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

在本文中，我们证明了用于随机人行天桥的Komlós-Major-Tusnády（KMT）嵌入定理的类似物。我们考虑的随机桥是通过具有iid跳的随机游走构造的，其条件取决于其端点的位置。我们证明，当在跳跃分布的某些温和技术假设下，跳跃为连续值或整数值时，此类桥可以与具有适当方差的布朗桥强耦合。我们的论点遵循与国民党最初的证明相似的二元方案，但是它们需要端点条件所必需的更精确的估计和更强的假设。特别是，我们的结果并非来自KMT嵌入定理，我们通过反例对此进行了说明。