Self-imitation learning motivated by lower-bound Q-learning is a novel and effective approach for off-policy learning. In this work, we propose a n-step lower bound which generalizes the original return-based lower-bound Q-learning, and introduce a new family of self-imitation learning algorithms. To provide a formal motivation for the potential performance gains provided by self-imitation learning, we show that n-step lower bound Q-learning achieves a trade-off between fixed point bias and contraction rate, drawing close connections to the popular uncorrected n-step Q-learning. We finally show that n-step lower bound Q-learning is a more robust alternative to return-based self-imitation learning and uncorrected n-step, over a wide range of benchmark tasks.