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If X(t) is a Wiener process with EX(t) = vt and var X(t) = σ 2 t (where v > 0) and if M(t) = max0≦τ≦t X(τ) and Ta is the time of first passage through level a (where a > 0) we show that where is the coefficient of variation of Ta . Applications of the result to the “maturity-time” representation of cell-growth and to queues with heavy traffic are discussed.

Methods for computing the asymptotic tensile strength distribution of a bundle of a finite number of fibres are presented for a variety of situations. The individual fibres are assumed to display random variations in load-extension behaviour over and above the usual variations in breaking elongation. Random fibre slack effects (resulting from uneven clamping and variations in fibre length) are shown to cause an additional loss in bundle strength over that already caused by variations in fibre breaking elongation. Bundles constructed of more than one generic type of elastic fibre in fixed blend ratios are shown to suffer strength losses over bundles constructed of only one fibre type. The method of approach differs from that used by other workers who have treated a special case, namely the classic fibre bundle problem. The solution evolves from reducing the problem to that of a Gaussian process boundary crossing problem, using recent advances in the theory of weak convergence of probability measures.

Let ( Ω , F , μ ) (\Omega ,\mathfrak {F},\mu ) be a measure space, μ ( Ω ) > ∞ \mu (\Omega ) > \infty . Let X n {X_n} be a sequence of measurable functions on Ω \Omega taking values in a compact metric space M M . The set of bounded stopping times τ \tau for the X n {X_n} is a directed set under the obvious ordering. The following theorem is proved: X n {X_n} converges pointwise almost everywhere if and only if the generalized sequence ∫ ϕ ( X τ ) d μ \int {\phi ({X_\tau })d\mu } converges for every continuous function ϕ \phi on M M . The martingale theorem is proved as an application.

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

In the theory of optimization, in connection with ordinary and partial differential equations, a number of closure and lower closure theorems have been obtained in different contexts and under a variety of conditions and modes of convergence. In particular "seminormali ty" conditions (property (Q) and its variants) have played different roles. In this paper we first prove closure and lower closure theorems for orientor fields in a rather abstract context, all based on weak convergence and MAZUR'S theorem (w167 and 5). In the context of orientor fields, these theorems can be given the most satisfactory formulation and simplest proofs (see, e.g., th. (4.i), (5.i)). Furthermore, in the present new approach, the interplay of "seminormali ty" conditions and modes of convergence can be easily seen: the stronger the mode of convergence, the weaker are the "seminormali ty" conditions that are needed. From these theorems we then derive, as corollaries, closure and lower closure statements for Mayer and Lagrange problems (~ 6 and 7) and lower semicontinuity statements for free problems (w Under suitable hypotheses, no seminormality condition (or property (Q)) is needed. Further theorems without seminormality conditions, as well as other details, are discussed in [3]. In particular, we show that seminormality conditions can be removed, not only under standard Lipschitz requirements, as expected, but also under much more satisfactory simple growth conditions, as proposed some time ago by E. H. ROTHE for free problems. Applications to multidimensional Lagrange problems are discussed in [8].

For appropriate metrics characterizing various modes of stochastic convergence, it is shown that rates of convergence are preserved by a large class of functions. For example, the extensions of a Lipschitz function on a separable metric space S to the space of all probability measures on S with the Prohorov metric and to the space of all S-valued random variables with the usual metric associated with convergence in probability inherit the Lipschitz property. Consequently, just as with the continuous mapping theorem associated with ordinary convergence, new rate of convergence theorems can sometimes be obtained from old ones by applying appropriate mappings.

In [4], Sen proved Theorems 22.1 and 22.2 of Billingsley [1] on the weak convergence of empirical distribution functions for sequences of Φ-mixing random variables to appropriate Gaussian random functions under less stringent regularity conditions. In this note, we shall prove the same theorems under weaker conditions than Sen's ones.

The distribution D ϕ of a correspondence ϕ is defined, and its connection with the set DL ϕ of distributions of its integrable selections is explored. The main result is that if ϕ1 and ϕ2 are equally distributed, i.e., if D ϕ1 = and D ϕ2, then DL ϕ1 and DL ϕ2 have the same closure in the weak convergence topology.

In this paper we prove that, for a Riesz space L, the order closure of each subset of L coincides with its pseudo order closure if and only if the order closure of each convex subset of L coincides with its pseudo order closure; moreover, each of these statements is equivalent to the strong Egoroff property. For Archimedean Riesz spaces, similar results hold for the relative uniform topology.

