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We extend the so-called Kotani Theory for a particular class of ergodic block Jacobi operators defined in l²(Z;Cl) by the law [Hωu]n:= D∗Tn-1ω)un-1 +D(Tnω)un+1 + V(Tnω)un, where T: Ω → Ω is an ergodic automorphism in the measure space (Ω,ν), the map D: Ω →GL(l,R) is bounded, and for each ω ∈ Ω, D(ω) is symmetric and D-1(ω) is bounded. Namely, it is shown that for each r ∈{1, . . ., l}, the essential closure of Zr := {x ∈ R | exactly 2r Lyapunov exponents of Az are zero} coincides with σac,2r(Hω), the absolutely continuous spectrum of multiplicity 2r, where Az is a Schr¨odinger-like cocycle induced by Hω . Moreover, if k ∈ {1, . . .,2l} is odd, then σac,k(Hω) = /0 for ν -a.e. ω ∈ Ω. We also provide a Thouless formula for such class of operators. [ABSTRACT FROM AUTHOR]

In this work we provide a new representation of isometric transformations on the group of -adic integers, then establish an appropriate ergodicity test. [ABSTRACT FROM AUTHOR]

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than 1. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure. [ABSTRACT FROM AUTHOR]

It is known that the Beverton-Holt equation with periodically varying carrying capacity has a globally attracting solution and the solution exhibits attenuation, i.e. the average of the solution over one period is strictly less than the average of the carrying capacity. Interpreted this means a periodically varying environment has a deleterious effect on the average of the solution. Also known is a randomly varying carrying capacity also yields attenuation. In this work the authors show that an almost periodic carrying capacity also yields attenuation. [ABSTRACT FROM AUTHOR]

Letting T denote an ergodic transformation of the unit interval and letting f:[0,1)→R denote an observable, we construct the f-weighted return time measure μy for a reference point y∈[0,1) as the weighted Dirac comb with support in Z and weights f∘Tz(y) at z∈Z, and if T is non-invertible, then we set the weights equal to zero for all z<0. Given such a Dirac comb, we are interested in its diffraction spectrum and analyse it for the dependence on the underlying transformation. Under certain regularity conditions imposed on the interval map and the observable we explicitly calculate the diffraction of μy which consists of a trivial atom and an absolutely continuous part, almost surely with respect to y. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider rigid rotations. In this situation we observe that the diffraction of μy is pure point, almost surely with respect to y and, if the rotation number is irrational and the observable is Riemann integrable, then the diffraction of μy is independent of y. Finally, for a converging sequence of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we develop and analyze a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are introduced to illustrate the analytical result. Highlights • A stochastic delayed SVEIR epidemic model with vaccination is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

We investigate the secrecy performance of a multiuser diversity scheme for an interference-limited wireless network with a base station (BS), $N$ legitimate users, and an eavesdropper in the presence of a single dominant interferer. Assuming interference dominates noise power at the eavesdropper and at each legitimate user's receiver, the BS transmits information to the legitimate user with the $k$ th best (highest) signal-to-interference ratio. We derive a closed-form expression for the secrecy outage probability for an arbitrary $N$ and an asymptotic expression for a fixed $k$ and large $N$. Furthermore, we derive a closed form asymptotic expression for the ergodic secrecy capacity of the $k$ th best user and shows that it scales like $O\left(\log (N)\right)$ for a fixed $k$ and large $N$. [ABSTRACT FROM AUTHOR]

We present a new approach to the statistical characterization of the second-order scattering fading (SOSF) channel model, which greatly simplifies its analysis. Exploiting the unadvertised fact that the SOSF channel can be seen as a continuous mixture of Rician fading channels, we obtain expressions for its probability density function and cumulative density function that are numerically better-behaved than those available in the literature. Our approach allows for obtaining new results for the SOSF model, such as a closed-form expression for its moment-generating function, as well as the characterization of the average channel capacity. Relevantly, and somehow counterintuitively, we observe that in the presence of a strong line-of-sight (LOS) component, the channel capacity of a LOS plus double-Rayleigh scattered diffuse component is larger than its LOS plus Rayleigh (i.e., Rician-like) counterpart. [ABSTRACT FROM AUTHOR]

