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

Convergence of Probability Measures

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

Space-Time Block Coding for Wireless Communications

In studying the performance of coded communications over memoryless channels (with or without fading), the results are given as upper bounds on the average bit error probability (BEP). In principle, there are three different approaches to arriving at these bounds, all of which employ obtaining the so-called pairwise error probability , or the probability of choosing one symbol sequence over another for a given pair of possible transmitted symbol sequences, followed by a weighted summation over all pairwise events. In this chapter, we will focus on the results obtained from the third approach since these provide the tightest upper bounds on the true performance. The first emphasis will be placed on evaluating the pairwise error probability with and without CSI, following which we shall discuss how the results of these evaluations can be used via the transfer bound approach to evaluate average BEP of coded modulation transmitted over the fading channel.

Coded Communication over Fading Channels

A Pareto-efficient, goal-driven, and distributed power control scheme for wireless networks is presented. We use a non-cooperative game-theoretic approach to propose a novel pricing scheme that is linearly proportional to the signal-to-interference ratio (SIR) and analytically show that with a proper choice of prices (proportionality constants), the outcome of the noncooperative power control game is a unique and Pareto-efficient Nash equilibrium (NE). This can be utilized for constrained-power control to satisfy specific goals (such as fairness, aggregate throughput optimization, or trading off between these two goals). For each one of the above goals, the dynamic price for each user is also analytically obtained. In a centralized (base station) price setting, users should inform the base station of their path gains and their maximum transmit-powers. In a distributed price setting, for each goal, an algorithm for users to update their transmit-powers is also presented that converges to a unique fixed-point in which the corresponding goal is satisfied. Simulation results confirm our analytical developments.

Pareto-efficient and goal-driven power control in wireless networks: a game-theoretic approach with a novel pricing scheme

In this paper we consider the compressive sensing based encryption and proposed the conditions in which the perfect secrecy is achievable. We prove that when the measurement matrix holds the Restricted Isometry Property (RIP) and the number of measurements is more than two times of the sparsity level, i.e., M ≥ 2k, the Shannon perfect secrecy condition is achievable either i) the cardinality of the message set tends to infinity or ii) the message set does not include the zero message. As an implicit assumption, we suppose that the eavesdropper has not access to the secret key during the transmission.

Perfect secrecy via compressed sensing

In this study the negative impacts of interference transmitters on automatic modulation classification (AMC) have been discussed. The authors proposed two approaches for AMC in the presence of interference: single user modulation classification (SUMC) and multiuser modulation classification (MUMC). When the received power of one transmitter is larger than the other transmitters, SUMC approach recognises the modulation type of the dominant transmitter and other transmitters are treated as interferences. Alternatively when the received powers of all the transmitters are close to each other the authors propose MUMC method to recognise the modulation type of all of the transmitted signals. The features being used to recognise the modulation types of transmitted signals for the both approaches, SUMC and MUMC are higher order cumulants. The superposition property of cumulants for independent random variables is utilised for SUMC and MUMC. The authors investigated the robustness of their classifier with respect to different powers of the received signals via analytical and simulation results and the authors have shown the analytical results will be confirmed by simulations. Also the authors studied the effects of signal synchronisation error, phase jitter and frequency offset on their proposed methods via simulations in the both condition for MUMC and SUMC.

Multiuser modulation classification based on cumulants in additive white Gaussian noise channel

In this article, we study the power control (PC) process in wireless cellular code division-multiple access systems under flat fading and propose a novel approach to find an optimum step size for closed-loop power control algorithms. In this approach, an optimum step size will be computed from a proposed function. This function depends on system parameters such as, the number of co-channel users, processing gain, the period of PC, Doppler frequency, channel attenuation and the order of diversity. Based on this computation, the mobile station (MS) adjusts its transmit power optimally to decrease interference for other co-channel users. Simulation results for different sets of system parameters show that the proposed algorithm decreases the bit error rate, the outage probability at the base station (BS), and increases the battery life of the MS compared with other values of the step size. The performance of the proposed algorithm is compared with the fixed-step-size power control algorithm and superiority of its performance is confirmed by simulation results. Moreover, the upper and lower bounds of the outage probability and the received signal-to-interference ratio for the proposed algorithm at the BS will be calculated.

Novel approach to adjust the step size for closed-loop power control in wireless cellular code division multiple access systems under flat fading

In this letter, we propose a new robust MUD, called /spl alpha/ detector, for non-Gaussian noise. We consider the Gaussian-mixture model for non-Gaussian or impulsive noise. Our technique outperforms the decorrelator and the minimax detectors in highly impulsive noise. The proposed method uses a parametric cost function, where the parameter /spl alpha/ is selected using the difference between the asymptotic variance of estimation error of the /spl alpha/ detector and that of the minimax detector.

Babak Seyfe

Papers

Pareto-efficient and goal-driven power control in wireless networks: a game-theoretic approach with a novel pricing scheme

Perfect secrecy via compressed sensing

Multiuser modulation classification based on cumulants in additive white Gaussian noise channel

Novel approach to adjust the step size for closed-loop power control in wireless cellular code division multiple access systems under flat fading

A new choice of penalty function for robust multiuser detection based on M-estimation