Volume 8, Issue 2

We study topological spaces with the strong Skorokhod property, i.e., spaces on which all Radon probability measures can be simultaneously represented as images of Lebesgue measure on the unit interval under certain Borel mappings so that weakly convergent sequences of measures correspond to almost everywhere convergent sequences of mappings. We construct nonmetrizable spaces with such a property and investigate the relations between the Skorokhod and Prokhorov properties. It is also shown that a dyadic compact has the strong Skorokhod property precisely when it is metrizable.

The problems of almost sure convergence to zero and almost sure boundedness of operator-normed sums of different sequences of random vectors are studied. The sequences of independent random vectors, orthogonal random vectors and the sequences of vector-valued martingale-differences are considered. General results are applied to the problem of asymptotic behaviour of multidimensional autoregression processes.

It is well known that for the Hilbert space H the minimum value of the functional F μ ( f ) = ∫ H ‖ f – g ‖ 2 dμ ( g ), f ∈ H , is achived at the mean of μ for any probability measure μ with strong second moment on H . We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

The main aim of this paper is to prove a bilateral inequality for , where t > 0, (𝑎 𝑘 ) are elements of a normed space, while (𝑟 𝑘 ) are Rademacher functions. Then this inequality is applied for estimation of . As another corollary we give a maximal inequality for exchangeable random variables that recently has been published in [Pruss, Proc. Amer. Math. Soc. 126: 1811–1819, 1998].

Given nonnegative real sequences and we study the generated summation operator regarded as a mapping from ℓ p (ℤ) to ℓ q (ℤ). We give necessary and sufficient conditions for the boundedness of S 𝑎, 𝑏 and prove optimal estimates for its entropy numbers relative to the summation properties of 𝑎 and 𝑏. Our results are applied to the investigation of the behaviour of as ε → 0, where is some nondecreasing sequence in [0, ∞) and ( W ( t )) t ≥ 0 denotes the Wiener process.

In this work, we consider the problem of moving average representations for random fields. As in the Kolmogorov–Wiener case, such representations lead to interesting questions in harmonic analysis in the polydisc. In particular, we study outer functions with respect to half-space, semigroup and quarterplane and their interrelations.

We give a sequential compactness criterion for the weak topology of vector measures with values in certain nuclear spaces, such as the space S of all rapidly decreasing, infinitely differentiable functions, the space D of all test functions, and the strong duals of those spaces. This result contains Prokhorov–LeCam's criterion for real measures and applies to cases which are not covered by März–Shortt's criterion for Banach space valued vector measures.

Set-parametric Brownian motion b in a star-shaped set G is considered when the values of b on the boundary of G are given. Under the conditional distribution given these boundary values the process b becomes some set-parametrics Gaussian process and not Brownian motion. We define the transformation of this Gaussian process into another Brownian motion which can be considered as “martingale part” of the conditional Brownian motion b and the transformation itself can be considered as Doob–Meyer decomposition of b . Some other boundary conditions and, in particular, the case of conditional Brownian motion on the unit square given its values on the whole of its boundary are considered.

We determine the asymptotic behaviour (as 𝑘 → ∞, up to multiplicative constants not depending on k) of the entropy numbers 𝑒 𝑘 ( D σ : l p → l q ), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = ( σ n ). This complements earlier results by Carl [J. Approx. Theory 32: 135–150, 1981] who investigated the case of power-like decay of the diagonal.

An asymptotic expansion for the approximation of standardized products of large numbers of smooth positive functions by exp(–𝑥 2 /2) is given. This result is closely related to the Bernstein–von Mises theorem on the normal approximation of posterior distributions.

The asymptotic behavior of the singular and entropy numbers is established for the Erdelyi–Köber and Hadamard integral operators (see, e.g., [Samko, Kilbas and Marichev, Integrals and derivatives. Theoryand Applications, Gordon and Breach Science Publishers, 1993]) acting in weighted L 2 spaces. In some cases singular value decompositions are obtained as well for these integral transforms.

It is shown that in an infinite-dimensional dually separated second category topological vector space X there does not exist a probability measure μ for which the kernel coincides with X . Moreover, we show that in “good” cases the kernel has the full measure if and only if it is finitedimensional. Also, the problem posed by S. Chevet [Kernel associated with a cylindrical measure, Springer-Verlag, 1981, p. 69] is solved by proving that the annihilator of the kernel of a measure μ coincides with the annihilator of μ if and only if the topology of μ -convergence in the dual space is essentially dually separated.

For rather general moduli of smoothness ρ , like ρ (ℎ)=ℎ α ln β (𝑐/ℎ), we consider the Hölder spaces H ρ ( B ) of functions [0,1] d → B is a separable Banach space. We establish an isomorphism between H ρ ( B ) and some sequence Banach space. With this analytical tool, we follow a very natural way to study, in terms of second differences , the existence of a version in H ρ ( B ) for a given random field.

A unified treatment of integral representations of conditional expectations, in terms of Bochner integrals, with specializations to Gaussian conditioning, is given. The integrals in the latter have deterministic kernels. Difficulties in actual calculations of (even regular) conditional probabilities, and the inadequacy of methods related to L'Hospital's procedures, are exemplified. Limitations of the methods used and some new problems for investigation are indicated.

The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space and for a p -orthogonal sequence of random elements in a Rademacher type p Banach space. The current work generalizes, simplifies, and unifies some of the recent results of Nam and Rosalsky [Teor. Īmovīr. ta Mat. Statist. 52: 120–131, 1995] and Rosalsky and Rosenblatt [Bull. Inst. Math. Acad. Sinica 11: 185–208, 1983, Nonlinear Anal. 30: 4237–4248, 1997].

Infinite systems of stochastic differential equations for randomly perturbed particle systems with pairwise interaction are considered. It is proved that under some reasonable assumption on the potential function there exists a local weak solution to the system and it is weakly locally unique for a wide class of initial conditions.

For a statistic S whose distribution can be approximated by χ 2 -distributions, there is a considerable interest in constructing improved χ 2 -approximations. A typical approach is to consider a transformation T = T ( S ) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale mixture of a χ 2 -variate or the distribution of S allows an asymptotic expansion in terms of χ 2 -distributions. For these statistics, we give sufficient conditions for T to have an improved χ 2 -approximation. Furthermore, we present a method for obtaining its error bound.

We study tractability in the worst case setting of tensor product linear operators defined over weighted tensor product Hilbert spaces. Tractability means that the minimal number of evaluations needed to reduce the initial error by a factor of ε in the d -dimensional case has a polynomial bound in both ε –1 and d . By one evaluation we mean the computation of an arbitrary continuous linear functional, and the initial error is the norm of the linear operator S d specifying the d -dimensional problem. We prove that nontrivial problems are tractable iff the dimension of the image under S 1 (the one-dimensional version of S d ) of the unweighted part of the Hilbert space is one, and the weights of the Hilbert spaces, as well as the singular values of the linear operator S 1 , go to zero polynomially fast with their indices.