The qualitative behaviour of control systems based on ordinary differential equations has been investigated with clarity and elegance using axiomatically defined General Control Systems. Here an attainablity set function, evolving in semigroup fashion, is the main entity of interest [1], [2], [3], [4]. In stochastic control theory quite complicated and messy mathematical machinery is required, usually obscuring the properties being investigated. It is felt that an analogous axiomatic approach could help circumvent this difficulty, at least for stochastic systems with perfect state observations and satisfying the Markov Principle, i.e. the future evolution is completely determined by the dynamics of the system and the current state, independently of how the current state was attained. In [5] the author showed that the attainability set function of stochastic control systems, governed by Ito stochastic equations and with perfect state observations, had similar properties to the attainability sets of deterministic systems. There the state space was a space of random variables with the topology of mean square convergence. However, any axiomatic attempt in this space would require additional measurability axioms. To avoid this it is necessary to consider probability measures as the basic state points. Then the attainability set function satisfies all but one (a minor one) of the axioms of a deterministic General Control System. The price paid for this simplification is the use of a considerably weaker topology, that of weak convergence of probability measures. The advantages though are quite tempting: the underlying space on which the probability measures are defined can be either finite or a continuum; there is need to specify the source or sources of randomisation, thus avoiding many of the modeling difficulties associated with stochastic control systems; the extensive qualitative theory developed for deterministic General Control Systems carries over with little modification.

The present article arose in connection with problems about the convergence of distributions of random processes. However the basic results of the paper as presented in w 3 and w 4 deal with the convergence of measures and charges in arbitrary (non metric) spaces and can be understood independently of :he theory of random processes. Related problems were studied in the paper [1] by different methods. In that paper a more detailed and selfcontained survey of the earlier results and references can also be found. The quoted theorems in this paper will be numbered with Roman numerals and the proved theorems will have arabic numerals.

In the last ten years the theory of weak convergence of probability measures has been used extensively in studying the models of applied probability. By far the greatest consumer of weak convergence has been the area of queueing theory. This survey paper represents an attempt to summarize the experience in queueing theory with the hope that it will prove helpful in other areas of applied probability. The paper is organized into the following sections: queues in light traffic, queues in heavy traffic, queues with a large number of servers, continuity of queues, rates of convergence, and special queueing models.

THE question of the mathematical justification of the method of spherical harmonics for the non-stationary kinetic transfer equation is studied. It is shown that the sequence of solutions {u?} of the equations of the method of spherical harmonics for the Vladimirov-Marshak boundary conditions, is weakly compact. This proves the weak convergence of the sequence (1~~) to the generalized solution of the mixed Cauchy problem for the non-stationary kinetic transfer equation. Estimates of the convergence are indicated in terms of trial functions.

In this paper, convergence theorems for heavy traffic queues are extended to multi-channel systems under general assumptions. Whitt (1968), and Iglehart and Whitt (1970) have proved weak convergence of the queue length process and the wait process for one-channel queues. Extensions to multi-server queues were established for the queue length process but only partly for the wait process (ρ = 1 only). We give here convergence theorems for the wait process when ρ > 1. Our approach uses weak convergence theory, but is different from previous ones in that we use the virtual delay as an intermediate result. The class of queues considered is more general than GI/G/m.

Kennedy (1972) showed that the standard single-server queueing model is continuous. These results are extended to the standard multi-server model here. Even when there is only one server, an additional condition is needed for the queue length process. CONTINUITY OF QUEUES; APPROXIMATIONS FOR QUEUES; MULTI-SERVER QUEUES; GI/GIs QUEUES; WEAK CONVERGENCE ON FUNCTION SPACES; THE CONTINUOUS MAPPING THEOREM

In the paper, the authors prove several convergence results for a numerical method in optimal stochastic control, based on finite difference approximations to the non-linear Bellman partial differential equation for the optimal cost. As a very useful by-product to the main development, the authors obtain new existence results for optimal controls, and interesting results on the approximation of a controlled diffusion by a controlled Markov chain. The methods and results are new, and depend on results concerning weak convergence of probability measures. The methods seem to be quite powerful, and have applications to many other problems in approximation and control. (Author)

Let Pn be a sequence of optimal control problems with fixed end times (described by ordinary differential equations, with pointwise and norm (of controls) constraints, and with general functional). Let P0 be an « unperturbed » problem (of the same type). In this paper theorems are obtained about the strong convergence (in some Lp) of optimal controls of Pn to some optimal control of P0, about the uniform convergence xn → x0 of states, and the property thatmin Pnmin P0. As corollaries, convergence theorems for some calculus of variations problems can be derived. Weak convergence theorems of optimal controls of Pn to an optimal one of P0 were considered in[7]. A general abstract theorem about strong convergence of minimum points, generalizing a result in[5], is proved.

