Event prediction in sequence databases is an important and challenging data mining task. We focus on the specific case of prediction of distant events. Our aim is to mine sequential association rules with consequents that are temporally distant from their antecedents. We therefore propose two new algorithms: D-SR-postMining and D-SR-in-Mining (D-SR stands for Distant Sequential Rules). The originality of these algorithms is that they integrate a minimal gap constraint between the antecedent and the consequent of existing rules, which, as we prove, has an anti-monotonicity property. This approach allows to predict events with enough time in advance (at least as much as the gap). Both algorithms are designed to coexist with legacy rule mining algorithms: D-SR-postMining can be used as a post-processing step of traditional mining algorithms, and D-SR-inMin-ing can be integrated into the mining process of such algorithms. Experiments on three data sets show that both algorithms are efficient for mining distant rules and scalable on large data sets. Even better, D-SR-inMining reduces execution time significantly (up to 9 times). Furthermore, an in-depth analysis of the rules mined from a real-world bank data set, demonstrates the efficiency of such rules for real-world applications such as churn analysis.