KMT coupling for random walk bridges

In this paper we prove an analogue of the Komlos–Major–Tusnady (KMT) embedding theorem for random walk bridges. The bridges consider are constructed through walks with i.i.d jumps that conditioned on locations their endpoints. We such can be strongly coupled to Brownian appropriate variance when either continuous or integer valued under some mild technical assumptions jump distributions. Our arguments follow a similar dyadic scheme KMT’s original proof, but they require more refined estimates and stronger necessitated by endpoint conditioning. particular, our result does not from KMT theorem, which illustrate via counterexample.