Complete Convergence Result Research Articles

Passive concentration dynamics incorporated into the library IB2d, a two-dimensional implementation of the immersed boundary method

In this paper, we present an open-source software library that can be used to numerically simulate the advection and diffusion of a chemical concentration or heat density in a viscous fluid where a moving, elastic boundary drives the fluid and acts as a source or sink. The fully-coupled fluid-structure interaction problem of an elastic boundary in a viscous fluid is solved using Peskin’s immersed boundary method. The addition or removal of the concentration or heat density from the boundary is solved using an immersed boundary-like approach in which the concentration is spread from the immersed boundary to the fluid using a regularized delta function. The concentration or density over time is then described by the advection-diffusion equation and numerically solved. This functionality has been added to our software library, IB2d, which provides an easy-to-use immersed boundary method in two dimensions with full implementations in MATLAB and Python. We provide four examples that illustrate the usefulness of the method. A simple rubber band that resists stretching and absorbs and releases a chemical concentration is simulated as a first example. Complete convergence results are presented for this benchmark case. Three more biological examples are presented: (1) an oscillating row of cylinders, representative of an idealized appendage used for filter-feeding or sniffing, (2) an oscillating plate in a background flow is considered to study the case of heat dissipation in a vibrating leaf, and (3) a simplified model of a pulsing soft coral where carbon dioxide is taken up and oxygen is released as a byproduct from the moving tentacles. This method is applicable to a broad range of problems in the life sciences, including chemical sensing by antennae, heat dissipation in plants and other structures, the advection-diffusion of morphogens during development, filter-feeding by marine organisms, and the release of waste products from organisms in flows.

On convergence properties for weighted sums of coordinatewise ANA random vectors in Hilbert spaces

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated random vectors are investigated. The results obtained in this article extend the corresponding ones for random variables in Chen et al. to H-valued random vectors. Weighted version of strong law of large numbers and complete integral convergence result are also presented.

Some strong laws of large numbers, L 2-convergence and complete convergence for m-AANA random vectors in Hilbert space

ABSTRACT In this paper, based on inequalities for the maximum of the partial sums of m-asymptotically almost negatively associated random vectors in Hilbert space, we establish various kinds of strong laws of large numbers, -convergence and complete convergence. In addition, by relating this dependence condition to residual Cesáro alpha-integrability and to strongly residual Cesáro alpha-integrability, we obtain some -convergence and complete convergence results.

Complete convergence and records for dynamically generated stochastic processes

We consider empirical multi-dimensional rare events point processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time, and record values point processes. An example where the clustering prevents the convergence of the record times point process is given.

Complete q-th moment convergence and its application in the dependent bootstrap

Let be an array of rowwise pairwise negatively quadrant dependent random variables with mean zero and as , where is a sequence of positive integers. The complete qth moment convergence results for and are established which are much stronger than the complete convergence results obtained in the literature. As applications of our main results, we obtain some complete moment theorems for the dependent bootstrap samples arising from a sequence of independent and identically distributed random variables.

Complete moment convergence of double-indexed randomly weighted sums of mixing sequences

In this paper, we study the complete moment convergence of the sums of ρ-mixing sequences which are double-indexed randomly weighted and stochastically dominated by a random variable X. Under the different moment conditions on X and weights, many complete moment convergence and complete convergence results are obtained. Moreover, some simulations are given for illustration.

Complete and complete moment convergence for i.i.d. random variables under exponential moment conditions

ABSTRACTIn this paper, we establish a complete convergence result and a complete moment convergence result for i.i.d. random variables under moment condition which is slightly weaker than the existence of the moment generating function. The main results extend and improve the related known results of Lanzinger (1998) and Gut and Stadtmüller (2011).

A complete convergence theorem for stationary regularly varying multivariate time series

For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.

Convergence for weighted sums of widely orthant dependent random variables

ABSTRACTIn this article, a complete convergence result and a complete moment convergence result are obtained for the weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results are also obtained under the extended negatively orthant dependent setup. In particular, the complete convergence result generalizes and improves the related known works in the literature.

