Inventory Accumulation with $k$ Products

TL;DR

This paper generalizes Sheffield's inventory accumulation model from two product types to any number of types, demonstrating that the scaled trajectories converge to a $k$-dimensional Brownian motion with a specific covariance structure.

Contribution

The work extends the inventory accumulation model to $k$ products, proving convergence to a $k$-dimensional Brownian motion, broadening the model's applicability.

Findings

Trajectory scales to a $k$-dimensional Brownian motion.

Convergence holds for any integer $k \,\geq 2$.

Identifies the covariance matrix for the limiting process.

Abstract

Sheffield (2011) proposed an inventory accumulation model with two types of products to study 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 $k \ge 2$, and prove that the corresponding trajectory scales to a $k$-dimensional Brownian motion with an appropriate covariance matrix.