Moment Convergence Research Articles

Initiation of prefrontal convection within a Northeast China cold vortex: The roles of elevated front and dry air intrusion

AbstractConvective initiation ahead of a surface cold front within the Northeast China cold vortex of June 1, 2021, is investigated using observations and a convection‐permitting simulation. The initiation of the convection is elevated without identifiable surface mesoscale forcing. Air feeding the initial convective cells originates from at least 1 km above the ground where parcel‐based convective available potential energy is sufficiently large. An elevated front with large equivalent potential temperature gradient and flow convergence across is primarily responsible for the initiation of elevated convection and later organization into a deep convective line. The elevated front is similar to the upper cold front within extratropic cyclones studied elsewhere, but its altitude is lower in this case. Backward‐trajectory analysis of air parcels on the cold side of the elevated front shows that dry air intrusion reinforces the elevated front by increasing the equivalent potential temperature gradient and moment convergence across the front. The convergence forcing combined with access to convective unstable air at the leading edge of the westerly flows east of the elevated front causes the initiation of convection. The development and spreading of a surface cold pool help push the convective line eastward away from the surface cold front located to the west. Orographic gravity waves provide additional forcing so that convection cells are initiated first at the upward branches of the gravity waves, but the overall organization of initial cells into a solid convective line is controlled mainly by mesoscale structures associated with the elevated front.

The convergence properties for randomly weighted sums of widely negative dependent random variables under sub-linear expectations with related statistical applications

In this paper, we study the complete convergence and complete moment convergence for randomly weighted sums of arrays of rowwise widely negative dependent random variables in sub-linear expectation space under some appropriate conditions, which extend and improve the corresponding ones in classical probability space to the case of sub-linear expectation space. And we obtain a strong law of large numbers for the randomly weighted sums of arrays of rowwise widely negative dependent random variables. As applications of our main results, we not only present a result on the complete consistency for the weighted estimator in a nonparametric regression model, but also obtain the complete consistency for the least squares estimators in errors-in-variables regression models based on widely negative dependent errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results and a real example is analysed for illustration.

Complete [formula omitted]th moment convergence of moving average processes for [formula omitted]-widely acceptable random variables under sub-linear expectations

In this article, we studied complete qth moment convergence of the moving average processes produced by m-widely acceptable (m-WA) random variables under sub-linear expectations. The results here extend those of the moving average processes generated by m-WOD random variables in probability.

Complete moment convergence for widely orthant dependent random variables

The author studies the complete moment convergence for widely orthant dependent random variables. The obtained result in this paper improves the corresponding one of Chen and Sung [A Spitzer-type law of large numbers for widely orthant dependent random variables, Statist. Probab. Lett. 154 (2019), pp. 108544] by obtaining a much stronger conclusion under the same conditions.

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

Abstract We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$ -horospheres in the space of affine lattices.

Conditioned Galton–Watson Trees: The Shape Functional, and More on the Sum of Powers of Subtree Sizes and Its Mean

For a complex number α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, we consider the sum of the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}th powers of subtree sizes in Galton–Watson trees conditioned to be of size n. Limiting distributions of this functional Xn(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_n(\\alpha )$$\\end{document} have been determined for Reα≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha \ e 0$$\\end{document}, revealing a transition between a complex normal limiting distribution for Reα<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha < 0$$\\end{document} and a non-normal limiting distribution for Reα>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha > 0$$\\end{document}. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case Reα=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha = 0$$\\end{document}. The same results are also established in the case of the so-called shape functional Xn′(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_n'(0)$$\\end{document}, which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. In addition, we prove convergence of all moments in the case Reα<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha < 0$$\\end{document}, where this result was previously missing, and establish new results about the asymptotic mean for real α<1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha < 1/2$$\\end{document}.A novel feature for Reα=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha =0$$\\end{document} is that we find joint convergence for several α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to independent limits, in contrast to the cases Reα≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha \ e 0$$\\end{document}, where the limit is known to be a continuous function of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. Another difference from the case Reα≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha \ e 0$$\\end{document} is that there is a logarithmic factor in the asymptotic variance when Reα=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Re}}\\alpha =0$$\\end{document}; this holds also for the shape functional.The proofs are largely based on singularity analysis of generating functions.

The Truncated EM Method for Stochastic Differential Equations Driven by Fractional Brownian Motion

We mainly focus on the numerical method of fractional Brownian motion in this paper. On the basis of the numerical method of general SDEs, an approximation scheme is obtained for the stochastic differential equations about fractional noise. And we get it by using the Lipschitz condition and combining with the truncation function f∆ and g∆. Furthermore, we also prove the moment boundedness and convergence of the solution by some lemma. At last, we apply this method to the Gilpin-Ayala model. The orbital image of the solution and the form of numerical solution are given. The error of solution also has been simulated by MATLAB.

On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation

On Complete Convergence of Moments in Exact Asymptotics under Normal Approximation

Some strong convergence properties for randomly weighted maximum partial sums of END random variables with statistical applications

This article mainly studies the strong convergence properties for randomly weighted maximum partial sums of arrays of row-wise extended negatively dependent (END, for short) random variables, including the complete moment convergence and complete f-moment convergence. The results obtained in this paper extend the corresponding ones in the literature. As an application, we investigate the complete consistency of the estimators of nonparametric regression models based on END random errors by using the complete convergence that we established. Then we provide comprehensive simulation studies to demonstrate the validity of the theoretical results based on the finite sample performance. Finally, an application to a real data set is illustrated.

