Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.

https://link.springer.com/content/pdf/10.1057%2Fpalgrave.jors.2600523.pdf

Rate control for communication networks: shadow prices, proportional fairness and stability

A survey of current continuous nonlinear multi-objective optimization (MOO) concepts and methods is presented. It consolidates and relates seemingly different terminology and methods. The methods are divided into three major categories: methods with a priori articulation of preferences, methods with a posteriori articulation of preferences, and methods with no articulation of preferences. Genetic algorithms are surveyed as well. Commentary is provided on three fronts, concerning the advantages and pitfalls of individual methods, the different classes of methods, and the field of MOO as a whole. The Characteristics of the most significant methods are summarized. Conclusions are drawn that reflect often-neglected ideas and applicability to engineering problems. It is found that no single approach is superior. Rather, the selection of a specific method depends on the type of information that is provided in the problem, the user’s preferences, the solution requirements, and the availability of software.

Survey of multi-objective optimization methods for engineering

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

It is shown that, particularly for terrestrial cellular telephony, the interference-suppression feature of CDMA (code division multiple access) can result in a many-fold increase in capacity over analog and even over competing digital techniques. A single-cell system, such as a hubbed satellite network, is addressed, and the basic expression for capacity is developed. The corresponding expressions for a multiple-cell system are derived. and the distribution on the number of users supportable per cell is determined. It is concluded that properly augmented and power-controlled multiple-cell CDMA promises a quantum increase in current cellular capacity. >

On the capacity of a cellular CDMA system

The robust form of the likelihood ratio for a signal in the presence of white noise has an additional term in the exponent called the correction factor which corresponds to the trace of the conditional covariance of the signal given the observations. The authors show that this correction term is nothing but the trace of the symmetrized Frechet derivative of the nonlinear filter map and hence the likelihood ratio can be completely represented in terms of the observations and the filter map. >

On the relation between filter maps and correction factors in likelihood ratios

There has been significant recent interest within the net- working research community to characterize the impact of mobility on the capacity and delay in mobile ad hoc networks.In this correspondence, the fundamental tradeoff between the capacity and delay for a mobile ad hoc network under the Brownian motion model is studied.It is shown that the two-hop relaying scheme proposed by Grossglauser and Tse (2001), while capable of achieving a per-node throughput of , incurs an expected packet delay of , where is the variance parameter of the Brownian motion model.It is then shown that an attempt to reduce the delay beyond this value results in the throughput dropping to its value under static settings.In particular, it is shown that under a large class of scheduling and relaying schemes, if the mean packet delay is , for any , then the per-node throughput must be .This result is in sharp contrast to other results that have recently been reported in the literature.

Degenerate Delay-Capacity Tradeoffs in Ad-Hoc Networks

THE STUDY of finitely-additive probability measures arises in problems of modelling and estimation of stochastic signals with bounded variation paths in L,[O, T]. This is because physical signals are of bounded variation and of finite energy for which L,[O, T] is the natural setting and results obtained via Ito theory hold only on a set of Wiener measure 1. It is well known that integrals of paths in &JO, T] lie in a set of Wiener measure zero. Hence, there is a need to construct a theory of white noise to model large bandwidth noise arising in signal analysis with the usual properties associated with it. In a series of papers [l-3], in which he advocated the use of such a framework, Balakrishnan showed that it could indeed be done and, thus, laid the basis for a such a theory. He considered white noise to be the identity map on a Hilbert space H with standard Gauss measure thereon. It is well known that such a measure is only finitely additive on the algebra of cylinder sets and cannot be extended to the Bore1 sets of H. This line of work culminated in the excellent treatise by Kallianpur and Karandikar [4] devoted to filtering and smoothing problems. The key assumption in the development of the theory in the nonlinear context so far, has been that the signal process is assumed to be defined on a countably-additive probability space with paths in H while the measurement noise process is defined on a cylindrical probability space associated with the Gauss measure. The formulation does not allow for signal-noise dependence and one is forced to work with a quasi-cylindrical probability space (a product space). Despite the fact that the mathematical difficulties are daunting, there nevertheless arises the need to develop a complete theory of white noise in order to study modelling issues as well as signal-noise dependence, since these issues arise quite naturally in physical problems. This paper presents one step in such a direction and is motivated by the issue of likelihood ratio evaluation for signals arising in differential systems driven by white noise. The specific problem addressed in this paper is the study of Radon-Nikodym derivatives (and their evaluation) of cylindrical (or finitely-additive) measures induced by nonlinear transformations on H with standard Gauss measure thereon. In the linear case this problem was solved by Balakrishnan [5]. Balakrishnan [2] also obtained some results in the nonlinear case when the transformation is given by I + K where K defines a homogeneous, finite “Volterra” polynomial by exploiting the connection with the pioneering work of Cameron and Martin [6].

On Radon-Nikodym derivatives of finitely additive measures induced by nonlinear transformations on Hilbert space

THE STUDY of finitely-additive probability measures arises in problems of modelling and estimation of stochastic signals with bounded variation paths in L,[O, T] This is because physical signals are of bounded variation and of finite energy for which L,[O, T] is the natural setting and results obtained via Ito theory hold only on a set of Wiener measure 1 It is well known that integrals of paths in &JO, T] lie in a set of Wiener measure zero Hence, there is a need to construct a theory of white noise to model large bandwidth noise arising in signal analysis with the usual properties associated with it In a series of papers [l-3], in which he advocated the use of such a framework, Balakrishnan showed that it could indeed be done and, thus, laid the basis for a such a theory He considered white noise to be the identity map on a Hilbert space H with standard Gauss measure thereon It is well known that such a measure is only finitely additive on the algebra of cylinder sets and cannot be extended to the Bore1 sets of H This line of work culminated in the excellent treatise by Kallianpur and Karandikar [4] devoted to filtering and smoothing problems The key assumption in the development of the theory in the nonlinear context so far, has been that the signal process is assumed to be defined on a countably-additive probability space with paths in H while the measurement noise process is defined on a cylindrical probability space associated with the Gauss measure The formulation does not allow for signal-noise dependence and one is forced to work with a quasi-cylindrical probability space (a product space) Despite the fact that the mathematical difficulties are daunting, there nevertheless arises the need to develop a complete theory of white noise in order to study modelling issues as well as signal-noise dependence, since these issues arise quite naturally in physical problems This paper presents one step in such a direction and is motivated by the issue of likelihood ratio evaluation for signals arising in differential systems driven by white noise The specific problem addressed in this paper is the study of Radon-Nikodym derivatives (and their evaluation) of cylindrical (or finitely-additive) measures induced by nonlinear transformations on H with standard Gauss measure thereon In the linear case this problem was solved by Balakrishnan [5] Balakrishnan [2] also obtained some results in the nonlinear case when the transformation is given by I + K where K defines a homogeneous, finite “Volterra” polynomial by exploiting the connection with the pioneering work of Cameron and Martin [6]

/pdf/on-radon-nikodym-derivatives-of-finitely-additive-measures-1aosskmlft.pdf

On Radon-Nikodym derivatives of finitely-additive measures induced by nonlinear transformations on Hilbert space

We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ? + n with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes.

Ravi R. Mazumdar

Papers

On the relation between filter maps and correction factors in likelihood ratios

Degenerate Delay-Capacity Tradeoffs in Ad-Hoc Networks

On Radon-Nikodym derivatives of finitely additive measures induced by nonlinear transformations on Hilbert space

On Radon-Nikodym derivatives of finitely-additive measures induced by nonlinear transformations on Hilbert space

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit