The strong approximation of extremal processes (II)

Summary In this paper, we extend the results of [4] to extremal processes generated by k ^th extremes. By working on inverse processes, we obtain a general representation formula for the k ^th extremal process and develop its main properties. In so doing, we obtain a 1-1 correspondance between a sequence of independent Poisson processes and an i.i.d. sequence, enabling us to associate to any such sequence a canonical Poisson structure.     

The strong approximation property

We introduce and investigate the strong approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the weak bounded approximation property. Among others, we show that the weak bounded approximation property is equivalent to a quantitative strengthening of the strong approximation property. Some recent results on the approximation property of Banach spaces and their dual spaces are improved.     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Semi-self-similar extremal processes

Let g be the distribution function (d.f.) of an extremal process Y. If g is invariant with respect to a continuous one-parameter group of time-space changes {η_α = (τ_α, L_α): α > 0}, i.e. g ∘ η_α = g ∀ α > 0, then g is self-similar. If g is invariant w.r.t. the cyclic group {η^∘(n), n ∈Z} of a time-space change ν, then g is semi-self-similar. The semi-self-similar extremal processes are limiting for sequences of extremal processes Y_n(t)=L _n ⁻¹ ∘ Y ∘ τ_n (t) if going along a geometrically increasing subsequence k_n ∼ ϕⁿ, ϕ > 1, n → ∞. The main properties of multivariate semi-self-similar extremal processes and some examples are discussed in the paper. The results presented are an analog of the theory of semi-self-similar processes with additive increments developed by Maejima and Sato in 1997. Supported by the Bulgarian Ministry of Education and Science (grant No. MM-705/97). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

On the strong approximation of bootstrapped empirical copula processes with applications

The purpose of the present paper is to provide a strong invariance principle for the generalized bootstrapped empirical copula processwith the rate of the approximation for multivariate empirical processes. As a by-product, we obtain a uniform-in-bandwidth consistency result for kernel-type estimators of copula derivatives, which is of its own interest. We introduce also the delta-sequence estimators of the copula derivatives. The applications discussed here are change-point detection in multivariate copula models, nonparametric tests of stochastic vectorial independence and the law of iterated logarithm for the generalized bootstrapped empirical copula process. Finally, a general notion of bootstrapped empirical copula process constructed by exchangeably weighting the sample is presented.     

Stationary self-similar extremal processes

Summary Let ( _k ) _k =– be a stationary sequence of random variables, and, forA, let where _n is an affine transformation of (has the forma _n·+b _n,a _n>0,b _n). ThenM _n is a random sup measure, that is, for arbitrary collections of open setsG . We show that the possible limiting random sup measures for such sequences (M _n) are those which are stationary (M(·+b)= _d M forb) and self-similar (M(a·)= _d ^loga (M) fora>0, where is an affine transformation of ). By applying simple transformations, we need only study stationaryM such thatM(a·)= _d aM fora>0. We show that these processes retain some but not all of the properties of the classical case. In particular, we display a nontrivial example such thattM (0,t] is continuous wp1. The classical planar point process representation of extremal processes is a special case of the present approach, but is not adequate for describing all possible limits.     

Some applications of the strong approximation of the integrated empirical copula processes

The purpose of the present paper is to provide a strong invariance principle for the integrated empirical copula process [introduced in a series of papers by Henze and Nikitin in the univariate setting] with the rate of the approximation for multivariate empirical processes. The applications discussed here are change-point detection in multivariate copula models and the integrated empirical copula process with estimated parameter. Finally, a general notion of bootstrapped integrated empirical copula process, constructed by exchangeably weighting sample, is presented.     

Optimal stopping for extremal processes

For an extremal process (Z_t)_t the optimal stopping problem for X_t = f(Z_t)?g(t) gives the continuous time analogue of the optimal stopping problem for max{Y₁,…,Y_k}?c_k where Y₁, Y₂,… are i.i.d. For the continuous time problem we derive optimal stopping times in explicit form and also show that the optimal stopping boundary is the limit of the optimal stopping boundaries for suitably standardized discrete problems.     

The strong approximation property and the weak bounded approximation property

We show that the strong approximation property (strong AP) (respectively, strong CAP) and the weak bounded approximation property (respectively, weak BCAP) are equivalent for every Banach space. This gives a negative answer to Oja's conjecture. As a consequence, we show that each of the spaces _c0$c_0$ and _?1${?}_1$ has a subspace which has the AP but fails to have the strong AP.     

A strong uniform approximation of fractional Brownian motion by means of transport processes

We construct a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals for any Hurst parameter H$H$, and we derive a rate of convergence, which becomes better when H$H$ approaches 1/2$1/2$. The construction is based on the Mandelbrot–van Ness stochastic integral representation of fractional Brownian motion and on a strong transport process approximation of Brownian motion. The objective of this method is to facilitate simulation.     

An approximation condition and extremal quasiconformal extensions

The possibility that the extremal dilatation of quasiconformal extensions from the circle is determined by quadrilaterals with vertices on the circle is related to an approximation question for holomorphic functions. This allows an alternative demonstration of a result of Anderson and Hinkkanen.

     

The strong derivative of the best approximation operator

ABSTRACT

This article is devoted to the derivation of sampling series associated with eigenvalue problems. No examples of sampling theorems associated with odd order problems are known except when the order is one. Here we give necessary and sufficient conditions for the boundary conditions to be self adjoint. Then sampling series associated with odd order self adjoint and non self adjoint problems are given. The sampling representations associated with non self adjoint odd order problems are derived provided that the boundary condition are regular.     

The influence of sequential extremal processes on the partial sum process

In this paper we derive general upper bounds for the total variation distance between the distributions of a partial sum process in row-wise independent, non-negative triangular arrays and the sum of a fixed number of corresponding extremal processes. As a special case we receive bounds for the supremum distance between the distribution functions of a partial sum and the sum of corresponding upper extremes which improve upon existing results. The outcome may be interpreted as the influence of large insurance claims on the total loss. Moreover, under an additional infinitesimal condition we also prove explicit bounds for limits of the above quantities. Thereby we give a didactic and elementary proof of the Ferguson–Klass representation of Lévy processes on ?_?≥?0 which reflects the influence of extremal processes in insurance.     

Pointwise strong and very strong approximation of Fourier series

We present an estimation of the and H _u ^λφ f means as approximation versions of the Totik type generalization (see [6, 7]) of the result of G. H. Hardy, J. E. Littlewood, considered by N. L. Pachulia in [5]. Some results on the norm approximation will also be given.      

Nested Artin strong approximation property

We study the Artin Approximation property with constraints in a different frame. As a consequence we give a nested Artin Strong Approximation property for algebraic power series rings over a field.