  We define a stochastic model of a two-sided limit order book in terms of its
key quantities \textit{best bid [ask] price} and the \textit{standing buy
[sell] volume density}. For a simple scaling of the discreteness parameters,
that keeps the expected volume rate over the considered price interval
invariant, we prove a limit theorem. The limit theorem states that, given
regularity conditions on the random order flow, the key quantities converge in
probability to a tractable continuous limiting model. In the limit model the
buy and sell volume densities are given as the unique solution to first-order
linear hyperbolic PDEs, specified by the expected order flow parameters.