On the convergence rate for arrays of row-wise NOD random variables

In this article, the complete convergence results of weighted sums for arrays of rowwise negatively orthant dependent (NOD) random variables are investigated. Some sufficient conditions for complete convergence for arrays of rowwise NOD random variables are presented without assumption of identical distribution.

Complete and complete moment convergence for weighted sums of widely orthant dependent random variables

In this paper, we establish a complete convergence result and a complete moment convergence result for weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results for weighted sums of extended negatively orthant dependent random variables are also obtained, which generalize and improve the related known works in the literature.

Strong Limit Theorems for Pairwise NQD Random Variables

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables.

On the strong convergence for weighted sums of ρ *-mixing random variables

A complete convergence result is obtained for weighted sums of identically distributed ρ*-mixing random variables with E|X1|α log(1 + |X1|) < ∞ for some 0 < α ≤ 2. This result partially extends the result of Sung (Stat Papers 52: 447–454, 2011) for negatively associated random variables to ρ*-mixing random variables. It also settles the open problem posed by Zhou et al. (J Inequal Appl, 2011, doi:10.1155/2011/157816).

Conditional limit theorems for conditionally linearly negative quadrant dependent random variables

From the ordinary notion of linearly negative quadrant dependence for a sequence of random variables, a new concept called conditionally linearly negative quadrant dependence is introduced. The relation between the two kinds of dependence is answered by examples, that is, the linearly negative quadrant dependence does not imply the conditionally linearly negative quadrant dependence, and vice versa. The fundamental properties of conditionally linearly negative quadrant dependence are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional exponential inequalities, conditionally complete convergence results and a conditional central limit theorem stated in terms of conditional characteristic functions are established.

Limiting Behavior of the Maximum of the Partial Sum for Linearly Negative Quadrant Dependent Random Variables under Residual Cesàro Alpha‐Integrability Assumption

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual Cesàro alpha‐integrability assumption, as well as to strongly residual Cesàro alpha‐integrability assumption, some Lp‐convergence and complete convergence results of the maximum of the partial sum are derived, respectively.

A note on the complete convergence for arrays of dependent random variables

A complete convergence result for an array of rowwise independent mean zero random variables was established by Kruglov et al. (2006). This result was partially extended to negatively associated and negatively dependent mean zero random variables by Chen et al. (2007) and Dehua et al. (2011), respectively. In this paper, we obtain complete extended versions of Kruglov et al. Mathematics Subject Classification 60F15

Marcinkiewicz–Zygmund inequality for nonnegative [formula omitted]-demimartingales and related results

In this paper the Marcinkiewicz–Zygmund inequality for mean zero independent random variables is extended to the case of nonnegative N -demimartingales. This result can be used for obtaining large deviation inequalities for nonnegative N -demimartingales as well as related complete convergence results.

Convergence of binomial-based derivative estimation for [formula omitted]2 noisy discretized curves

We present a derivative estimator for discrete curves and discretized functions which uses convolutions with integer-only binomial masks. The convergence results work for C 2 functions, and as a consequence we obtain a complete uniform convergence result for parameterized C 2 curves for derivatives of any order.

Limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alpha-integrability assumption

Asymptotically negative association is a special dependence structure. By relating such dependence condition to residual Cesàro alpha-integrability and to strongly residual Cesàro alpha-integrability, some Lp-convergence and complete convergence results of the maximum of the partial sum are derived, respectively. In addition, some of these conclusions are based on a new Rosenthal type inequality concerning asymptotically negatively associated random variables, which is of independent interest.

Complete moment convergence for sequence of identically distributed φ-mixing random variables

In the paper we extend and generalize some results of complete moment convergence results (or the refinement of complete convergence) obtained by Chow [On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Academia Sinica, 16, 177–201 (1988)] and Li & Spătaru [Refinement of convergence rates for tail probabilities. J. Theor. Probab., 18, 933–947 (2005)] to sequences of identically distributed φ-mixing random variables.