مرکز منطقه ای اطلاع رساني علوم و فناوري - On the <img src="/images/tex/96.gif" alt="\\epsilon"> -entropy and the rate-distortion function of certain non-Gaussian processes

Abstract :

Let $\\xi = {\\xi(t), 0 \\leq t \\leq T}$ be a process with covariance function $K(s,t)$ and $E \\int_0^T \\xi^2(t) dt < \\infty$ . It is proved that for every $\\varepsilon > 0$ the $\\varepsilon$ -entropy $H_{\\varepsilon }(\\xi)$ satisfies begin{equation} H_{varepsilon}(xi_g) - mathcal{H}_{xi_g} (xi) leq H_{varepsilon}(xi) leq H_{varepsilon}(xi_g) end{equation} where $\\xi_g$ is a Gaussian process with the covarianee $K(s,t)$ and $mathcal{H}_{\\xi_g}(\\xi)$ is the entropy of the measure induced by $\\xi$ (in function space) with respect to that induced by $\\xi_g$ . It is also shown that if $mathcal{H}_{\\xi_g}(\\xi) < \\infty$ then, as $\\varepsilon \\rightarrow 0$ begin{equation} H_{varepsilon}(xi) = H_{varepsilon}(xi_g) - mathcal{H}_{xi_g}(xi) + o(1). end{equation} Furthermore, ff there exists a Gaussian process $g = { g(t); 0 \\leq t \\leq T }$ such that $mathcal{H}_g(\\xi) < \\infty$ , then the ratio between $H_{\\varepsilon }(\\xi)$ and $H_{\\varepsilon }(g)$ goes to one as $\\varepsilon$ goes to zero. Similar results are given for the rate-distortion function, and some particular examples are worked out in detail. Some cases for which $mathcal_{\\xi_g}(\\xi) = \\infty$ are discussed, and asymptotic bounds on $H_{\\varepsilon }(\\xi)$ , expressed in terms of $H_{\\varepsilon }(\\xi_g)$ , are derived.