Information Rates in Certain Stationary Non-Gaussian Channels in Weak-Signal Transmission

Let $\xi=\{\xi\}$ and $\zeta=\{\zeta_j\}$ be independent discrete-time second-order stationary processes obtained by means of an invertible linear transformation $L$ from a stationary entropy-regular process $X=\{X_j\}$ and a sequence of i.i.d. random variables $Z=\{Z_j\}$ such that $\xi=LX$, and $\zeta=LZ$. Under the assumption that the Fisher information $J(Z_1)$ exists and some additional assumptions on the properties of the linear transformation $L$ and on the density function of $Z_1$, it is shown that the following equality for the information rate $\overline I(\varepsilon\xi;\varepsilon\xi+\zeta)$ holds: $\overline I(\varepsilon\xi;\varepsilon\xi+\zeta)=\frac{1}{2}J(Z_1)\mathbf DX_1\varepsilon^2+o(\varepsilon^2$, $\varepsilon\to\infty$. This result is a generalization of the corresponding results of [M. S. Pinsker et al., IEEE Trans. Inf. Theory, 41, No. 6, 1877–1888 (1995); M. S. Pinsker and V. V. Prelov, Probl. Peredachi Inf., 30, No. 4, 3–11 (1994)], where $\zeta$ was assumed to be Gaussian.