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity. SAMPLE RANGE; ORDER STATISTICS; WEAK CONVERGENCE

We employ the theory of weak convergence of cumulative sums to the Wiener Process to obtain large sample theory for cusum tests. These results provide at theoretical basis for studying the effects of serial correlation on the performance of the one-sided cusum test proposed by Page (1955). Particular attention is placed on the first, order auto-regressive and first order moving average models. In order to treat the sequential version of the test, we employ the same Wiener process approximation. This enables us to study the effect of correlation not only on the average run length but, more importantly, on the run length distribution itself. These theoretical distributions are shown to compare quite favorably with the true distribution on the basis of a Monte Carlo study using normal observations. The results on the changes in the shape of the run length distributions show that more than average run length should be considered. Our primary conclusion is that the cusum test is not robust with respect, to dep...

In the present paper we show that the wave operator in the Dirichlet problem for an elliptic differential operator is the limit of those in the appropriate whole space problems. The proof of this fact consists of three parts; the convergence of resolvents [Lemmas 1 and 2], the convergence of semi-groups [Lemma 3] and the convergence of wave operators [Theorem 1]. In proving the convergence of resolvents, we generalize the technique used by Oeda [7]. As for the convergence of semi-groups, we apply a standard method (Kato [2]). And we prove the convergence of wave operators by means of the time-dependent scattering theory. The convergence of scattering operators is also considered [Corollary 1]. We give a comment on the Neumann problem for — t^-\-q(x) [Remark 3].

This note discusses the conditions for convergence of algorithms for finding the minimum of a function of several variables which are based on solving a sequence of one-variable minimization problems. Theorems are given which contrast the weakest conditions for convergence of gradient-related algorithms with those for more general algorithms, including those which minimize in turn along a sequence of uniformly linearly independent search directions.

A convergence theorem of Billingsley for the empirical process of stationary, real valued radom variables under a mixing condition is generalized to the k -dimensional and nonstationary case. Further a more general empirical process is treated, including the upper summation boundary as argument. Some applications are given to a Kolmogorov-Smirnov and a Cramer-von Mises type statistic.

In the paper various types of convergence for sequences of unbounded self-adjoint operators are considered. One of them, the so-called ρ-convergence , seems to be very natural and important. This convergence may be characterized in terms of convergence of groups of unitary operators.

1. SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables. The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable. In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞. A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration. The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities. The paper concludes with a discussion of various related results. It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits. Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman. Transient processes such as epidemics are also shown to exhibit a “central limit” behavior.

The problem of the speed of convergence in the method of Fejer approximations is examined, with an application to the problem of finding at least one solution of the system of inequalitiesfj(x) ≤ 0, j=1,..., m, where thefj(x) are smooth convex functions defined on a real Hilbert space.

We are primarily concerned with minimal P {\text {P}} convergence spaces, where P {\text {P}} is one of the following convergence space properties: Hausdorff, T 2 , λ {{\text {T}}_2}, \lambda -regular, λ \lambda -Urysohn, and first countable, λ \lambda an infinite cardinal number. Our conclusions usually resemble the corresponding topological results, but with some deviations ; for instance, a minimal Hausdorff convergence space is always compact, whereas a countable minimal regular convergence space need not be compact.

1. Introduction. Let (X, C) be a Riesz space (or vector lattice) with positive cone C. A subset B of X is said to be solid if it follows from |x| ≤ |b| with b in B that x is in B (where |x| denotes the supremum of x and − x). The solid hull of B (absolute envelope of B in the terminology of Roberts (2)) is denoted to be the smallest solid set containing B, and is denoted by SB. A locally convex Hausdorff topology on (X, C) is called a locally solid topology if admits a neighbourhood-base of 0 consisting of solid and convex sets in X; and (X, C, ), where is a locally solid topology, is called a locally convex Riesz space.

Bounds on the distribution of the maximum of the absolute values of the "partial sums" constructed from a multi-dimensional array of independent random variables are derived and used to establish the weak convergence of certain random functions defined in terms of such "partial sums".

If ^ = {7^: t ^0} is a one-parameter semigroup of operators on a Banach space X , an element x of X is called ergodic if T t X has a generalized limit as t -> oo. It is shown, for a wide class of semigroups, that the use of Abel or Cesaro limits, and of weak or strong convergence, leads to four equivalent definitions of ergodicity. When the resolvent operator of G has suitable compactness properties, every element of X is ergodic. The ergodic properties of G can be completely determined when its infinitesimal generator is known. Some of these results can be extended to more generaltypes of weak convergence in X , and this leads to a discussion of ergodic properties of the semigroup adjoint to G