Output Statistics of Random Binning: Tsallis Divergence and Its Applications

Random binning is a widely used technique in information theory with diverse applications. In this paper, we focus on the output statistics of random binning (OSRB) using the Tsallis divergence $T_\alpha$. We analyze all values of $\alpha \in (0, \infty)\cup\{\infty\}$ and consider three scenarios: (i) the binned sequence is generated i.i.d., (ii) the sequence is randomly chosen from an $\epsilon$-typical set, and (iii) the sequence originates from an $\epsilon$-typical set and is passed through a non-memoryless virtual channel. Our proofs cover both achievability and converse results. To address the unbounded nature of $T_\infty$, we extend the OSRB framework using R\'enyi's divergence with order infinity, denoted $D_\infty$. As part of our exploration, we analyze a specific form of R\'enyi's conditional entropy and its properties. Additionally, we demonstrate the application of this framework in deriving achievability results for the wiretap channel, where Tsallis divergence serves as a security measure. The secure rate we obtain through the OSRB analysis matches the secure capacity for $\alpha \in (0, 2]\cup\{{\infty}\}$ and serves as a potential candidate for the secure capacity when $\alpha \in (2, \infty)$.