  We consider a general asynchronous Stochastic Approximation (SA) scheme
featuring a weighted infinity-norm contractive operator, and prove a bound on
its finite-time convergence rate on a single trajectory. Additionally, we
specialize the result to asynchronous $Q$-learning. The resulting bound matches
the sharpest available bound for synchronous $Q$-learning, and improves over
previous known bounds for asynchronous $Q$-learning.
