Let X1, X2,… be i.i.d. random points in ℝ2 with distribution ν, and let Nn denote the number of points spanning the convex hull of X1, X2,…,Xn. We obtain lim infn→∞E(Nn)n-1/3 ≥ γ1 and E(Nn) ≤ γ2n1/3(logn)2/3 for some positive constants γ1, γ2 and sufficiently large n under the assumption that ν is a certain self-similar measure on the unit disk. Our main tool consists in a geometric application of the renewal theorem. Exactly the same approach can be adopted to prove the analogous result in ℝd.

This paper introduces a new stochastic process in which the iterates of a dynamical system evolving in discrete time coincide with the events of a Poisson process. The autocovariance function of the stochastic process is studied and a necessary and sufficient condition for it to vanish is deduced. It is shown that the mean function of this process comprises a continuous-time semidynamical system if the underlying dynamical map is linear. The flow of probability density functions generated by the stochastic process is analysed in detail, and the relationship between the flow and the solutions of the linear Boltzmann equation is investigated. It is shown that the flow is a semigroup if and only if the point process defining the stochastic process is Poisson, thereby providing a new characterization of the Poisson process.

We give some explicit constructions of type III product measures with various properties, resolving some conjectures of Brown, Dooley and Lake. We also define a family of Markov odometers of type III0 and show that the associated flow is approximately transitive.

Let m and n be integers with 0<m<n and let μ be a Radon measure on ℝn with compact support. For the Hausdorff dimension, dimH, of sections of measures we have the following equality: for almost all (n − m)-dimensional linear subspaces V

provided that dimH μ > m. Here μv,a is the sliced measure and V⊥ is the orthogonal complement of V. If the (m + d)-energy of the measure μ is finite for some d>0, then for almost all (n − m)-dimensional linear subspaces V we have

In the present paper we show that a non-zero, continuous, locally finite Borel measure on R cannot be b-concentrated for any 1 <b≤2.

The relationship between the topological dimension of a separable metric space and the Hausdorff dimensions of its homeomorphic images has been known for some time. In this note we consider topological and packing dimensions, and show that if X is a separable metric space, then

where and denote the topological and packing dimensions of X, respectively.

We provide a simple and direct proof of the completeness of the L1-space of any vector measure taking its values in the class of Fréchet spaces which do not contain a copy of the sequence space ω.

Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.

We show that a measure on ℝd is linearly rectifiable if, and only if, the lower l-density is positive and finite and agrees with the lower average l-density almost everywhere.

It is known that if A and B are nontriangular 2 × 2 non-negative integral matrices similar over the integers and –tr A ≤det A, then A and B are strongly shift equivalent. Suppose that A and B are 2 × 2 non-negative integral matrices similar over the integers. In this article it is shown that if –2 tr A≤det A <– tr A and if | det A | is not a prime, then A and B are strongly shift equivalent.

The aim of this paper is to resolve Taylor's question concerning certain regularity conditions on a Borel measure. The proposed solution is given in the framework of Brown, Michon and Peyrière, and Olsen.

We provide a unified and simplified proof that for any partition of (0, 1] into sets that are measurable or have the property of Baire, one cell will contain an infinite sequence together with all of its sums (finite or infinite) without repetition. In fact any set which is large around 0 in the sense of measure or category will contain such a sequence. We show that sets with 0 as a density point have very rich structure. Call a sequence and its resulting all-sums set structured provided for each We show further that structured all-sums sets with positive measure are not partition regular even if one allows shifted all-sums sets. That is, we produce a two cell measurable partition of (0, 1 ] such that neither set contains a translate of any structured all-sums set with positive measure.

We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

We consider the Riesz product with a constant coefficient and odometer action over infinite product spaces. By studying the ratio set we can conclude the type of the above dynamical systems is III1.

Dealing with a problem posed by Kupka we give results concerning the permanence of the almost strong lifting property (respectively of the universal strong lifting property) under finite and countable products of topological probability spaces. As a basis we prove a theorem on the existence of liftings compatible with products for general probability spaces, and in addition we use this theorem for discussing finite products of lifting topologies.

Throughout this paper we assume that k is a given positive integer. As usual, B(x, r) denotes the closed ball with centre at x∈ℝk and radius r > 0. Let μ be a Radon measure on ℝk, that is, μ is locally finite and Borel regular. For s ≥ 0, the lower and upper s–dimensional densities of μ at x are denned respectively by

and

A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

Let be a sequence of mutually disjoint open balls, with centres xj and corresponding radii aj, each contained in the closed unit ball in d-dimensional euclidean space, ℝd. Further we suppose, for simplicity, that the balls Bj are indexed so that aj≥aj+1. The set

obtained by removing, from the balls {Bj} is called the residual set. We say that the balls {Bj} constitute a packing of provided that λ(ℛ)=0, where λ denotes the d-dimensional Lebesgue measure. Thus it follows that henceforth denoted by c(d), whilst the packing restraint ensures that Larman [11] has noted that, under these circumstances, one also has .

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n2). The general case is discussed in terms of operator semigroups.

For a compact group G, we compute the Kazhdan constants κ(G, G) obtained by taking G itself as a generating subset. We get κ(G, G) = if G is finite of order n, and κ(G, G) = if G is infinite.