Rapid increase in number of cellular users and high demand for data has lead to the formation of multi-tier networks. Non-orthogonal multiple access (NOMA) has proved to be an efficient method to cater to the paradigm shift from 4G to 5G. This paper employs NOMA in an heterogeneous cellular network consisting of a macro base station (MBS) tier underlaid with femto base station (FBS) tier and device-to-device (D2D) tier, where NOMA is employed in FBS and D2D tier only. The congestion at the MBS tier is relieved by offloading macro users (MU) to the FBS tier. The offloaded MU are further supported by the D2D tier when the FBS tier fails to find a corresponding pairing user for the incoming offloaded MU. Since, absence of pairing user means outage for offloaded MU, D2D cooperation is employed, which decreases the rate outage probability by $86.87\%$ for the MU offloaded as cell edge user (CEU) in comparison to no cooperation. Also, a three times increase in ergodic rate and four times increase in sum ergodic rate for MU offloaded as CEU is achieved using cooperation from D2D tier. Verification of the results is done using Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

Dual-hop communication systems are known to extend coverage and reduce transmit power in scenarios where one-hop communication is not possible. The $\alpha -\eta -\mu$ fading distribution is a general fading distribution that can be reduced to other fading distributions like Rayleigh, Nakagami- $m$ , $\alpha$ - $\mu$ , and $\eta$ - $\mu$. In this paper, we study the statistics of the end-to-end signal to noise ratio (SNR) of a dual-hop amplify and forward system with semi-blind relay over independent but not necessarily identical $\alpha -\eta -\mu$ fading channels. Specifically, we derive an expression for the $n$ th moment of the end-to-end SNR. The derived expression is then utilized to find the average value, the amount of fading, and the ergodic capacity of the end-to-end SNR. Numerical results are provided and compared to simulations to validate the derived expressions. [ABSTRACT FROM AUTHOR]

We consider a cooperative non-orthogonal multiple access system with radio frequency energy harvesting, in which a user with good channel harvests energy from its received signal and serves as a decode-and-forward relay for enhancing the performance of a user with poor channel. We here aim at maximizing the weighted sum rate of the system by optimizing the power allocation coefficient used at the source and the power splitting coefficient used at the user with good channel. By exploiting the specific structure of the considered problem, we propose a low-complexity one-dimensional search algorithm, which can provide optimal solution to the problem. As a benchmark comparison, we derive analytic expressions and simple high signal-to-noise ratio (SNR) approximations of the ergodic rates achieved at the two users and their weighted sum with fixed values of the power allocation and the power splitting coefficients, from which the scaling of the weighted sum in the high SNR region is revealed. Finally, we provide representative numerical results to demonstrate the validity of our results. [ABSTRACT FROM AUTHOR]

An ergodicity condition for the first component ηt of a controlled Poisson process without boundary is found. The Laplace transform of the same component ηt, t ≥ 0, is obtained from the given transition probabilities of the process {ηt, ξt}, t ≥ 0. It is essential that the given process {ηt, ξt}, t ≥ 0, is a Markov process homogeneous in the second component. [ABSTRACT FROM AUTHOR]

This is a sequel of our previous work on characterization of maximum sum rate of slotted Aloha networks. By extending the analysis to incorporate the capacity-achieving receiver structure, successive interference cancellation (SIC), this paper aims to identify the rate loss due to random access. Specifically, two representative SIC receivers are considered, i.e., ordered SIC, where packets are decoded in a descending order of their received power, and unordered SIC, where packets are decoded in a random order. The maximum sum rate and the corresponding optimal parameter setting including the transmission probability and the information encoding rate in both cases are obtained as the functions of the mean received signal-to-noise ratio (SNR). The comparison to the capture model shows that the gains are significant only with the ordered SIC at moderate values of the mean received SNR $\rho $. With a large $\rho $ , the rate gap diminishes, and they all have the same high-SNR slope of $e^{-1}$ , which is far below that of the ergodic sum capacity of fading channels. The effect of multipacket reception (MPR) on the sum rate performance is also studied by comparing the MPR receivers including SIC and the capture model to the classical collision model. [ABSTRACT FROM AUTHOR]

