Approximations for the Stieltjes integral with (φ,Φ)-Lipschitzian integrators are given. Applications for the Riemann integral of a product and for the generalized trapezoid and Ostrowski inequalities are also provided.

Let F(x ) denote a distribution function in R d and let F *n (x ) denote the nth convolution power of F(x ). In this paper we discuss the asymptotic behaviour of 1 - F *n (x ) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F *n (x ) ∼ n(1 - F(x )). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

Continuous-time random walks incorporate a random waiting time between random jumps. They are used in physics to model particle motion. A physically realistic rescaling uses two different time scales for the mean waiting time and the deviation from the mean. This paper derives the scaling limits for such processes. These limit processes are governed by fractional partial differential equations that may be useful in physics. A transfer theorem for weak convergence of finite-dimensional distributions of stochastic processes is also obtained.

It is a stylized fact that estimators in extreme-value theory suffer from serious bias. Moreover, graphical representations of extremal data often show erratic behaviour. In the statistical literature it is advised to use a Box–Cox transformation in order to make data more suitable for statistical analysis. We provide some of the theoretical background to see the effect of such transformations and to investigate under what circumstances they might be helpful.

Let X and Y be separable metrizable spaces, and f: X→Y a function. It is wished to recover f from its values on a small set via a simple algorithm. It is shown that this is possible if f is Baire class one, and in fact a characterization is obtained. This leads to the study of sets of Baire class one functions and to a characterization of the separability of the dual space of an arbitrary Banach space.

We extend classical renewal theorems to the weighted case. A hierarchical chain of successive sharpenings of asymptotic statements on the weighted renewal functions is obtained by imposing stronger conditions on the weighting coefficients.

We consider spectrally positive Lévy processes with regularly varying Lévy measure and study conditional limit theorems that describe the way that various rare events occur. Specifically, we are interested in the asymptotic behaviour of the distribution of the path of the Lévy process (appropriately scaled) up to some fixed time, conditionally on the event that the process exceeds a (large) positive value at that time. Another rare event we study is the occurrence of a large maximum value up to a fixed time, and the corresponding asymptotic behaviour of the (scaled) Lévy process path. We study these distributional limit theorems both for a centred Lévy process and for one with negative drift. In the latter case, we also look at the reflected process, which is of importance in applications. Our techniques are based on the explicit representation of the Lévy process in terms of a two-dimensional Poisson random measure and merely use the Poissonian properties and regular variation estimates. We also provide a proof for the asymptotic behaviour of the tail of the stationary distribution for the reflected process. The work is motivated by earlier results for discrete-time random walks (e.g. Durrett (1980) and Asmussen (1996)) and also by their applications in risk and queueing theory.

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

A near-maximum is an observation which falls within a distance a of the maximum observation in an i.i.d. sample of size n. The asymptotic behaviour of the number K n (a) of near-maxima is known for the cases where the right extremity of the population distribution function is finite, and where it is infinite and the right hand tail is exponentially small, or fatter than exponential. This paper completes the picture for thin tails, i.e., tails which decay faster than exponential. Limit theorems are derived and used to find the large-sample behaviour of the sum of near-maxima.

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

In this paper we shall be concerned with the growth of functions in the class ℳp, where we write f ∈ ℳp 1 <p<∞, if:

(i) f is absolutely continuous on bounded subintervals of [0, ∞]);

(ii) f(0) = 0; and

(iii) .

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

Ramsey's theorem implies that every function f:0, 1ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.

The main aim of this note is the proof of the following

Let −∞ ≤ a > b ≤ ∞ and let A ⊂ (a, b) be a measurable set such that λ((a, b)\A) = 0, where λ denotes Lebesgue measure on ℝ. Let f: A→ℝ be a measurable and midconvex function, i.e.

whenever. Then there exists a convex functionsuch that.