On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.

Method for the Calculation of all Zeros of an Analytic Function Based on the Kantorovich Theorem

We propose a method to calculate all zeros of an analytic function in a given rectangle. The main idea of our method is to construct a covering of the initial rectangle by subsets where either there are no zeros or the zero uniqueness test based on the Kantorovich theorem is satisfied. The algorithm for the construction of such covering is presented. The implementation of the method is shown on different examples.

MODIFICATION OF THE KANTOROVICH-TYPE CONDITIONS FOR NEWTON'S METHOD INVOLVING TWICE FRECHET DIFFERENTIABLE OPERATORS

We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal.11 (2004) 103–119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math.79 (1997) 131–145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math.47 (1993) 211–217].