  In this paper, we study various new Hawkes processes, namely, so-called
general compound and regime-switching general compound Hawkes processes to
model the price processes in the limit order books. We prove Law of Large
Numbers (LLN) and Functional Central Limit Theorems (FCLT) for these processes.
The latter two FCLTs are applied to limit order books where we use these
asymptotic methods to study the link between price volatility and order flow in
our two models by studying the diffusion limits of these price processes. The
volatilities of price changes are expressed in terms of parameters describing
the arrival rates and price changes.
