For a time in the mid-1970s probabilists were tantalized by a seemingly simple problem posed by Araki and Kakutani[3]. An interval is repeatedly divided by points chosen successively at random, the nth point being uniformly distributed over the largest of the n intervals formed by the first n − 1 points. Is this sequence of points asymptotically uniformly distributed?

Let D be the unit disc in the complex plane ℂ with centre 0 and let ∂D be its boundary. By Möb (D) we denote the group of all Möbius transformations which leave D invariant. A Fuchsian group G acting on D is a discrete subgroup of Möb (D). The limit set of G is in ∂D. We decompose ∂D into the following three disjoint sets:

We determine the rate of growth of the expected number of real zeros of a class of random polynomials whose coefficients form a stationary uniformly mixing sequence.

The relationship between the local growth of a continuous function and the packing measure of its level sets and of its graph is studied. For the Weierstrass function with b an integer such that b ≥ 2 and with 0 < α < 1, and for x ∈ Range (W) outside a set of first category, the level set W−1(x) has packing dimension at least 1 − α. Furthermore, for almost all x ∈ Range (W), the packing dimension of f is at most 1 − α. Finer results on the occupation measure and the size of the graph of a continuous function satisfying the Zygmund Λ-condition are obtained.

The connection problems associated with the one-dimensional Schrödinger equation in the presence of a general isolated cluster containing an unspecified number of complex transition points in unspecified positions can be studied by means of the phase-integral method developed by Fröman and Fröman. Any anti-Stokes line, i.e. any line in the complex z-plane on which the solutions behave as travelling waves with constant flow, must asymptotically (i.e. in the limit of large values of |z|) point in one of m +2 possible directions, which divide the region around the cluster into m +2 sectors, where m is the degree of the cluster. The tracing of these waves from an anti-Stokes line, bounding a sector, to an anti-Stokes line constituting the other boundary of the same sector is expressed by means of the Stokes constant for the sector in question. This paper examines the relation between these m + 2 Stokes constants in the general case when the transition points in the cluster may also be close-lying in the sense that it is impossible to treat them individually, when the solutions are traced. Under the assumption that the effective potential in the Schrodinger equation is a regular analytic function in a sufficiently large region containing the cluster, it is shown that the m + 2 Stokes constants are in general constrained by three algebraic relations, which are obtained for arbitrary m. The cases m = 1, 2, 3 and 4 are worked out in detail.

Reade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.

The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

In many cases of practical interest there is some kind of symmetry associated with an ordinary second-order differential equation of the Schrödinger type. This symmetry is reflected in certain relations between different F-matrices or different elements of the same F-matrix of the phase-integral method. These symmetry relations imply in turn further relations between the Stokes constants, in addition to those which exist for a general cluster of transition points, and thereby reduce the number of independent Stokes constants. The present paper deals with two important kinds of symmetry relations.

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean that

for all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

We describe the spectrum of a self-adjoint pseudo-differential operator on L2 (– ∞, ∞). We show that the essential spectrum coincides with the interval ([1, ∞) and give a lower bound for the lowest eigenvalue in (– ∞, 1). A sufficient condition for the existence of an eigenvalue in (– ∞, 1) is also given.

In this note we characterize those sequences kn such that the Hill estimator of the tail index based on the kn upper order statistics of a sample of size n from a Pareto-type distribution is strongly consistent.

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.

A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.

At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.

It is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

In recent years it has become clear that there are several ways in which complex Banach spaces can differ quite markedly from their real counterparts, and many of these concern martingales. Thus, in [6] complex uniform convexity was related to martingale inequalities, in [3] and [7] the convergence of L1-bounded analytic martingales was considered and in [8] this property was related to the analytic Radon–Nikodym property.

It is well known that the trace X([0, ∞) of the 3-dimensional Brownian motion X has positive capacity and, therefore, the harmonic measure is well defined on this set. It is shown that a.s., this harmonic measure is singular with respect to the occupation measure Lebesgue ο X−1.

A Markov branching process in either discrete time (the Galton–Watson process) or continuous time is modified by the introduction of a process of catastrophes which remove some individuals (and, by implication, their descendants) from the population. The catastrophe process is independent of the reproduction mechanism and takes the form of a sequence of independent identically distributed non-negative integer-valued random variables. In the continuous time case, these catastrophes occur at the points of an independent Poisson process with constant rate. If at any time the size of a catastrophe is at least the current population size, then the population becomes extinct. Thus in both discrete and continuous time we still have a Markov chain with stationary transition probabilities and an absorbing state at zero. Some authors use the term ‘emigration’ as an alternative to ‘catastrophe’.