This monograph is devoted to random-set theory, and allows unordered collections of random elements, drawn from an arbitrary space, to be handled. After illustrating its foundations, we focus on Random Finite Sets, i.e., unordered collections of random cardinality of points from an arbitrary space, and show how this theory can be applied to a number of problems arising in wireless communication systems. Three of these problems are: (1) neighbor discovery in wireless networks, (2) multiuser detection in which the number of active users is unknown and time-varying, and (3) estimation of multipath channels where the number of paths is not known a priori and which are possibly time-varying. Standard solutions to these problems are intrinsically suboptimum as they proceed either by assuming a fixed number of vector components, or by first estimating this number and then the values taken on by the components. It is shown how random-set theory provides optimum solutions to all these problems. The complexity issue is also examined, and suboptimum solutions are presented and discussed.
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