On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

&lt;abstract&gt;&lt;p&gt;The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty &amp;lt; i &amp;lt; \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty &amp;lt; i &amp;lt; \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.&lt;/p&gt;&lt;/abstract&gt;

Complete convergence and complete moment convergence for weighted sums of random variables satisfying generalized Rosenthal type inequalities and an application

This article mainly investigates the complete convergence and complete moment convergence for weighted sums of random variables satisfying generalized Rosenthal type inequalities, which improves the corresponding ones of the literature. As an application, we further research the errors-in-variables (EV, for short) regression models and exhibit the strong consistency for the least square (LS, for short) estimators. In addition, simulation studies are conducted to verify the finite sample performance of theoretical results.

Convergence properties for randomly weighted sums of ρ-mixing sequences with related statistical applications

In this paper, we investigate some convergence properties, such as complete convergence, complete moment convergence, complete f-moment convergence and strong law of large numbers, for partial sums of randomly weighted sums of -mixing sequences by using the moment inequality, which extend and improve the corresponding ones in the literature. As applications, we obtain the complete consistency for the randomly weighted estimator in a nonparametric regression models and the estimators for the unknown parameters in the simple linear errors-in-variables regression models based on -mixing errors.

Some convergence properties for arrays of rowwise asymptotically almost negatively associated random variables under sub-linear expectations

. In this article, under some suitable conditions, we study the L q convergence and complete q-th moment convergence for arrays of rowwise asymptotically almost negatively associated (AANA, for short) random variables under sub-linear expectations. Some general results on L q convergence and complete q-th moment convergence for arrays of rowwise AANA random variables under sub-linear expectations are established, which extend the corresponding ones from classic probability space to the case of sub-linear expectation space. As applications of our main results, we not only present a result on the complete consistency of the weighted estimator in a nonparametric regression model but also obtain the complete consistency of the estimators in errors-in-variables (EV, for short) regression models based on AANA errors under sub-linear expectations. We perform some numerical simulations to verify the validity of the theoretical results.

The number of descendants in a random directed acyclic graph

AbstractWe consider a well‐known model of random directed acyclic graphs of order , obtained by recursively adding vertices, where each new vertex has a fixed outdegree and the endpoints of the edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number of vertices that are descendants of . We show that converges in distribution; the limit distribution is, up to a constant factor, given by the th root of a Gamma distributed variable with distribution . When , the limit distribution can also be described as a chi distribution . We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.

On the strong approximation of the non-overlapping k-spacings process with application to the moment convergence rates

Abstract In the present work, we establish the strong approximations of the empirical k-spacings process {αn(x): 0 ≤ x&lt; ∞} (cf. (3)). We state the moment convergence rates results for this process.

Complete f-moment convergence for randomly weighted sums of extended negatively dependent random variables

In this article, some results on the complete f-moment convergence for randomly weighted sums of extended negatively dependent (END, for short) random variables are established. The results extend the results of Xu and Yan (2021) from complete moment convergence to complete f-moment convergence for randomly weighted sums of END random variables.

On a central limit theorem in renewal theory

Serfozo (2009, Theorem 2.65) gives a useful central limit theorem for processes with regenerative increments. Unfortunately, there is a gap in the proof. We fill this gap, and at the same time we weaken the assumptions. Furthermore, we give conditions for moment convergence in this setting. We give also further results complementing results in Serfozo (2009) on the law of large numbers and estimates for the mean; in particular, we show that there is a gap between conditions for the weak and strong laws of large numbers.

Complete f-moment convergence for a class of random variables with related statistical applications

. In this article, we establish the complete f-moment convergence for a class of random variables satisfying a Rosenthal-type maximal inequality and a weak mean dominating condition with a mean dominating variable. As corollaries, the complete moment convergence and complete convergence for a class of random variables are also obtained. In addition, an application of main results to nonparametric regression models is provided. Finally, we provide a numerical simulation to verify the validity of our theoretical results based on finite samples.

An Asymptotically Optimal Two-Part Fixed-Rate Coding Scheme for Networked Control With Unbounded Noise

It is known that under fixed-rate information constraints, adaptive quantizers can be used to stabilize an open-loop-unstable linear system on Rn driven by unbounded noise. These adaptive schemes can be designed so that they have near-optimal rate, and the resulting system will be stable in the sense of having an invariant probability measure, or ergodicity, as well as boundedness of the state second moment. Although structural results and information theoretic bounds of encoders have been studied, the performance of such adaptive fixed-rate quantizers beyond stabilization has not been addressed. In this paper, we propose a two-part adaptive (fixed-rate) coding scheme that achieves state second moment convergence to the classical optimum (i.e., for the fully observed setting) under mild moment conditions on the noise process. The first part, as in prior work, leads to ergodicity (via positive Harris recurrence) and the second part ensures that the state second moment converges to the classical optimum at high rates. These results are established using an intricate analysis which uses random-time state-dependent Lyapunov stochastic drift criteria as a core tool.

Complete [formula omitted]-moment convergence for maximal randomly weighted sums of arrays of rowwise widely orthant dependent random variables and its statistical applications

In this paper, we study the complete f-moment convergence for maximal randomly weighted sums of arrays of rowwise widely orthant dependent random variables under some general conditions. The results obtained in the paper extend and improve some previous known ones. As applications of our main results, we not only present convergence for the state observers of linear-time-invariant systems but also obtain complete consistency for the estimators of errors-in-variables regression models based on widely orthant dependent errors. We further perform a numerical simulation to verify the validity of the theoretical results.