A Comparison of Bounds on Three Sets of Copulas with Given Degree of Non-exchangeability

We establish best-possible supremum bounds of copulas with the degree of non-exchangeabilityt=3/4, t=3/5 and t=3/6=1/2, and study the structures of these sets of copulas. The volumes between the upper and lower bounds are calculated to illustrate that the supremum bounds are specific practical and effective in narrowing the Fr\'{e}chet-Hoeffding bounds.     

On the Zero-Outage Secrecy-Capacity of Dependent Fading Wiretap Channels

It is known that for a slow fading Gaussian wiretap channel without channel state information at the transmitter and with statistically independent fading channels, the outage probability of any given target secrecy rate is non-zero, in general. This implies that the so-called zero-outage secrecy capacity (ZOSC) is zero and we cannot transmit at any positive data rate reliably and confidentially. When the fading legitimate and eavesdropper channels are statistically dependent, this conclusion changes significantly. Our work shows that there exist dependency structures for which positive zero-outage secrecy rates (ZOSR) are achievable. In this paper, we are interested in the characterization of these dependency structures and we study the system parameters in terms of the number of observations at legitimate receiver and eavesdropper as well as average channel gains for which positive ZOSR are achieved. First, we consider the setting that there are two paths from the transmitter to the legitimate receiver and one path to the eavesdropper. We show that by introducing a proper dependence structure among the fading gains of the three paths, we can achieve a zero secrecy outage probability (SOP) for some positive secrecy rate. In this way, we can achieve a non-zero ZOSR. We conjecture that the proposed dependency structure achieves maximum ZOSR. To better understand the underlying dependence structure, we further consider the case where the channel gains are from finite alphabets and systematically and globally solve the ZOSC. In addition, we apply the rearrangement algorithm to solve the ZOSR for continuous channel gains. The results indicate that the legitimate link must have an advantage in terms of the number of antennas and average channel gains to obtain positive ZOSR. The results motivate further studies into the optimal dependency structures.