Correction to "Polar codes for Covert Communications over Asynchronous Discrete Memoryless Channels"
This is a small correction to the proof of Lemma 11 in [1]. Many thanks to Ishaque Ashar for discovering the error.
Erroneous statement
The erroneous statement is after Eq. (79) in the proof of Lemma 11. We claimed that the Markov chain \(Z_{1:i-1}^{1:n}-CW'_i-Z_{i}^{1:n}\) holds. However, looking at the dependencies introduced through the chaining in Fig. 3, we see that the correct Markov chain is instead \(Z_{i}^{1:n}-CW'_i-Z_{i+1:b_n}^{1:n}\).
Fixing the error
Fortunately, there is an easy correction. By Pinsker’s inequality,
\[\mathbb{V}\left(\tilde{p}_{Z_{1:b_n}^{1:n}},\prod_{i=1}^{b_n}\tilde{p}_{Z_i^{1:n}}\right)\leq \mathbb{D}\left(\tilde{p}_{Z_{1:b_n}^{1:n}}\bigg\Vert\prod_{i=1}^{b_n}\tilde{p}_{Z_i^{1:n}}\right).\]The relative entropy can then be bounded as
\[\begin{align} \mathbb{D}\left(\tilde{p}_{Z_{1:b_n}^{1:n}}\bigg\Vert\prod_{i=1}^{b_n}\tilde{p}_{Z_i^{1:n}}\right) &= \sum_{i=1}^{b_n}\mathbb{D}\left(\tilde{p}_{Z_{i}^{1:n}|Z_{i+1:b_n}^{1:n}}\bigg\Vert\tilde{p}_{Z_i^{1:n}}\bigg\vert\tilde{p}_{Z_{i+1:b_n}^{1:n}}\right)\\ &=\sum_{i=1}^{b_n}\mathbb{I}\left(\tilde{Z}_{i}^{1:n};\tilde{Z}_{i+1:b_n}^{1:n}\right)\\ &\leq \sum_{i=1}^{b_n}\mathbb{I}\left(\tilde{Z}_{i}^{1:n};CW'_i\tilde{Z}_{i+1:b_n}^{1:n}\right)\\ &= \sum_{i=1}^{b_n}\left[\mathbb{I}\left(\tilde{Z}_{i}^{1:n};CW'_i\right)+\underbrace{\mathbb{I}\left(\tilde{Z}_{i}^{1:n};\tilde{Z}_{i+1:b_n}^{1:n}|CW'_i\right)}_{=0}\right]\\ & = \sum_{i=1}^{b_n}\mathbb{I}\left(\tilde{Z}_{i}^{1:n};CW'_i\right)\\ &\leq b_n\delta_n^{(5)} \end{align}\]Therefore, \(\mathbb{V}\left(\tilde{p}_{Z_{1:b_n}^{1:n}},\prod_{i=1}^{b_n}\tilde{p}_{Z_i^{1:n}}\right)\leq \sqrt{b_n\delta_n^{(5)}} \leq b_n\sqrt{\delta_n^{(5)}}.\)