polynomial, we conclude that the modified Ishikawa, Agarwal et al., SP, Noor are faster than simple Ishikawa, Agarwal et al., SP, Noor; but if we compare modified Ishikawa, Agarwal et al., SP, Noor with decreasing order of **rate** of **convergence** of modified Agarwal et al., SP, Noor, Ishikawa, modified new Agarwal et al. have con- sistent **rate** of **convergence**. The graphs drawn are based on data formed from C ++ program and plot the data in mathematica to show the fixed point.

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Abstract. In this paper, we first construct a new iteration method for approximating fixed points of a class of weak contractions in a Banach space and then prove strong **convergence** theorem of the proposed method under some control conditions. It is shown that our iteration method converges faster than Noor iteration. Moreover, we give some numerical example for comparing the **rate** of **convergence** between the Noor iteration and our iteration. Keywords: strong **convergence**; **rate** of **convergence**; Noor iterations; weak contractions; Banach space. 2010 AMS Subject Classification: 47H10, 54H25.

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Throughout the document, we will be interested in **rate** of **convergence** results for the class F of (1,C,ρ,σ 2 )-smooth distributions (X,Y ) such that X has compact support with diameter 2ρ, the regression function r is Lipschitz with constant C and, for all x ∈ R d , σ 2 (x) = V [Y | X = x] ≤ σ 2 (the symbol V denotes variance). It is known (see, for example Ibragimov and Khasminskii, 1980, 1981, 1982) that for the class F , the sequence (n − d+2 2 ) is the optimal minimax **rate** of **convergence**.

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In [7], the second author and Ladas obtained both local and global stability results for (1.2), (1.3), and (1.4) and found the region in the space of parameters where the equilib- rium solution is globally asymptotically stable. In this paper, we will precisely determine the **rate** of **convergence** of all solutions in this region by using Poincar´e’s theorem and an improvement of Perron’s theorem.

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In [20], the solution theory was extended to Burgers-type equations with multi- plicative noise (i.e. when the multiplier of the noise term is a nonlinear local function θ(u) of the solution). Analysis of numerical schemes approximating the equation in the multiplicative case was performed in [21], where the appearance of a correction term was observed and the **rate** of **convergence** in the uniform topology was shown to be of order 1 6 − κ , for every κ > 0.

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There are many articles have been published on the iterative methods using for approximation of fixed points of nonlinear mappings, see for instance [1-7]. However, there are only a few articles concerning comparison of those iterative methods in order to establish which one converges faster. As far as we know, there are two ways for comparison of the **rate** of **convergence**. The first one was introduced by Berinde [27]. He used this idea to compare the **rate** of **convergence** of Picard and Mann itera- tions for a class of Zamfirescu operators in arbitrary Banach spaces. Popescu [28] also used this concept to compare the **rate** of **convergence** of Picard and Mann iterations for a class of quasi-contractive operators. It was shown in [29] that the Mann and Ishi- kawa iterations are equivalent for the class of Zamfirescu operators. In 2006, Babu and Prasad [30] showed that the Mann iteration converges faster than the Ishikawa itera- tion for this class of operators. Two years later, Qing and Rhoades [31] provided an example to show that the claim of Babu and Prasad [30] is false.

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For a dynamic matching and bargaining market, the question of how small frictions need to be for equilibria to be approximately competitive is equally important. In this paper, we ll this gap by proving a **rate** of **convergence** result for a decentralized model of trade. We study the steady state of a market with incoming cohorts of buyers and sellers who are randomly matched pairwise and bargain without knowing each other's reservation value. The model is in discrete time and shares several features with the model in Satterthwaite and Shneyerov (2007). Exactly as in that paper, a friction parameter is , the length of the time period until the next meeting. There are per-period participation costs, B for

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A binary classification problem is considered. The excess error probability of the k-nearest- neighbor classification rule according to the error probability of the Bayes decision is revis- ited by a decomposition of the excess error probability into approximation and estimation errors. Under a weak margin condition and under a modified Lipschitz condition or a local Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded. The concept of modified Lipschitz condition is applied for discrete distributions, too. As a consequence of both concepts, we present the **rate** of **convergence** of L 2 error for the corresponding nearest neighbor regression estimate.

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We introduce a new sequence of linear positive operators B n,α (f , x), which is the Bezier variant of the well-known Baskakov Beta operators and estimate the **rate** of **convergence** of B n,α (f , x) for functions of bounded variation. We also propose an open problem for the readers.

