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In this paper, a lognormal approximation is proposed for the sum of lognormal processes weighted by binary processes. The analytical approach moves from the method early proposed by Wilkinson for approximating first-order statistics of a sum of lognormal components, and extends to incorporate second-order statistics and the presence of both time-correlated random binary weights and cross-correlated lognormal components in moments' matching. Since the sum of weighted lognormal processes models the signal-to-interference-plus-noise ratio (SINR) of wireless systems, the method can be applied to evaluate in an effective and accurate way the outage occurrence rate and outage duration for different wireless systems of practical interest. In a frequency-reuse-based cellular system, the method is applied for various propagation scenarios, characterized by different shadowing correlation decay distances and correlations among shadowing components. A further case of relevant interest is related to power-controlled wideband wireless systems, where the random weights are binary random variables denoting the activity status of each interfering source. Finally, simulation results are used to confirm the validity of the analysis.