Timeline for order of convergence of the conditional entropy (2)
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 14, 2013 at 7:54 | comment | added | Stéphane Laurent | Actually I'm interested in this example mathoverflow.net/questions/144783/…. Note that I'm expecting that $(**)$ holds for any choice of the $Y_n$. | |
| Oct 14, 2013 at 7:24 | comment | added | Anthony Quas | Notice that $H(X_n|Y_n)=H(X_n,Y_n)-H(Y_n)=H(X_n)-H(Y_n)$, so that condition (**) is asking when $\limsup (H(X_n)-H(Y_n))/H(X_n) < 1$. Equivalently you're asking when is $\liminf H(Y_n)/H(X_n)>0$. I'm not sure what kind of asymptotic conditions you're thinking of, but this really tells you exactly what you need... | |
| Oct 13, 2013 at 21:32 | comment | added | Stéphane Laurent | Do you think we could dream about a sufficient asymptotic condition on the laws of the $X_n$ for $(**)$ to hold ? | |
| Oct 13, 2013 at 17:27 | comment | added | Stéphane Laurent | Thank you. The point $H(X_n) \to \infty$ does not hold with your choice of $N_n$ but one can take $N_n=2^{n^2}$. | |
| Oct 13, 2013 at 17:24 | vote | accept | Stéphane Laurent | ||
| Oct 13, 2013 at 16:07 | history | answered | Anthony Quas | CC BY-SA 3.0 |