One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be arbitrarily approximated by ones with zero Lyapunov exponent in the space of Cl(1≤l≤∞) smooth quasi-periodic cocycles, which shows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open in smooth topology. [ABSTRACT FROM AUTHOR]

The Fejér sums of periodic measures and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating in terms of corresponding Fourier coefficients, in fact, using the same formulas. As a result, well-known estimates for the rates of convergence in the von Neumann ergodic theorem can be restated as estimates for the Fejér sums at a point for periodic measures. In this way, natural sufficient conditions for the polynomial growth and polynomial decay of these sums can be obtained in terms of Fourier coefficients. Besides, for example, it is shown that every continuous 2π-periodic function is uniquely determined by its sequence of Fejér sums at any two points whose difference is incommensurable with π. [ABSTRACT FROM AUTHOR]

Abstract: When Gaussian errors are inappropriate in a multivariate linear regression setting, it is often assumed that the errors are iid from a distribution that is a scale mixture of multivariate normals. Combining this robust regression model with a default prior on the unknown parameters results in a highly intractable posterior density. Fortunately, there is a simple data augmentation (DA) algorithm and a corresponding Haar PX‐DA algorithm that can be used to explore this posterior. This paper provides conditions (on the mixing density) for geometric ergodicity of the Markov chains underlying these Markov chain Monte Carlo algorithms. Letting d denote the dimension of the response, the main result shows that the DA and Haar PX‐DA Markov chains are geometrically ergodic whenever the mixing density is generalized inverse Gaussian, log‐normal, inverted Gamma (with shape parameter larger than d/2) or Fréchet (with shape parameter larger than d/2). The results also apply to certain subsets of the Gamma, F and Weibull families. [ABSTRACT FROM AUTHOR]

We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the d-bar metric. As a consequence the set of ergodic measures of such a shift is also arcwise connected in the weak-star topology, and the entropy function over this set attains all values in the interval between zero and the topological entropy of the shift (inclusive). The latter result is motivated by a conjecture of A. Katok. [ABSTRACT FROM AUTHOR]

This paper considers a simultaneous wireless information and power transfer (SWIPT) based decode-and-forward relaying network, where the “harvest-then-forward” strategy is employed. We focus on designing optimal static and dynamic transmission schemes of joint time allocation and power splitting to conduct the relay in terms of outage performance and ergodic performance, respectively. In particular, the analytical expressions for the outage probability and ergodic capacity are derived to determine the optimal static power splitting (PS) and time allocation (TA) ratios. Moreover, we study the dynamic transmission scheme and formulate two joint optimization problems to minimize the outage probability and maximize the instantaneous channel capacity. Considering that the two optimization problems are nonconvex, a split-step iterative method is proposed to obtain the optimal dynamic PS ratio and TA ratio for minimizing the outage probability, and an alternate convex optimization method is employed to solve the problem of maximizing the instantaneous channel capacity. Simulation results verify the advantages of the proposed schemes over three peer schemes. [ABSTRACT FROM AUTHOR]

Consider the problem of a multiple-input multiple-output multiple-access channel at the limit of large number of users. Clearly, in practical scenarios, only a small subset of the users can be scheduled to utilize the channel simultaneously. Thus, a problem of user selection arises. However, since solutions which collect channel state information from all users and decide on the best subset to transmit in each slot do not scale when the number of users is large, distributed algorithms for user selection are advantageous. In this paper, we analyze a distributed user selection algorithm, which selects a group of users to transmit without coordinating between users and without all users sending CSI to the base station. This threshold-based algorithm is analyzed for both zero-forcing and minimum mean square error receivers, and its expected sum rate in the limit of large number of users is investigated. It is shown that for large number of users, it achieves the same scaling laws as the optimal centralized scheme. [ABSTRACT FROM AUTHOR]

The characteristics of the wireless medium create difficulty to shield the data transmission from unauthorized recipients. In this paper, power optimization in an amplify-and-forward untrusted relay network is presented, using cooperative jamming transmission to prevent the untrusted relay from intercepting the confidential signals. Considering imperfect channel estimation error at the destination, an optimal power allocation (OPA) is designed to maximize the achievable secrecy rate for the network. Simplified OPA is derived for high signal-to-noise ratio regime with imperfect CSI and the ergodic secrecy rate is also analyzed to evaluate the achievable average secrecy rate for different scenarios as a common performance metric. The numerical results show that when the error of CSI is considered, the proposed OPA generates limited and acceptable degradation on the secrecy rate. [ABSTRACT FROM AUTHOR]

We show that for every ergodic system ( X , μ , T 1 , … , T d ) with commuting transformations, the average 1 N d + 1 ∑ 0 ≤ n 1 , … , n d ≤ N − 1 ∑ 0 ≤ n ≤ N − 1 f 1 ( T 1 n ∏ j = 1 d T j n j x ) f 2 ( T 2 n ∏ j = 1 d T j n j x ) ⋯ f d ( T d n ∏ j = 1 d T j n j x ) converges for μ -a.e. x ∈ X as N → ∞ . If X is distal, we prove that the average 1 N ∑ n = 0 N − 1 f 1 ( T 1 n x ) f 2 ( T 2 n x ) ⋯ f d ( T d n x ) converges for μ -a.e. x ∈ X as N → ∞ . We also establish the pointwise convergence of averages along cubical configurations arising from a system with commuting transformations. Our methods combine the existence of sated and magic extensions introduced by Austin and Host respectively with ideas on topological models by Huang, Shao and Ye. [ABSTRACT FROM AUTHOR]

Abstract: Unlike for ℓ p, 1 < p ≤ ∞, the discrete Cesàro operator C does not map ℓ1 into itself. We identify precisely those weights w such that C does map ℓ 1 ( w ) continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of C are presented. It is also possible to identify all w such that C is a compact operator in ℓ 1 ( w ). The final section investigates the mean ergodic properties of C in ℓ 1 ( w ). Many examples are presented in order to supplement the results and to illustrate the phenomena that occur. [ABSTRACT FROM AUTHOR]

In this note we identify the distributional limits of non-negative, ergodic stationary processes, showing that all are possible. Consequences for infinite ergodic theory are also explored and new examples of distributionally stable -- and α-rationally ergodic -- transformations are presented. [ABSTRACT FROM AUTHOR]

The main mathematical argument of the universal framework for local equilibrium proposed in Analysis 36, 49 (2016) is condensed and formulated as a fundamental dichotomy between subsets of positive measure and subsets of zero measure in ergodic theory. The physical interpretation of the dichotomy in terms of local equilibria rests on the universality of time scale separation in an appropriate long-time limit. [ABSTRACT FROM AUTHOR]

Quantized eigenenergies and their associated wave functions provide extensive information for predicting the physics of quantum many-body systems. Using a chain of nine superconducting qubits, we implement a technique for resolving the energy levels of interacting photons. We benchmark this method by capturing the main features of the intricate energy spectrum predicted for two-dimensional electrons in a magnetic field—the Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels of the system as it undergoes the transition from a thermalized to a localized phase. Our work introduces a many-body spectroscopy technique to study quantum phases of matter. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to estimate how multiple ergodic averages appart from a given quantity. This problem can be studied by describing a large deviation process for empirical measures as obtained by using the contraction principle. The case of single ergodic averages for empirical measures was already studied by Pfister and Sullivan [Nonlinarity, 10 (2005) 237-261]. To have a more complete picture on empirical measures andV– statistics, we estimate the size of the setsGK= {x:Lr(x) ⊂K}, whereLr(x) is the limit-point set of the sequence of empirical measures andKis a compact subset ofℳ(Xr)withℳ(X)the set of measures onX. In pasrticular, we obtain a variational formula for the topological entropy ofGk. The result of this work about the dimension of the setsGkcan be compared with the one recently circulated by Fan, Schemeling and Wu [arXiv:1206.3214v1 (2012)]. [ABSTRACT FROM PUBLISHER]

