Although the fertilization of Cd2+- contaminated soil with cellulose stimulated the proliferation of microorganisms, it failed to mitigate the adverse effects of Cd2+ on bacterial diversity. Bacteria from the Sphingomonas, Sphingobium, Achromobacter, and Pseudomonas genera represented the core microbiome of the soils sown with two plant species, contaminated with Cd2+ and fertilized with cellulose. Stimulation of the growth and development of these bacteria may boost the efficacy of phytoremediation of cadmium-contaminated soils with Sinapis alba L. and Avena sativa L.

Complete Convergence for M-Pairwise Negatively Dependent Random Variables

Hsu and Robbins (1947) introduce the concept complete convergence as follows. A sequence of random variables is said to converge completely to a constant if for all The converse is true if the are independent. They also show that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erdös proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. In this paper, let be a sequence of positive constants with and be a sequence of m-pairwise negatively dependent random variables. We study the complete convergence for m-pairwise negatively dependent random variables under mild condition . Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables and also pairwise negatively dependent random variables.

On λ-statistical convergence of order α for complex uncertain sequences

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of λ-statistical convergence almost surely of order α, λ-statistical convergence in measure of order α, λ-statistical convergence in mean of order α, λ-statistical convergence in distribution of order α in complex uncertain theory. Also establish some relationships among them. The results established generalize and unify different existing notions of the concepts of convergence of sequences of complex uncertain variables.

Weighted Davis Inequalities for Martingale Square Functions

For a Hilbert space-valued martingale (f_{n}) and an adapted sequence of positive random variables (w_{n}), we show the weighted Davis-type inequality E|f0|w0+14∑n=1N|dfn|2fn∗wn≤E(fN∗wN∗).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\left( {|}f_{0}{|} w_{0} + \\frac{1}{4} \\sum _{n=1}^{N} \\frac{{|}df_{n}{|}^{2}}{f^{*}_{n}} w_{n} \\right) \\le {\\mathbb {E}}( f^{*}_{N} w^{*}_{N}). \\end{aligned}$$\\end{document}More generally, for a martingale (f_{n}) with values in a (q,delta )-uniformly convex Banach space, we show that E|f0|w0+δ∑n=1∞|dfn|q(fn∗)q-1wn≤CqE(f∗w∗).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathbb {E}}\\left( {|}f_{0}{|} w_{0} + \\delta \\sum _{n=1}^{\\infty } \\frac{{|}df_{n}{|}^{q}}{(f^{*}_{n})^{q-1}} w_{n} \\right) \\le C_{q} {\\mathbb {E}}( f^{*} w^{*}). \\end{aligned}$$\\end{document}These inequalities unify several results about the martingale square function.

Multivariable sequential model predictive control of LCL‐type grid connected inverter

The conventional single variable indirect model predictive control (SIMPC) of LCL-type three-level grid- connected inverter (GCI) is to indirectly control the grid current by controlling the output current of the inverter, which usually produces large errors and reduces the system anti-disturbance performance. Although the previously proposed multivariable weighted model predictive control (MWMPC) method of LCL-type GCI can directly control many variables, such as grid current, it requires setting the weight coefficient for each variable and involves a substantial amount of calculation. To due with these problems, this paper proposes a multivariable sequential model predictive control (MSMPC) method for LCL-type three-level GCI. The inverter output current, the filter capacitor voltage, and the grid current are gradually predicted and optimized under the variable control sequence of the LCL-type GCI mathematical model. Under the premise of achieving direct control of the variables above, this method can reduce the number of weighting coefficients and gradually contract the optimization space to reduce the computation required for multivariable prediction and optimization. This paper compares the proposed MSMPC to the conventional SIMPC and MWMPC in steady state, transient state performance, and anti-disturbance performance. The effectiveness of the proposed MSMPC is demonstrated through simulation and experimentation.

Monte Carlo simulation of present value of cash flows

This paper is concerned with financial application of Monte Carlo simulation of infinite series. In fact, present value of a cash flow is represented as infinite series. Hence, the Monte Carlo may be applied to approximate the present value. It is seen that the Monte Carlo method is sample approximation of the mean of cash flow computed at a stopping time. The stopping time, first, is considered as an up-crossing time point of sequence of uniform random variables and then it is replaced by up-crossing of macro-economic variables (called as factor) which brings the dependency structure to the problem which is modeled by auto-regressive to any things (ARTA) models. Here, two different Monte Carlo simulation methods are presented. A financial application is also given. Continuous time extensions of results are also proposed. Finally, a concluding remark section is given.

