Nonuniform Rates Of Convergence Research Articles

An Asymptotic Ratio Characterization of Input-to-State Stability

For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic ratio input-to-state stability (ISS) Lyapunov function. The derivative of such a function along solutions is upper-bounded by the difference of two terms whose ratio is asymptotically smaller than 1 for large states. This asymptotic ratio condition is sometimes more convenient to check than standard ISS Lyapunov function conditions. We show that the existence of an asymptotic ratio ISS Lyapunov function is equivalent to ISS. A related notion of ISS with nonuniform convergence rate is also explored.

An exponential nonuniform convergence rate for a class of normalized L-statistics

In this paper, using the change of measure method (Petrov, 1975; Wang and Jing, 1999) and Rényi’s representation (Rényi, 1953), we derive the exponential nonuniform convergence rate in the central limit theorem for a class of normalized L-statistics. Moreover, our results are applied to Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967).

On Nonuniform Convergence Rate Estimates in the Central Limit Theorem

We sharpen the upper bounds for the absolute constant in nonuniform convergence rate estimates in the central limit theorem for sums of independent identically distributed random variables possessing absolute moments of the order $2+\delta$ with some $0<\delta\le1$. In particular, it is demonstrated that under the existence of the third moment this constant does not exceed $18.2$. Also it is shown that the absolute constant in the estimates under consideration can be replaced by a function $C^*(|x|,\delta)$ of the argument $x$ of the difference of the prelimit and limit normal distribution functions for which a positive bounded nonincreasing majorant is found. Moreover, for $\delta=1$ this majorant is asymptotically exact (unimprovable) as $x\to\infty$ and sharpens the estimates due to Nikulin [preprint, arXiv:1004.0552v1 [math.ST], 2010] for all $x$. For the first time a similar result is obtained for the case $\delta\in(0,1)$. As a corollary, we obtain upper estimates for the Kolmogorov functions which are the analogues of the exact and the asymptotically exact constants in the (uniform) Berry--Esseen inequality.

Structural improvements of nonuniform convergence rate estimates in the central limit theorem with applications to poisson random sums

Structural improvements of nonuniform convergence rate estimates in the central limit theorem with applications to poisson random sums

A generalized BPX multigrid framework covering nonnested V-cycle methods

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or non-inherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far. This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

Non-uniform rates of convergence for double arrays of independent random variables with applications

Abstract)Let be a double array of consteds and be a sequence of independent random variables. DenoteIt is well known thatunder some conditions, where is the standard normal distributionfunction.This paper studies the rate of convergence of Fn(x) toΦ(x) depending on n and x wherex is in a neighborhood of the origin under minimal conditions. Explicitly, we establish thefollowing theorems.Theorem 1. Suppose thatReceived January 20, 1994. Revised February 22, 1995.where 0 1. Then, there exist n) b 0 (not depending on n) such that for all n alld allx E R with lxl 5 "L.'/',Theorem 2. Suppose that (4) holds, 0 L. - 0 and t.n E N is a sequence ofconstants with t. ￣ co such that t. ￣ o(L.'/'). ThenRemark 1. Noting that L. S sup la.kl/A. ￣ L;, therefore, Theorems 1 and 2 alsokhold if L. whenever it appears, is replaced by L;.For the particular caseSa reduces to the partial sum Z:=, uk, which has been studied extensively in the literature(see [1,2]). In this cajse, A: = n, L. = .--i. It is well known that Theorem 2 can't beremoved under (4).In time series analysis, an important class of covariance stationary models is defined bywhere {Ek, --co k co) is a sequence of independent random variables and Cd areconstants. The sequence {Z j, j 2 1} is called lillear process generated by {Ek, --co k co}. Since Z:=, Zj = ZZ=--co ankEk where ark = Z:=, Cj--k, making use of Theorems 1and 2 we obtain.Theorem 3. Let (4) hold, Cd be constants such thatThen, there exist n, b 0 such that for all n all x E R with lxl 5 un'/6,Theorem 4. Assume that the conditions of Theorem 3 hold and t. is a sequence ofconstants with t. ￣ co such that t. ￣ o(n'/'). ThenRemark 2. for the following Ck,where the function l(x) 2 0 is slowly varying at infinity, a 2 1 and in or = t, ZC=, k--'l(k) co, Halll61 obtained the rate of convergence of F.(x) to .(x). Applying the same methodas that of dealing with aam ￣ Z:=, sj--k in [6], we can obtain some results similar to (10),(11) for Cd satisfying (12).TO conclude this paper, we give some classical examples (the so-called summabilitymethods) including the most importallt ones from the paint of view of probability (see [3--5]for details. In [51, the authors have dealt with the rate of convergence for SummabilityMethods "lider an assumption of finite third moments).1. Cesaro Sum:2. Abel Sum:3. Borel Sum:4. Euler Sum: (0 p 1)5. Delayed Sum (or RI Sum): (e 0)6. Valiron Sum:7. Taylor Sum: (0 or 1)8. Meyer-K5nig Sum: (0 a 1)

