The Expected Logarithm of a Noncentral Chi-Square Random Variable
In my Ph.D. thesis I have published a closed-form expression for the expected value of the logarithm of a noncentral chi-square random variable with even degrees of freedom. Recently I have now been able to generalize this to the expected value of the logarithm and also to negative integer moments of a noncentral chi-square random variable with arbitrary (i.e., even or odd) degrees of freedom. Moreover, I have found tight elementary upper and lower bounds to this expression.
Details can be found in
Chi-square, chi-squared, negative moments, noncentral chi-square, noncentral chi-squared, expected logarithm, Rayleigh, Rice, Ricean, Rician.
The following lemma has been proven in (Appendix A, Lemma 3)
Lemma: Consider an random matrix with , where is deterministic while the entries of are zero-mean unit-variance IID complex Gaussian. Denoting by the eigenvalues of we have
where is an matrix with entries
-||- _|_ _|_ / __|__ Stefan M. Moser
Last modified: Wed Sep 21 12:33:50 UTC 2022