The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...

We define the universal 1-adic thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process. We show that the transposition of the perfection process at the real archimedean place is identical to the “dequantization” process and yields Viro’s tropical real hyperfield T R. Then we pr...

We will extend Ostrowki’s theorem from Q to the quadratic field Q(i). On Q, every nonarchimedean absolute value is equivalent to the p-adic absolute value for a unique prime number p, and the archimedean absolute values are all equivalent to the usual absolute value on Q. We will see a similar thing happens in Q(i): any non-archimedean absolute value is associated to a prime in Z[i] (unique up ...

In the classical multivariate time series models the residuals are assumed to be normally distributed. However the assumption of normality is rarely consistent with the empirical evidence and leads to possibly incorrect inferences from financial models. The copula theory allows us to extend the classical time series models to nonelliptically distributed residuals. In this paper we analyze the t...

In this paper, we solve the additive ρ -functional inequalities ‖ f (x+ y)− f (x)− f (y)‖ ∥∥∥ρ ( 2 f ( x+ y 2 ) − f (x)− f (y) ∥∥∥ (0.1) and ∥∥∥2 f ( x+ y 2 ) − f (x)− f (y) ∥∥∥ ‖ρ ( f (x+ y)− f (x)− f (y))‖ , (0.2) where ρ is a fixed non-Archimedean number with |ρ| < 1 . Furthermore, we prove the Hyers-Ulam stability of the additive ρ -functional inequalities (0.1) and (0.2) in non-Archimedean...

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009)) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is d...

One of the features inherent in nested Archimedean copulas, also called hierarchical Archimedean copulas, is their rooted tree structure. In this paper, a nonparametric, rank-based method to estimate this structure is developed. Our approach consists in representing the rooted tree structure as a set of trivariate structures that can be estimated individually. Indeed, for any triple of variable...

We introduce and discuss a condition generalizing one of the Archimedean properties characterizing parabolas. Archimedes was familiar with the following property of parabolas: If for any two points A, B on a parabola we denote by S the area of the region between the parabola and the secant AB, and by T the maximum of the area of the triangle ABC, where C is a point on the parabola between A and...

There is a natural analytification functor from the category of locally separated algebraic spaces locally of finite type over C to the category of complex-analytic spaces [Kn, Ch. I, 5.17ff]. (Recall that a map of algebraic spaces X → S is locally separated if the diagonal ∆X/S : X → X ×S X is an immersion. We require algebraic spaces to have quasi-compact diagonal over SpecZ.) It is natural t...