Non-Uniform Speed of Convergence to Normality for Some Stationary m-Dependent Processes

Nonuniform rates of convergence to normality are studied for standardised sample sum for some stationary m dependent processes when i) some finite moment of order⩾ 2 exists, ii) when all the finite order moments exist but moment generating function of the individual random variables may not exist. These rates are then used to compute the probabilities of deviations and certain moment type convergences. Possible extensions of the results to non-stationary m-dependent processes are indicated.

Bounds for the Convergence Rate of Numerical Characteristics in Limit Theorems with Stable Limit Law

Previous article Next article Bounds for the Convergence Rate of Numerical Characteristics in Limit Theorems with Stable Limit LawA. I. DaugavetA. I. Daugavethttps://doi.org/10.1137/1132079PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. I. Daugavet, Convergence of certain numerical characteristics of sums of independent random variables to the corresponding characteristics of stable laws, Deposited with the VINITI, No. 1681–1684. (In Russian.) Google Scholar[2] V. M. Kruglov, Convergence of numerical characteristics of sums of independent random variables with values in Hilbert space, Theory Probab. Appl., 18 (1973), 694–712 10.1137/1118091 0321.60045 LinkGoogle Scholar[3] V. I. Paulauskas, Estimates of the remainder term in limit theorems in the case of stable limit law, Lithuanian Math. J., 14 (1974), 127–146 0306.60012 CrossrefGoogle Scholar[4] V. I. Paulauskas, Uniform and nonuniform estimates of the remainder term in a limit theorem with a stable limit law, Lithuanian Math J., 14 (1974), 661–672 0334.60014 CrossrefGoogle Scholar[5] S. Steishunas, On the nonuniform convergence rate to a stable limit law, Summaries of Reports of the 24th Conference of the Lithuanian Mathematical Society (Vilnius, June 22–23, 1983), Inst. Math. and Cybern. Lith. Acad. Sci, Vilnius, 1983, 184–185, (In Russian.) Google Scholar[6] P. L. Butzer and , L. Hahn, General theorems on rates of convergence in distribution of random variables. II. Applications to the stable limit laws and weak law of large numbers, J. Multivariate Anal., 8 (1978), 202–221 10.1016/0047-259X(78)90072-6 80a:60026b 0383.60025 CrossrefGoogle Scholar[7] Michael J. Klass, A method of approximating expectations of functions of sums of independent random variables, Ann. Probab., 9 (1981), 413–428 82f:60119 0463.60023 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 32, Issue 3| 1988Theory of Probability & Its Applications History Submitted:22 October 1985Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132079Article page range:pp. 531-535ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Certain non-uniform rates of convergence to normality for a restricted class of martingales

Non-uniform rates of convergence to normality are derived for standardized sums of random variables which form a martingale difference array, with the sum of conditional variances being a constant in each row. As a consequence, we obtain probabilities of moderate deviations, convergence of absolute moments and (nonuniform) L p versions of the Berry-Esseen theorem.

Certain non-uniform rates of convergence to normality for martingale differences

Some non-uniform rates of convergence to normality are derived for standardised sums of martingale differences and certain ‘naturally’ stopped martingales, under the main assumption of uniform boundedness of the conditional m.g.f. around a neighbourhood of the origin. Non-uniform L p -versions of the Berry-Esseen theorem, convergence of absolute moments and probabilities of moderate deviation (in some restricted cases) follow as consequences.

5. Limit theory : Nonuniform rates of convergence to normality of m-dependent and linear processes

5. Limit theory : Nonuniform rates of convergence to normality of m-dependent and linear processes

Two-sided bounds for nonuniform rates of convergence in the central limit theorem

Two-sided bounds are derived for nonuniform rates of convergence in the central limit theorem, and are applied to solve several problems on rates of convergence in nonuniform metrics. In particular, an expression for the fastest rate of convergence is obtained, and rates of convergence using different norming constants are compared. A useful estimate of the rate of convergence in the (1+¦x¦2+ɛ)-metric is derived, which does not require the assumption of (2+ɛ)'th order moments.

Some Limit Theorems for Large Deviations

Some Limit Theorems for Large Deviations