Markov random processes are neither bandlimited nor recoverable from samples or after quantization

This paper considers basic questions regarding Markov random processes. It shows that continuous-time, continuous-valued, wide-sense stationary, Markov processes that have absolutely continuous second-order distribution and finite second moment are not bandlimited. It also shows that continuous-time, stationary, Markov processes that are continuous-valued or discrete-valued and satisfy additional mild conditions cannot be recovered from uniform sampling. Further it shows that continuous-time, continuous-valued, stationary, Markov processes that have absolutely continuous second-order distributions and are continuous almost surely, cannot be recovered without error after quantization. Finally, it provides necessary and sufficient conditions for stationary, discrete-time, Markov processes to have zero entropy rate, and relates this to information singularity

Curvature-induced Resolution of Anti-brane Singularities

We study AdS$_7$ vacua of massive type IIA string theory compactified on a 3-sphere with $H_3$ flux and anti-D6-branes. In such backgrounds, the anti-brane backreaction is known to generate a singularity in the $H_3$ energy density, whose interpretation has not been understood so far. We first consider supersymmetric solutions of this setup and give an analytic proof that the flux singularity is resolved there by a polarization of the anti-D6-branes into a D8-brane, which wraps a finite 2-sphere inside of the compact space. To this end, we compute the potential for a spherical probe D8-brane on top of a background with backreacting anti-D6-branes and show that it has a local maximum at zero radius and a local minimum at a finite radius of the 2-sphere. The polarization is triggered by a term in the potential due to the AdS curvature and does therefore not occur in non-compact setups where the 7d external spacetime is Minkowski. We furthermore find numerical evidence for the existence of non-supersymmetric solutions in our setup. This is supported by the observation that the general solution to the equations of motion has a continuous parameter that is suggestive of a modulus and appears to control supersymmetry breaking. Analyzing the polarization potential for the non-supersymmetric solutions, we find that the flux singularities are resolved there by brane polarization as well.Comment: 25 pages, 9 figures. v2: minor changes, discussion of scalar masses adde

A partial solution for lossless source coding with coded side information

This paper considers the problem, first introduced by Ahlswede and Körner in 1975, of lossless source coding with coded side information. Specifically, let X and Y be two random variables such that X is desired losslessly at the decoder while Y serves as side information. The random variables are encoded independently, and both descriptions are used by the decoder to reconstruct X. Ahlswede and Körner describe the achievable rate region in terms of an auxiliary random variable. This paper gives a partial solution for the optimal auxiliary random variable, thereby describing part of the rate region explicitly in terms of the distribution of X and Y

Real Output Costs of Financial Crises: A Loss Distribution Approach

We study cross-country GDP losses due to financial crises in terms of frequency (number of loss events per period) and severity (loss per occurrence). We perform the Loss Distribution Approach (LDA) to estimate a multi-country aggregate GDP loss probability density function and the percentiles associated to extreme events due to financial crises. We find that output losses arising from financial crises are strongly heterogeneous and that currency crises lead to smaller output losses than debt and banking crises. Extreme global financial crises episodes, occurring with a one percent probability every five years, lead to losses between 2.95% and 4.54% of world GDP.Comment: 31 pages, 10 figure

On Lossless Coding With Coded Side Information

This paper considers the problem, first introduced by Ahlswede and Korner in 1975, of lossless source coding with coded side information. Specifically, let X and Y be two random variables such that X is desired losslessly at the decoder while Y serves as side information. The random variables are encoded independently, and both descriptions are used by the decoder to reconstruct X. Ahlswede and Korner describe the achievable rate region in terms of an auxiliary random variable. This paper gives a partial solution for an optimal auxiliary random variable, thereby describing part of the rate region explicitly in terms of the distribution of X and Y

Entropy of Highly Correlated Quantized Data

This paper considers the entropy of highly correlated quantized samples. Two results are shown. The first concerns sampling and identically scalar quantizing a stationary continuous-time random process over a finite interval. It is shown that if the process crosses a quantization threshold with positive probability, then the joint entropy of the quantized samples tends to infinity as the sampling rate goes to infinity. The second result provides an upper bound to the rate at which the joint entropy tends to infinity, in the case of an infinite-level uniform threshold scalar quantizer and a stationary Gaussian random process. Specifically, an asymptotic formula for the conditional entropy of one quantized sample conditioned on the previous quantized sample is derived. At high sampling rates, these results indicate a sharp contrast between the large encoding rate (in bits/sec) required by a lossy source code consisting of a fixed scalar quantizer and an ideal, sampling-rate-adapted lossless code, and the bounded encoding rate required by an ideal lossy source code operating at the same distortion

Low-Resolution Scalar Quantization for Gaussian Sources and Absolute Error

This correspondence considers low-resolution scalar quantization for a memoryless Gaussian source with respect to absolute error distortion. It shows that slope of the operational rate-distortion function of scalar quantization is infinite at the point Dmax where the rate becomes zero. Thus, unlike the situation for squared error distortion, or for Laplacian and exponential sources with squared or absolute error distortion, for a Gaussian source and absolute error, scalar quantization at low rates is far from the Shannon rate-distortion function, i.e., far from the performance of the best lossy coding technique