A bounded operator A on a real or complex separable infinite-dimensional Banach space Z is universal in the sense of Glasner and Weiss if for every invertible ergodic measure-preserving transformation T of a standard Lebesgue probability space (X, B, μ), there exists an A-invariant probability measure v on Z with full support such that the two dynamical systems (X, B, μ; T) and (Z,BZ, v;A) are isomorphic. We present a general and simple criterion for an operator to be universal, which allows us to characterize universal operators among unilateral or bilateral weighted shifts on lp or c0, to show the existence of universal operators on a large class of Banach spaces and to give a criterion for universality in terms of unimodular eigenvectors. We also obtain similar results for operators which are universal for all ergodic systems (not only for invertible ones) and study necessary conditions for an operator on a Hilbert space to be universal. [ABSTRACT FROM AUTHOR]

In this paper, the channel capacity of secondary user is investigated for opportunistic spectrum sharing with primary user in a Rayleigh fading environment. In the proposed communication scenario, on finding transmission opportunities in licensed band, secondary user utilizes the band as long as the interference power inflicted on primary receiver is below the predefined threshold, and adjusts its transmission power and data rate based on the sensing information available from spectrum sensor. In this context, two different adaptation schemes namely adaptive transmission power scheme and adaptive rate and transmission power scheme are investigated under joint peak and average received power constraints at primary receiver for multilevel quadrature amplitude modulation format. The closed form expressions are derived for the ergodic channel capacities of these schemes and numerical results are presented to validate the theoretical results. Moreover, a comparison between channel capacities is given to illustrate the benefit of using soft sensing information under said constraints. [ABSTRACT FROM AUTHOR]

In this paper, the ergodic channel capacity of the downlink is analyzed for a hybrid satellite-terrestrial cooperative system, which consists of a satellite (the source), a mobile terminal (the destination), and several gap fillers (the relays) located at the ground. The links between the satellite and the relays and the link between the satellite and the destination experience independent shadowed Rician fading, and the links between the relays and the destination experience Rayleigh fading. The maximal ratio combining technique is used at the destination to combine the direct signal received from the satellite and the relayed signals from relays with different cooperative protocols, namely amplify-and-forward (AF) and decode-and-forward (DF). The moment generating function (MGF)-based approach is adopted to derive the closed-form expressions of the ergodic downlink channel capacity of the hybrid satellite-terrestrial cooperative system. The numerical results are compared with Monte Carlo simulations and numerical results calculated with the existing analytical expressions. Comparison results show that the analytical expression derived with the MGF-based approach can achieve a higher accuracy in the low signal-to-noise ratio (SNR) regime for the single relay scenario, and significantly reduce the computational complexity with a little loss of accuracy for multiple relays scenario. On the other hand, the ergodic downlink channel capacity of the hybrid satellite-terrestrial DF cooperative system is generally higher than that of the AF cooperative system. Moreover, the ergodic channel capacity of the hybrid satellite-terrestrial cooperative system decreases as the number of participating relays increases, which can be overcome using the best relay selection strategy. In addition, the ergodic downlink channel capacity of the hybrid satellite-terrestrial cooperative system increases when the channel condition of the link between the satellite and the relay goes better, and is larger than that of the no relay land mobile satellite system when the transmitted SNR is below a certain value. [ABSTRACT FROM AUTHOR]

