Abstract
We consider the independent sum of a given random variable with a Gaussian variable and an infinitely divisible one. We find a novel tight upper bound on the entropy of the sum which still holds when the variable possibly has an infinite second moment. The proven bound has several implications on both information theoretic problems and infinitely divisible noise channels’ transmission rates.
Highlights
Information inequalities have been investigated since the foundation of information theory
The first one is an upper bound on the entropy of Random Variables (RV)s having a finite second moment by virtue of the fact that Gaussian distributions maximize entropy under a second moment constraint (the entropy h(Y) of a random variable Y having a probability density function p(y) is defined as:
We find a tight upper bound on the entropy of the independent sum of a RV X not necessarily having a finite variance with an infinitely divisible variable having a Gaussian component
Summary
Information inequalities have been investigated since the foundation of information theory. The rest of this section is dedicated to the implications of identity Equation (10) which are fivefold: 1- While the usefulness of this upper bound is clear for RVs X having an infinite second moment for which Equation (1) fails, it can in some cases, present a tighter upper bound than the one provided by Shannon for finite second moment variables X. This is the case, for example, when Z ∼ N (μ1, σ12) and X is a RV having the following PDF: pX(x) =. Equation (10) provides the answer: I(Y; X) ≤ 1 ln (1 + pJ(Z)) , 2 is maximized whenever Z ∼ N
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