High Frequency Asymptotics for the Limit Order Book We study the one-sided limit order book for sell (or buy) orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy (or sell) periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We then consider the order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders places their limit sell orders close to the current best price. Our first set of results provide weak limits for the price process and the properly scaled measure-valued order book process in the high frequency regime. In particular, we characterize the limiting measure-valued order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting order book process.