Sensitivity of the $\varepsilon$-Entropy of Stationary Continuous-Time Gaussian Processes

Let $N=N(t)$ and $Z=Z(t)$ be independent continuous-time stationary random processes, and let $N$ be Gaussian. Denote by $\overline H_\varepsilon (N+\theta Z)$ the $\varepsilon$-entropy (relative to the mean-square-error criterion) of the process $N+\theta Z$. We prove that for any entropy-regular process $Z$, the limit

$$ S_{\overline H_\varepsilon}(N,Z)=\lim_{\theta\to 0}{1\over\theta^2}[\overline H_\varepsilon (N+\theta Z)-\overline H_\varepsilon (N)], $$

called the sensitivity of the $\varepsilon$-entropy, exists. Moreover, in this case, the equality $S_{\overline H_\varepsilon}(N,Z)=S_{\overline H_\varepsilon}(N,\overline Z)$ holds, where $\overline Z=\overline Z(t)$ is a stationary Gaussian process with the same autocorrelation function as $Z$. An explicit expression for $S_{\overline H_\varepsilon}(N,Z)$ in terms of the spectral densities of $N$ and $Z$ is also derived. Similar results for discrete-time processes have been obtained in [1, 2].