Before revision:   For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges to a continuous-time limit when the order arrival rates tend to infinity , the impact of an individual order arrival on the book as well as the tick size tend to zero.
After revision:  For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a coupled SDE-SPDE system when the order arrival rates tend to infinity , the impact of an individual order arrival on the book as well as the tick size tend to zero.