St. Petersburg Paradoxes in Performance Analysis of Adaptive Wireless Systems

In this paper we present an adaptive transmission scheme that leads to a surprising situation in analysis of a mathematical model of wireless communications. Namely, the average received energy tends to infinity despite the fact that the average transmitted energy and the representative energy gain of the channel are finite. Yet, it is reasonable to postulate that such a scheme should not be used in any practical wireless system. This anomaly is commonly referred to as St. Petersburg paradox. We discuss possible reasons for the St. Petersburg paradox to occur in analysis of wireless systems and how it can be resolved using suitable performance metrics and fading models. For example, the paradox is resolved by optimizing some performance metric which is either bounded or concave function of the received signal-to-noise ratio. The analysis of St. Petersburg paradox brings new intuition to the normalization of fading channels and optimization of wireless systems.