Limit theorems in financial market models

Invariance principle states that a scaled simple random walk converges to the standard Brownian motion. In this article, we present a discrete time stochastic process, which reflects a microstructure of market dynamics, and prove a convergence to a scaling limit process with a drift term and a jump term. These terms are derived from a macroscopic condition on volumes traded in some time intervals. The mathematical tools for obtaining our results are Dobrushin–Hryniv theory and the method of cluster expansion developed in mathematical studies of statistical mechanics.