Contents Prologue xi 1 Information Sources 1 1.1 Introduction .............................. 1 1.2 Probability Spaces and Random Variables ............. 1 1.3 Random Processes and Dynamical Systems ............ 5 1.4 Distributions ............................. 6 1.5 Standard Alphabets ......................... 10 1.6 Expectation .............................. 11 1.7 Asymptotic Mean Stationarity ................... 14 1.8 Ergodic Properties .......................... 15 2 Entropy and Information 17 2.1 Introduction .............................. 17 2.2 Entropy and Entropy Rate ..................... 17 2.3 Basic Properties of Entropy ..................... 20 2.4 Entropy Rate ............................. 31 2.5 Conditional Entropy and Information . . ............. 35 2.6 Entropy Rate Revisited ....................... 41 2.7 Relative Entropy Densities ...................... 44 3 The Entropy Ergodic Theorem 47 3.1 Introduction ..........

Information theorists frequently use the ergodic theorem; likewise entropy concepts are often used in information theory. Recently the two subjects have become partially intertwined as deeper results from each discipline find use in the other. A brief history of this interaction is presented in this paper, together with a more detailed look at three areas of connection, namely, recurrence theory, blowing-up bounds, and direct sample-path methods.

ar expended. A more idealistic motivation was that the presentation had merit as filling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash's excellent text [1], which was likely influenced by his original training as an electrical engineer. Still, even that text devotes little e#ort to ergodic theorems, perhaps the most fundamentally important family of results for applying probability theory to real problems. In addition, there are many other special topics that are given little space (or none at all) in most texts on advanced probability and ran

. We consider a metric for probability densities with finite variance on R d , and compare it with other metrics. We use it for several applications, including a uniqueness result for the solution of the spatially homogeneous Boltzmann equation for a gas of true Maxwell molecules. 1. Introduction Denote by P s (R d ), s ? 0, the class of all probability distributions F on R d , d 1, such that Z R d jvj s dF (v) ! 1: We introduce a metric on P s (R d ) by d s (F; G) = sup ¸2R d j b f (¸) \Gamma bg(¸)j j¸j s (1) where b f is the Fourier transform of F , b f(¸) = Z R d e \Gammai¸ \Deltav dF (v): Let us write s = m + ff, where m is an integer and 0 ff ! 1. In order that d s (F; G) be finite, it suffices that F and G have the same moments up to order m. The norm (1) has been introduced in [6] to investigate the trend to equilibrium of the solutions to the Boltzmann equation for Maxwell molecules. There, the case s = 2 + ff, ff ? 0, was considered. Further a...

We introduce and discuss certain models of dilute granular systems of spheres with dissipative collisions and variable coe#cient of restitution, under the assumption of weak inelasticity. The dissipation is taken into account by introducing a correction to the Boltzmann collision operator in the form of a nonlinear friction type operator. Using this correction we obtain formally from the Boltzmann equation in a direct way a hydrodynamic description of a system of nearly elastic particles colliding with a variable coe#cient of restitution. In one dimension of the velocity variable, the correction reduces to the nonlinear friction operator obtained in [24] as the quasi-elastic limit of a model Boltzmann equation for partially anelastic spheres. The large--time asymptotic of this one--dimensional model can be described in detail.

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