In the scalar dirty multiple-access channel, in addition to Gaussian noise, two additive interference signals are present, each known non-causally to a single transmitter. It was shown by Philosof et al. that for strong interferences, an independent identically distributed ensemble of codes does not achieve the capacity region. Rather, a structured-codes approach was presented that was shown to be optimal in the limit of high signal-to-noise ratios, where the sum capacity is dictated by the minimal (“bottleneck”) channel gain. In this paper, we consider the multiple-input multiple-output (MIMO) variant of this setting. In order to incorporate structured codes in this case, one can utilize matrix decompositions that transform the channel into effective parallel scalar dirty multiple-access channels. This approach, however, suffers from a “bottleneck” effect for each effective scalar channel and, therefore, the achievable rates strongly depend on the chosen decomposition. It is shown that a recently proposed decomposition, where the diagonals of the effective channel matrices are equal up to a scaling factor, is optimal at high signal-to-noise ratios, under an equal rank assumption. This approach is then extended to any number of transmitters. Finally, an application to physical-layer network coding for the MIMO two-way relay channel is presented. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate the ergodic sum rate (ESR) performance of cellular multiuser two-way relaying networks (CMTWRNs), where a multiantenna base station (BS) exchanges information bidirectionally with one of the several single-antenna mobile stations (MSs) with the help of a single-antenna relay terminal. Specifically, we adopt two transmission schemes (i.e., beamforming (BF) and antenna selection (AS)) at the BS and user selection at MSs to maximize the end-to-end signal-to-noise ratios. Under such transmission schemes, we derive new tight closed-form ESR expressions for the CMTWRNs in the presence of Nakagami- m fading environment. Further, based on the numerical results, we conduct a comparative study between the ESR performances of the two schemes, which indicates that the AS scheme provides approximately equal ESR performance as BF. Therefore, AS scheme can be a good alternative to the BF for the CMTWRNs as it reduces the transceiver complexity and signaling cost. Finally, simulation results are presented to corroborate our theoretical findings. [ABSTRACT FROM AUTHOR]

The main subject of the paper, motivated by a question raised by Boshernitzan, is to give criteria for a bounded complex-valued sequence to be uncorrelated to any strictly ergodic sequence. As a tool developed to study this problem we introduce the notion of correlation between two shiftinvariant measures supported by the symbolic space with complex symbols. We also prove a "lifting lemma" for generic points: given a joining ξ of two shift-invariant measures μ and v, every point x generic for μ lifts to a pair (x, y) generic for ξ (such y exists in the full symbolic space). This lemma allows us to translate correlation between bounded sequences to the language of correlation of measures. Finally, to establish that the property of an invariant measure being uncorrelated to any ergodic measure is essentially weaker than the property of being disjoint from any ergodic measure, we develop and apply criteria for ergodicity of four-jump cocycles over irrational rotations. We believe that apart from the applications to studying the notion of correlation, the two developed tools, the lifting lemma and the criteria for ergodicity of four-jump cocycles, are of independent interest. This is why we announce them also in the title. In the Appendix we also introduce the notion of conditional disjointness. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the ergodicity of stochastic cou- pled fractional Ginzburg-Landau equations driven by α-stable noise on the Torus T. By the maximal inequality for stochastic α-stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero. [ABSTRACT FROM AUTHOR]

We study a continuous-time power policy to maximize the ergodic channel throughput of an energy harvesting transmitter over a Markov fading channel. In particular, we consider transmission power policies that are adapted to the fading process of the channel as well as the storage process of the battery. We obtain a set of equations that determine the probability density of the energy in the battery at each channel state. Specifically, for an ergodic battery storage process, these equations describe the relation between the probability density of stored energy and the transmission power at each channel state. From these equations, we derive an upper bound on the average transmission power and an upper bound on the average transmission rate. To compute a lower bound on the average transmission rate, we apply a calculus of variations technique to a non-linear throughput maximization problem. As a result, we obtain a system of coupled ordinary differential equations for locally optimal power policies. We then focus on the Gilbert–Elliot channel as a special case and derive some structural results for specific classes of fast and slow fading channels. Furthermore, we numerically find a locally optimal transmission power policy for the two channel state scenario. [ABSTRACT FROM PUBLISHER]

Due to the difficulty in manipulating the distribution of Wishart random matrices, the performance analysis of multiple-input-multiple-output (MIMO) channels has mainly focused on deriving capacity bounds via Jensen’s inequality. However, to the best of our knowledge, the tightness of Jensen’s bounds has not yet been rigorously quantified in the general MIMO context. This paper proposes a new methodology for measuring the tightness of Jensen’s bounds via the sandwich theorem. In particular, we first compare the tightness of two different pairs of upper/lower bounds for a general class of MIMO channels based on the unordered eigenvalue of the instantaneous correlation matrix and for arbitrary numbers of antennas. The tightness of Jensen’s bounds in different channel scenarios is investigated including multiuser MIMO with maximal ratio combining. Our analysis is facilitated by deriving some new results for finite-dimensional Wishart matrices, i.e., for the arbitrary moments of the unordered eigenvalue of central and non-central Wishart matrices. Our results provide very interesting insights into the implications of the system parameters, such as the number of antennas, and signal-to-noise ratio, on the tightness of Jensen’s bounds, and showcase the suitability and limitations of Jensen’s bounds. [ABSTRACT FROM PUBLISHER]