Statistical Measurement and Influencing Factors of Green Total Factor Productivity of China’s Tourism Industry Based on DEA-EBM Model

In order to explore the statistical measurement and influencing factors of green total factor productivity in tourism, this paper proposes the use of gray absolute correlation to measure the similarity between input variable sequences. Moreover, this paper obtains the key parameters of the combination of radial and nonradial in the model based on the proximity index of gray correlation and calculates the economic efficiency of the decision-making unit according to the steps. In addition, this paper combines the DEA-EBM model to carry out the statistical measurement of China’s tourism industry green total factor productivity and the analysis of influencing factors and verify it through data. Through research, it can be seen that the DEA-EBM model proposed in this paper can play an important role in the statistical measurement and analysis of influencing factors of green total factor production in the tourism industry.

Elongated Poisson-Voronoi cells in an empty half-plane

The Voronoi tessellation of a homogeneous Poisson point process in the lower half-plane of R2 gives rise to a family of vertical elongated cells in the upper half-plane. The set of edges of these cells is ruled by a Markovian branching mechanism which is asymptotically described by two sequences of iid variables which are respectively Beta and exponentially distributed. This leads to a precise description of the scaling limit of a so-called typical cell. The limit object is a random apeirogon that we name menhir in reference to the Gallic huge stones. We also deduce from the aforementioned branching mechanism that the number of vertices of a cell of height λ is asymptotically equal to 45log⁡λ.

Maximal inequalities and the strong law of large numbers for strong demimartingales

The maximal inequalities for demimartingales have been extensively studied. In this article, we establish the Chow type maximal inequality for strong demimartingales which are new sequences of random variables completely different from demimartingales. Then, we obtain the Doob type maximal inequality together with a strong law of large numbers for strong demimartingales by taking the Chow type maximal inequality as a “source result”.

A note on an average run length calculation for the EWMA and other charts

AbstractIn the last decade numerous papers have appeared in the statistical process monitoring literature where the in‐control and out‐of‐control average run length (ARL) profiles of the exponentially weighted moving average chart, and some of its variants, have been approximated under the assumption that the signal event can be represented by a sequence of independent Bernoulli random variables. In this short note we show that these approximations, presented in the literature as being exact, can be very inaccurate when compared with the true ARL values. Moreover, the use of these poor ARL approximations has led to many flawed articles, which should be either corrected or retracted.

40P The detection and quantification of different sequence-variable NPM1 mutations using RNase H-dependent PCR (rhPCR)

Molecular diagnostics for the detection and characterization of genetic variants is based in the majority of cases on PCR technology. Since the primers and probes are designed according to the nucleotide sequence of the DNA region for the specific mutation, PCR technology can only be applied if the mutation is well known. In cases where the mutated region of a gene may have variable sequences, different between patients, a set of primers for each sequence variant is necessary, which involves a large volume of work.

Simulation research on the coordinated development path of urbanization and real estate market using system dynamics in Chongqing City, Southwest China

• Coupling link of urbanization (UR) and real estate market (REM) is studied as systems. • To build a coordinated growth simulation model of UR and REM using system dynamics. • To predict future evolution trends of UR and REM and propose predictive measures. Urbanization (UR) and the real estate market (REM) are closely related to the improvement of human living quality. However, there are fewer simulation studies on the future coordinated development path of UR and REM. Thus, this paper takes Chongqing, China as an example, and considers UR and REM as systems, where the former is divided into demographic, economic, spatial, and social UR, and the latter contains investment, supply, demand, and price. This study estimates the coordinated development links between UR and REM in 2001–2019 utilizing a VAR model and a coupling coordination evaluation model, and simulates the coordinated development path of both in 2020–2030 using system dynamics. Specifically, natural continuation type (NCT), urbanization priority type (UPT), real estate priority type (REPT), and coordinated development type (CDT) are installed using system dynamics. The results reveal that (1) the coordinated development level between UR and REM improves from serious imbalance in 2001 to good coordination in 2019; (2) the sequences of target variables under four scenarios are disparate; (3) the sequence of comprehensive development index regarding UR and REM in 2020–2030 is NCT > CDT > UPT > REPT, and CDT > REPT > UP ≈ NCT, respectively; (4) the order of coordinated development degree of both in 2020–2030 is CDT > REPT > NCT > UPT, i.e., the first-best development path is CDT. The study provides insights and methods for other Chinese inland cities or developing countries with increasing REM.

Fatigue damage development of grouted connection under varying cyclic loading

• Axial fatigue test of grouted connection under variable load range and sequencing. • Concrete damage plasticity-based model parameters used for numerical simulation. • Damage accumulation rate increased after sudden high load. • Load sequencing affects damage accumulation rate. • The damage accumulation of the real sized model was similar to the experimental model. The jacket structures of offshore wind turbines connect the foundation piles through a grout annulus. The grouted connections of the jacket structures are predominantly under the action of low-range fluctuating axial fatigue loads. However, they are sporadically subject to sudden large loads. The process of recurrent loading of the connection can lead to lose of capacity due to the gradual grout damage. Therefore, it is essential to study the damage accumulation process under variable fatigue loading. The cyclic compression tests of grouted connection scale model were performed and the influence of load-range, loading-sequence, and sudden large load on the fatigue damage accumulation was studied. The finite element models of experimental specimens and actual sized model were numerically simulated. An initial constant low load range yielded constant value of damage index and a sudden large load applied caused damage acceleration of the grout, which enhanced the development of damage under the subsequent low-range cyclic load. The larger load range prior to smaller load range created greater damage accumulation and reduced the residual vertical bearing capacity of the connection. The work offers a numerical basis for damage assessment of grouted connections of offshore wind turbines under variable axial fatigue load using the scalar damage degradation parameter in Abaqus finite element software.

Push-Sum Based Algorithm for Constrained Convex Optimization Problem and Its Potential Application in Smart Grid

Dear Editor, This letter aims to investigate the optimization problem where the decision variable is contained in a closed convex set. By combing the gradient decent-like method with the push-sum algorithm framework, we design the distributed iterative formulas under the condition that the considered graphs sequence is time-varying and unbalanced. Under some standard assumptions on the problem setting and the decaying step-sizes, we analyze the convergence property of the generated variables sequence under the proposed iterative formulas.

The laws of large numbers for Pareto-type random variables under sub-linear expectation

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.

A new RCAR(1) model based on explanatory variables and observations

The random coefficient autoregressive (RCAR) processes are very useful to model time series in applications. It is commonly observed that the random autoregressive coefficient is assumed to be an independent identically distributed (i.i.d.) random variable sequence. To make the RCAR model more practical, this paper considers a new RCAR(1) model driven by explanatory variable and observations. We use the conditional least squares, the quantile regression and the conditional maximum likelihood methods to estimate the model parameters. The consistency and asymptotic normality of the proposed estimates are established. Simulation studies are conducted for the evaluation of the developed approaches and two applications to real-data examples are provided. The results show that the proposed procedures perform well for the simulations and application.

Multiaxial fatigue life prediction by equivalent energy‐based critical plane damage parameter under variable amplitude loading

AbstractA new path‐independent multiaxial fatigue damage parameter , which is defined as the largest normal stress range between adjacent turning points of the maximum shear stress on the critical plane, is proposed to describe multiaxial fatigue damage. By combining the maximum normal stress or the largest normal stress range with the maximum shear stress range on the critical plane, two axial equivalent stress modification factors, and , are proposed, which can account for the influence of non‐proportional additional hardening. The sensitivity of the proposed axial equivalent stress modification factors to multiaxial variable amplitude loading sequences is analyzed. Furthermore, two new multiaxial fatigue damage models based on the axial equivalent stress modification factors are proposed to estimate fatigue life. The applicability of the presented methodology was verified by the experimental data of En15R steel and 7050‐T7451 aluminum alloy, and the results showed that the predicted fatigue lifetimes agree well with the experimental data under variable amplitude multiaxial loading.

Regression methods for the appearances of extremes in climate data

AbstractFor any given city, on any calendar day, there will be record high and low temperatures. Which record occurred earlier? If there is a trend towards warming then, intuitively, there should be a preponderance of record highs occurring more recently than the record lows for each of the 365 calendar days. We are interested in modeling the joint distribution of appearances of the extremes but not these values themselves. We develop a bivariate discrete distribution modeling the joint indices of maximum and minimum in a sequence of independent random variables sampled from different distributions. We assume these distributions share a proportional hazard rate and develop regression methods for these paired values. This approach has reasonable power to detect a small mean change over a decade. Using readily available public data, we examine the daily calendar extreme values of five US cities for the decade 2011–2020. We develop linear regression models for these data, describe models to account for calendar‐date dependence, and use diagnostic measures to detect remarkable observations.