Our techniques build upon earlier work on the **rate** of **convergence** of AdaBoost, which have mainly considered two particular cases. In the first case, the weak learning assumption holds, that is, the edge in each round is at least some fixed constant. In this situation, Freund and Schapire (1997) and Schapire and Singer (1999) show that the optimal loss is zero, that no solution with finite size can achieve this loss, but AdaBoost achieves at most ε loss within O(ln(1/ε)) rounds. In the second case some finite combination of the weak classifiers achieves the optimal loss, and R¨atsch et al. (2002), using results from Luo and Tseng (1992), show that AdaBoost achieves within ε of the optimal loss again within O(ln(1/ε)) rounds.

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about cost functions following Theorem 1), and hence Theorem 1 can be applied for the noisy distribution. For many distributions, the performance deteriorates by adding noise, but at least the **rate** of **convergence** is guaranteed to stay the same, and only the value of the constant C will be affected. Unfortunately, it is impossible to test whether η is bounded away from zero or not, and it may be safe to add a little noise. Of course, the level of the added noise (i.e., the probability of flipping the labels in the training set) does not need to be the 1 / 4 described above. Any strictly positive value may be used and Corollary 10 remains true. While a more precise study is out of the scope of this paper, let us just remark that a sensible choice of the noise level based on the present bounds should be able to find a tradeoff between the improvement of the bias in the A-risk and the performance degradation as appearing in Equation (4).

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Despite the popularity of ABC methods, theoretical analysis is still in its infancy. The aim of this article is to provide a foundation for such analysis by providing rigorous results about the **convergence** properties of the ABC method. Here, we restrict discussion to the most basic variant, to set a baseline to which different ABC variants can be compared. We consider Monte Carlo estimates of posterior expectations, using the ABC samples for the estimate. Proposition 3.1 shows that such ABC estimates converge to the true value under very weak conditions; once this is established we investigate the **rate** of **convergence** in theorem 3.3. Similar results, but in the context of estimating posterior densities rather than posterior expectations can be found in Blum (2010) and Biau et al. (2013). Further studies will be required to establish analogous results for more

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One caveat is that the non-convergent examples we have constructed for the double auction are all quite special in that they require a great deal of coordination among the traders. Additional assumptions (e.g. continuity of strategies and boundedness of the ratio of buyers to sellers) could be imposed to restrict the set of equilibria with the purpose of proving their **convergence** at the linear **rate**, but this has not yielded to our eﬀorts. In addition, allowing a multilateral matching technology may also restore **convergence** of all equilibria of the double auction mechanism. Also, we have only explored specific bargaining protocols. Characterizing a set of protocols for which the **rate** of **convergence** is (optimal) linear is an open question. 25

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In what follows we shall be concerned so much with the probability distance approach as a new mathematical tool to establish the estimates for rates of **convergence** in limit theorems for random sums, we need to recall the deﬁnition of Trotter-distance with some needed properties (see [, ] and []).

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The detailed study of the local dimensions of measures is known as multifractal analysis and has received enormous interest during the past 20 years; the reader is refereed to the texts [r]

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Since the Durrmeyer-Bézier operators Dn,α are an important approximation operator of new type, the purpose of this paper is to continue studying the approximation properties of the oper[r]

Based on Taberski’s study [], Gadjiev [] investigated both the pointwise **convergence** theorems and the order of pointwise **convergence** theorems for operators type () at a gen- eralized Lebesgue point. Then Rydzewska [] conducted a similar study by changing the point to a μ-generalized Lebesgue point of f ∈ L (–π, π) instead of a generalized Lebesgue

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Universit´ a di Milano Bicocca, October 22, 2007. Joint work with I.[r]

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[3] T.-C. Hu and R. L. Taylor, On the strong law for arrays and for the bootstrap mean and vari- ance, Int. J. Math. Math. Sci. 20 (1997), no. 2, 375–382. MR 97k:60011. Zbl 883.60024. [4] D. Li, A. Rosalsky, and S. E. Ahmed, Complete **convergence** of bootstrapped means and moments of the supremum of normed bootstrapped sums, Stochastic Anal. Appl. 17 (1999), no. 5, 799–814. CMP 1 714 899. Zbl 990.78202.

In [9], Qing and Rhoades , by taking a counter example , showed that the Ishikawa iteration process is faster than the Mann iteration process for Zamfirescu operators.. Thus , Theorem in[r]