In the paper, the semigroup of weak limits of the powers of an infinite transformation of rank one of Chacon type is completely described. [ABSTRACT FROM AUTHOR]

We prove some results on the behavior of infinite sums of the form ∑ f o Tn(x) 1/n, where T: S¹ → S¹ is an irrational circle rotation and f is a mean-zero function on S¹. In particular, we show that for a certain class of functions f, there are Liouville α for which this sum diverges everywhere and Liouville α for which the sum converges everywhere. [ABSTRACT FROM AUTHOR]

We define the entropy of an infinite family C of measurable sets in a probability space, and show that a family has zero entropy if and only if it is totally bounded under the symmetric difference semi-metric. Our principal result is that the mean ergodic theorem holds uniformly for C under every ergodic transformation if and only if C has zero entropy. When the entropy of C is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly strong mixing on C if and only if the entropy of C is zero, and we obtain a corresponding result for weak mixing transformations. [ABSTRACT FROM AUTHOR]

We prove several noncommutative maximal inequalities associated with convex functions, including a Doob type inequality for a convex function of maximal operators on noncommutative martingales, and noncommutative Dunford-Schwartz and Stein maximal ergodic inequalities for a convex function of positive and symmetric positive contractions. The key ingredient in our proofs is a Marcinkiewicz type interpolation theorem for a convex function of maximal operators in the noncommutative setting, which we establish in this paper. These generalize the results of Junge and Xu in the Lp case to the case of convex functions. [ABSTRACT FROM AUTHOR]

In this note, taking the killed Brownian motion as an illustrative model, we derive a conditional deviation inequality for ∫ t 0 V (Xs)ds for certain (unbounded) functions V . Then we apply it to prove a quasi L1-ergodic theorem for the killed process. [ABSTRACT FROM AUTHOR]

In this paper, a simple primal-dual method named PDL is proposed for a convex concave saddle problem and applied to total variational image deblurring. Introduction of linear mapping on proximal term relaxes convergence requirement on pairwise primal-dual stepsize. Simple proof is presented for O(1/N) convergence rate in ergodic sense. Experiments show that performance of PDL is comparable with proximal PDHG (Zhu et al., 2010; Bonettini and Ruggiero, 2012) and PDCP (Chambolle and Pock, 2011) on Gaussian or Salt-Pepper noisy image deblurring. [ABSTRACT FROM AUTHOR]

We begin by presenting a simple lossy compressor operating at near-zero rate: The encoder merely describes the indices of the few maximal source components, while the decoder’s reconstruction is a natural estimate of the source components based on this information. This scheme turns out to be near optimal for the memoryless Gaussian source in the sense of achieving the zero-rate slope of its distortion-rate function. Motivated by this finding, we then propose a scheme comprised of iterating the above lossy compressor on an appropriately transformed version of the difference between the source and its reconstruction from the previous iteration. The proposed scheme achieves the rate distortion function of the Gaussian memoryless source (under squared error distortion) when employed on any finite-variance ergodic source. It further possesses desirable properties, and we, respectively, refer to as infinitesimal successive refinability, ratelessness, and complete separability. Its storage and computation requirements are of order no more than (n^2)/(\log ^\beta n) per source symbol for $\beta >0$ at both the encoder and the decoder. Though the details of its derivation, construction, and analysis differ considerably, we discuss similarities between the proposed scheme and the recently introduced Sparse Regression Codes of Venkataramanan et al. [ABSTRACT FROM AUTHOR]

A view on the physical meaning of the so called ergodic hypothesis: its role on the foundations of equilibrium statistical mechanics in mid '1800, its interpretations and hints at its relevance for modern nonequilibrium statistical mechanics. Followed by appendices with detailed comments on the original papers. [ABSTRACT FROM AUTHOR]

We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms. [ABSTRACT FROM AUTHOR]

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]