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

This paper provides a comprehensive review of the domain of physical layer security in multiuser wireless networks. The essential premise of physical layer security is to enable the exchange of confidential messages over a wireless medium in the presence of unauthorized eavesdroppers, without relying on higher-layer encryption. This can be achieved primarily in two ways: without the need for a secret key by intelligently designing transmit coding strategies, or by exploiting the wireless communication medium to develop secret keys over public channels. The survey begins with an overview of the foundations dating back to the pioneering work of Shannon and Wyner on information-theoretic security. We then describe the evolution of secure transmission strategies from point-to-point channels to multiple-antenna systems, followed by generalizations to multiuser broadcast, multiple-access, interference, and relay networks. Secret-key generation and establishment protocols based on physical layer mechanisms are subsequently covered. Approaches for secrecy based on channel coding design are then examined, along with a description of inter-disciplinary approaches based on game theory and stochastic geometry. The associated problem of physical layer message authentication is also briefly introduced. The survey concludes with observations on potential research directions in this area.

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Principles of Physical Layer Security in Multiuser Wireless Networks: A Survey

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

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Stochastic Processes And Filtering Theory

This balanced and comprehensive study presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light Alongside the core subjects in matrix analysis, such as singular value analysis, the solution of matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book The very topical subjects of gradient analysis and optimization play a central role here Also included are subspace analysis, projection analysis and tensor analysis, subjects which are often neglected in other books Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis has in science and engineering, making this book suitable for scientists, engineers and graduate students alike

Matrix Analysis and Applications

It was recently shown that low rank Matrix Completion (MC) theory can support the design of new sampling schemes in the context of MIMO radars, enabling significant reduction of the volume of data required for accurate target detection and estimation. Based on the data received, a matrix can be formulated, which can then be used in standard array processing methods for target detection and estimation. For a small number of targets relative to the number of transmission and reception antennas, the aforementioned data matrix is low-rank and thus can be recovered from a small subset of its elements using MC. This allows for a sampling scheme that populates the data matrix in a uniformly sparse fashion. This paper studies the applicability of MC theory on the type of data matrices that arise in colocated MIMO radar systems. In particular, for the case in which uniform linear arrays are considered for transmission and reception, it is shown that the coherence of the data matrix is both asymptotically and approximately optimal with respect to the number of antennas, and further, the data matrix is recoverable using a subset of its entries with minimal cardinality. Sufficient conditions guaranteeing low matrix coherence and consequently satisfactory matrix completion performance are also presented. These results are then generalized to the arbitrary 2-dimensional array case, providing more general but yet easy to use sufficient conditions ensuring low matrix coherence.

Matrix Completion in Colocated MIMO Radar: Recoverability, Bounds & Theoretical Guarantees

We develop recursive, data-driven, stochastic subgradient methods for optimizing a new, versatile, and application-driven class of convex risk measures, termed here as mean-semideviations, strictly generalizing the well-known and popular mean-upper-semideviation. We introduce the MESSAGEp algorithm, which is an efficient compositional subgradient procedure for iteratively solving convex mean-semideviation risk-averse problems to optimality. We analyze the asymptotic behavior of the MESSAGEp algorithm under a flexible and structure-exploiting set of problem assumptions. In particular: 1) Under appropriate stepsize rules, we establish pathwise convergence of the MESSAGEp algorithm in a strong technical sense, confirming its asymptotic consistency. 2) Assuming a strongly convex cost, we show that, for fixed semideviation order $p>1$ and for $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm achieves a squared-${\cal L}_{2}$ solution suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/2})$ iterations, where, for $\epsilon>0$, pathwise convergence is simultaneously guaranteed. This result establishes a rate of order arbitrarily close to ${\cal O}(n^{-1/2})$, while ensuring strongly stable pathwise operation. For $p\equiv1$, the rate order improves to ${\cal O}(n^{-2/3})$, which also suffices for pathwise convergence, and matches previous results. 3) Likewise, in the general case of a convex cost, we show that, for any $\epsilon\in\left[0,1\right)$, the MESSAGEp algorithm with iterate smoothing achieves an ${\cal L}_{1}$ objective suboptimality rate of the order of ${\cal O}(n^{-\left(1-\epsilon\right)/\left(4\bf{1}_{\left\{ p>1\right\} }+4\right)})$ iterations. This result provides maximal rates of ${\cal O}(n^{-1/4})$, if $p\equiv1$, and ${\cal O}(n^{-1/8})$, if $p>1$, matching the state of the art, as well.

Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models

We consider a source (Alice) trying to communicate with a destination (Bob), in a way that an unauthorized node (Eve) cannot infer, based on her observations, the information that is being transmitted. The communication is assisted by multiple multi-antenna cooperating nodes (helpers) who have the ability to move. While Alice transmits, the helpers transmit noise that is designed to affect the entire space except Bob. We consider the problem of selecting the helper weights and positions that maximize the system secrecy rate. It turns out that this optimization problem can be efficiently solved, leading to a novel decentralized helper motion control scheme. Simulations indicate that introducing helper mobility leads to considerable savings in terms of helper transmit power, as well as total number of helpers required for secrecy communications.

Mobile jammers for secrecy rate maximization in cooperative networks

We consider the problem of enhancing Quality-of-Service (QoS) in mobile relay beamforming networks, by optimally controlling relay motion, in the presence of a dynamic channel We assume a time-slotted system, where the relays update their positions before the beginning of each slot Modeling the wireless channel as a Gaussian spatiotemporal field, we propose a novel 2-stage stochastic programming approach for optimally specifying relay positions and beamforming weights, such that the expected QoS of the network is maximized, based on causal channel state information and under a total relay power budget This results in a scheme where, at each time slot, apart from optimally beamforming to the destination, the relays also optimally decide their positions at the next slot, based on causal experience The stochastic program considered is shown to be equivalent to a set of simple subproblems, which may be solved in a naturally distributed fashion, one at each relay However, exact evaluation of the objective of each subproblem is impossible To mitigate this issue, we propose three efficient, theoretically grounded surrogates to the original subproblems, which rely on the Sample Average Approximation method, the Gauss-Hermite Quadrature, and the Method of Statistical Differentials, respectively The efficacy and several properties of the proposed approach are demonstrated via simulations In particular, we report a substantial improvement of about ${\text{80}{\%}}$ on the average network QoS at steady state, compared to randomized relay motion This shows that strategic relay motion control can result in substantial performance gains, as far as QoS maximization is concerned

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Spatially Controlled Relay Beamforming

This paper revisits grid based recursive approximate filtering of general Markov processes in discrete time, partially observed in conditionally Gaussian noise. The grid based filters considered rely on two types of state quantization, namely, the Markovian type and the marginal type. A set of novel, relaxed sufficient conditions is proposed, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, the notion of conditional regularity of stochastic kernels is introduced which, to the best of the authors' knowledge, constitutes the most relaxed condition under which asymptotic optimality of the respective grid based filters is guaranteed. Further, the convergence results are extended to include filtering of bounded and continuous functionals of the state, as well as recursive approximate state prediction. For both Markovian and marginal quantizations, the whole development of the respective grid-based filters relies more on linear-algebraic techniques and less on measure theoretic arguments, making the presentation considerably shorter and technically simpler.

Dionysios S. Kalogerias

Papers

Matrix Completion in Colocated MIMO Radar: Recoverability, Bounds & Theoretical Guarantees

Recursive Optimization of Convex Risk Measures: Mean-Semideviation Models

Mobile jammers for secrecy rate maximization in cooperative networks

Spatially Controlled Relay Beamforming

Grid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality