Indoor localization is a key enabler for pervasive computing and network optimization. Wireless local area network (WLAN) positioning systems typically rely on fingerprints of received signal strength (RSS) measures from access points. In this paper, we review approaches for modeling full distributions of Wi-Fi signals, including Bayesian graphical models, smoothing, compressive sensing, and random field differentiation and concentrate on the Kullback-Leibler divergence metric that compares multivariate RSS distributions. We provide theoretical insights on the required spatial density of fingerprints and on the number of samples necessary, during tracking or during signal map building, to differentiate among signal distributions and to provide accurate location estimates. We validate our methods on contrasting datasets where we obtain state-of-the-art localization results. Finally, we exploit datasets collected by a self-localizing mobile robot that continuously records Wi-Fi along with ground truth position, where we define increasingly denser fingerprint grids and study asymptotic localization accuracy.