The Gauss-Markov source produces Ui = aUi−1 + Zi for i ≥ 1, where U0 = 0, |a| < 1 and Zi ∼ N (0, σ 2 ) are i.i.d. Gaussian random variables. We consider lossy compression of a block of n samples of the Gauss-Markov source under squared error distortion. We obtain the Gaussian approximation for the Gauss-Markov source with excess-distortion criterion for any distortion d > 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n, d, ) approaches the rate-distortion function R(d)is the dispersion, ∈ (0, 1) is the excess-distortion probability, and Q −1 is the inverse Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 < d ≤ σ 2 (1+|a|) 2 , R(n, d, ) of the Gauss-Markov source coincides with that of Zi, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations.

We consider the problem of communication over a network containing a hidden and malicious adversary that can control a subset of network resources, and aims to disrupt communications. We focus on omniscient nodebased adversaries, i.e., the adversaries can control a subset of nodes, and know the message, network code and packets on all links. Characterizing information-theoretically optimal communication rates as a function of network parameters and bounds on the adversarially controlled network is in general open, even for unicast (single source, single destination) problems. In this work we characterize the information-theoretically optimal randomized capacity of such problems, i.e., under the assumption that the source node shares (an asymptotically negligible amount of) independent common randomness with each network node a priori (for instance, as part of network design). We propose a novel computationally-efficient communication scheme whose rate matches a natural informationtheoretically "erasure outer bound" on the optimal rate. Our schemes require no prior knowledge of network topology, and can be implemented in a distributed manner as an overlay on top of classical distributed linear network coding. I. INTRODUCTIONNetwork coding allows routers in networks to mix packets. This helps attain information-theoretically throughput for a variety of network communication problems; in particular for network multicast [1], [2], often via linear coding operations [3], [4]. Throughput-optimal network codes can be efficiently designed [5], and may even be implemented distributedly [6]. Also, network-coded communication is more robust to packet losses/link-failures [2], [4], [7].However, when the network contains malicious nodes/links, due to the mixing nature of network coding, even a single erroneous packet can cause all packets at the receivers being corrupted. This motivates the problem of network error correction, which was first studied by Cai and Yeung in [8], [9]. They considered an omniscient adversary capable of injecting errors on any z links, and showed that C − 2z was both an inner and outer bound on the optimal throughput, where C is the network-multicast min-cut. Jaggi et al. [10] proposed efficient network codes to achieve this rate. In parallel, Kötter and Kschischang [11] developed a different and elegant approach based on subspace/rank-metric codes to achieve the same rate. Furthermore, when the adversary is of "limited-view" in some manner (for instance, adversary can observe only a sufficiently small subset of transmissions, or is computationally bounded, or is "causal/cannot predict future transmissions"), a higher rate is possible, and in fact [10], [12], [13] proposed a suite of network codes that achieve C − z, all of which meet the network Hamming bound in [8]. A more refined adversary model is considered in [14].Although communication in the presence of link-based adversaries is now relatively well-understood, problems where the adversaries are "node-based" seem to be much more challenging. ...

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This paper provides a precise error analysis for the maximum likelihood estimateâ(u u u) of the parameter a given samples u u u = (u1, . . . , un) drawn from a nonstationary Gauss-Markov process Ui = aUi−1 + Zi, i ≥ 1, where a > 1, U0 = 0, and Zi's are independent Gaussian random variables with zero mean and variance σ 2 . We show a tight nonasymptotic exponentially decaying bound on the tail probability of the estimation error. Unlike previous works, our bound is tight already for a sample size of the order of hundreds. We apply the new estimation bound to find the dispersion for lossy compression of nonstationary Gauss-Markov sources. We show that the dispersion is given by the same integral formula derived in our previous work [1] for the (asymptotically) stationary Gauss-Markov sources, i.e., |a| < 1. New ideas in the nonstationary case include a deeper understanding of the scaling of the maximum eigenvalue of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound.

The Gauss-Markov source produces Ui = aUi−1 + Zi for i ≥ 1, where U0 = 0, |a| < 1 and Zi ∼ N (0, σ 2) are i.i.d. Gaussian random variables. We consider lossy compression of a block of n samples of the Gauss-Markov source under squared error distortion. We obtain the Gaussian approximation for the Gauss-Markov source with excess-distortion criterion for any distortion d > 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n, d,) approaches the rate-distortion function R(d) as R(n, d,) = R(d) + V (d) n Q −1 () + o 1 √ n , where V (d) is the dispersion, ∈ (0, 1) is the excess-distortion probability, and Q −1 is the inverse of the Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 < d ≤ σ 2 (1+|a|) 2 , R(n, d,) of the Gauss-Markov source coincides with that of Zi, the i.i.d. Gaussian noise driving the process (whose dispersion is 1/2), up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations.