Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC

The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over the (skew-)fields $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space $G//K$ is identified with the Weyl chamber $C_q^B\subset \mathbb R^q$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. R\"osler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $p\in[2q-1,\infty[$, and that associated commutative convolution structures $*_p$ on $C_q^B$ exist. In this paper we introduce moment functions and the dispersion of probability measures on $C_q^B$ depending on $*_p$ and study these functions with the aid of this generalized integral representation. Moreover, we derive strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $(C_q^B, *_p)$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitely. For integers $p$, all results have interpretations for $G$-invariant random walks on the Grassmannians $G/K$. Besides the BC-cases we also study the spaces $GL(q,\mathbb F)/U(q,\mathbb F)$, which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $q=1$, the results of this paper are well-known in the context of Jacobi-type hypergroups on $[0,\infty[$.Comment: Extended version of arXiv:1205.4866; some corrections to prior version. Accepted for publication in J. Theor. Proba

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Anomalous dispersion in correlated porous media: A coupled continuous time random walk approach

We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy's law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of heterogeneity point values. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials

Gaussian Multiple and Random Access in the Finite Blocklength Regime

This paper presents finite-blocklength achievabil- ity bounds for the Gaussian multiple access channel (MAC) and random access channel (RAC) under average-error and maximal-power constraints. Using random codewords uniformly distributed on a sphere and a maximum likelihood decoder, the derived MAC bound on each transmitter’s rate matches the MolavianJazi-Laneman bound (2015) in its first- and second-order terms, improving the remaining terms to ½ log n/n + O(1/n) bits per channel use. The result then extends to a RAC model in which neither the encoders nor the decoder knows which of K possible transmitters are active. In the proposed rateless coding strategy, decoding occurs at a time n t that depends on the decoder’s estimate t of the number of active transmitters k. Single-bit feedback from the decoder to all encoders at each potential decoding time n_i, i ≤ t, informs the encoders when to stop transmitting. For this RAC model, the proposed code achieves the same first-, second-, and third-order performance as the best known result for the Gaussian MAC in operation