We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao’s theorem (Shao, Q.-M., 1995. On a conjecture of Révész. Proc. Amer. Math. Soc. 123, 575-582) in the gaussian case.

Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is often not desirable to base the estimator of the variance function on the residuals from an optimal est...

Riemann’s theorems on the rearrangement of absolutely convergent and conditionally convergent series of real numbers are analysed within Bishop-style constructive mathematics. The constructive proof that every rearrangement of an absolutely convergent series has the same sum is relatively straightforward; but the proof that a conditionally convergent series can be rearranged to converge to what...

Davidsons method is widely used for finding the lowest eigenvalues of large matrices. Recently, mathematicians have shown that Davidsons derivation could be improved. They have corrected the derivation yielding a new iteration method. In this article this new method is adapted for realistic MRCI and MRCEPA calculations. Results show that the new method converges significantly faster in H2O and ...

where Ω = R \ O, and O is a bounded open set in R with smooth boundary. Throughout this paper, we suppose that n is an odd integer with n ≥ 3. We assume that Ω is connected and that the initial data ~ f = (f0, f1) belongs to the associated energy space HD(Ω). Here and in the following, for an open set Y ⊂ R,HD(Y ) stands for the completion of (C 0 (Y )) 2 with respect to ‖~ f‖HD(Y ) = ‖∇f0‖L2(Y...

We propose a new matrix geometric mean satisfying the ten properties given by Ando, Li and Mathias [Linear Alg. Appl. 2004]. This mean is the limit of a sequence which converges superlinearly with convergence of order 3 whereas the mean introduced by Ando, Li and Mathias is the limit of a sequence having order of convergence 1. This makes this new mean very easily computable. We provide a geome...

If S is a cofinite set of positive integers, an “S-restricted composition of n” is a sequence of elements of S, denoted ~λ = (λ1, λ2, . . . ), whose sum is n. For uniform random S-restricted compositions, the random variable B(~λ) = ∏ i λi is asymptotically lognormal. (A precise statement of the theorem includes an error term to bound the rate of convergence.) The proof is based upon a combinat...

We introduce and consider a proximal point algorithm for solving minimization problems using the technique of Güler. This proximal point algorithm is obtained by substituting the usual quadratic proximal term by a class of convex nonquadratic distance-like functions. It can be seen as an extragradient iterative scheme. We prove the convergence rate of this new proximal point method under mild a...