Inventory Accumulation with Multiple Products

Sheffield (2016) proposed an inventory accumulation model with two types of products which encodes the critical Fortuin-Kasteleyn model on a random planar map, and showed that a two-dimensional inventory accumulation trajectory in the discrete model scales to a correlated planar Brownian motion. In this work, we generalize the inventory model to k types of products for any integer k ≥ 2, and prove that the corresponding trajectory scales to a k-dimensional Brownian motion with a certain covariance matrix. We also discuss implications of the scaling limit result in inventory theory and beyond.