In this paper a method is developed for the asymptotic expansion of some classes of integral as a parameter k → 0+. The procedure is analogous to the method of inner and outer sums for treating certain types of infinite series whose terms contain a small parameter, and can involve heavy algebra. However, this aspect of the process can be delegated to a symbolic manipulation package.

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

We establish the existence and approximation of solutions to the operator inclusion y ∈ Ty for deterministic and random cases for a nonexpansive and *-nonexpansive multivalued mapping T defined on a closed bounded (not necessarily convex) subset C of a Banach space. We also prover random fixed points and approximation results for*-nonexpansive random operators defined on an unbounded subject C of a uniformly convex Banach space.

In this paper, we obtain sharp estimates for the expected payoffs and prices of European call options on an asset with an absolutely continuous price in terms of the price density characteristics. These techniques and results complement other approaches to the derivative pricing problem. Exact analytical solutions to option-pricing problems and to Monte-Carlo techniques make strong assumptions on the underlying asset's distribution. In contrast, our results are semi-parametric. This allows the derivation of results without knowing the entire distribution of the underlying asset's returns. Our results can be used to test different modelling assumptions. Finally, we derive bounds in the multiperiod binomial option-pricing model with time-varying moments. Our bounds reduce the multiperiod setup to a two-period setting, which is advantageous from a computational perspective.

We give an example of a holomorphic function, admitting Gérard-Sibuya asymptotic expansion on a polysector of Cn, and such that none of its derivatives admits such an expansion. This motivates the study of the relationship between the concepts of asymptotic expansion in several variables respectively given by Gérard-Sibuya and Majima. For a function f, Majima's notion is proved to be equivalent, on the one hand, to the existence of Gérard-Sibuya asymptotic expansion for f and its derivatives, and on the other hand, to the boundedness of the derivatives of f on bounded proper subpolysectors of S.

In this paper, we consider positive linear operators L representable in terms of stochastic processes Z having right-continuous non-decreasing paths. We introduce the equivalent notions of derived operator and derived process of order n of L and Z, respectively. When acting on absolutely continuous functions of order n, we obtain a Taylor's formula of the same order for such operators, thus extending to a positive linear operator setting the classical Taylor's formula for differentiable functions. It is also shown that the operators satisfying Taylor's formula are those which preserve generalized convexity of order n. We illustrate the preceding results by considering discrete time processes, counting and renewal processes, centred subordinators and the Yule birth process.

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.

The aim of this paper is to continue our investigation of the Lebesgue function of weighted Lagrange interpolation by considering Erdős weights on ℝ and weights on [−1, 1]. The main results give lower bounds for the Lebesgue function on large subsets of the relevant domains.

An asymptotic expression for the expected area of the union of n random rectangles is derived by Mellin transforms, where their two diagonal corners are independently and uniformly distributed over (0,1)2. The general result for d-dimensional hyper-rectangles is also stated.

In this paper the method of inner and outer sums [5], together with the computational power of computer symbolic manipulation, are used to extend to high order the asymptotic expansions in an appropriate limit of some infinite series arising in low Reynolds-number fluid mechanics. The enhanced applicability of the expansions is demonstrated, and the method is extended to treat alternating series.

We present a systematic method of approximating, to an arbitrary accuracy, a probability measure µ on x = [0,1]q , q 1, with invariant measures for iterated function systems by matching its moments. There are two novel features in our treatment. 1. An infinite set of fixed affine contraction maps on , w 2, · ·· }, subject to an ‘ϵ-contractivity' condition, is employed. Thus, only an optimization over the associated probabilities pi is required. 2. We prove a collage theorem for moments which reduces the moment matching problem to that of minimizing the collage distance between moment vectors. The minimization procedure is a standard quadratic programming problem in the pi which can be solved in a finite number of steps. Some numerical calculations for the approximation of measures on [0, 1] are presented.

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.

In this paper, we derive the MacLaurin series of the mean waiting time in light traffic for a GI/G/1 queue. The light traffic is defined by random thinning of the arrival process. The MacLaurin series is derived with respect to the admission probability, and we prove that it has a positive radius of convergence. In the numerical examples, we use the MacLaurin series to approximate the mean waiting time beyond light traffic by means of Padé approximation.

We generalise the classical Bernstein's inequality: . Moreover we obtain a new representation formula for entire functions of exponential type.

Let (ℋ, G, U) be a continuous representation of the Lie group G by bounded operators g ↦ U(g) on the Banach space ℋ and let (ℋ, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1,…, ad′ is a Lie algebra basis of g and Ai = dU(ai) then we examine elliptic regularity properties of the subelliptic operators where (cij) is a real-valued strictly positive-definite matrix and c0, c1,…, cd′ ∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between the Cn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closure of H and the group representation. Thirdly, for sufficiently large values of Re c0 the fractional powers of the closure of H are defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/r where r is the rank of the basis. Further we establish that 2γ/r is the optimal regularity value and it is attained for unitary representations or for the representations obtained by restricting U to ℋγ. Many other regularity properties are obtained.

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

Some new results concerning tubular sets are presented, with applications to the convergence of the Polya algorithm in the contexts of simultaneous approximation and approximation of multivariate functions by univariate functions. (The Polya algorithm constructs a best uniform approximation from the limit, as p → ∞, of best Lp approximations.)

Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.