This paper introduces a prototype model in an attempt to capture some aspects of limit order book dynamics simulating market trading mechanisms. We start with a discrete time/space Markov process and then perform a re-scaling procedure leading to a deterministic dynamical system controlled by non-linear ODEs. This allows us to introduce approximants for the equilibrium distribution of the model represented by fixed points of deterministic dynamics.

A system with two infinite-buffer FCFS servers is analyzed and the arrival is formed by three independent Poisson flows /spl Xi//sub i/, of rates /spl lambda//sub i/, i = 0,1,2, each with IID task service times. The tasks from /spl Xi//sub i/ are directed to server i, i = 1,2. The tasks from /spl Xi//sub 0/ are directed to the server that has the shorter workload in the buffer at the time of arrival. We analyze the large deviation probabilities for the virtual waiting time in flow /spl Xi//sub 0/ in the stationary regime.

We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblin-type mixing condition. The estimators are Cesaro averages of longest match-lengths, and their consistency follows from a generalized ergodic theorem due to Maker (1940). We provide examples of their performance on English text, and we generalize our results to countable alphabet processes and to random fields.

We prove an asymptotic relationship between certain longest match-lengths along a single realization of a stationary process and its entropy rate: Given a process X={X/sub n/;n/spl isin/Z} and a realization x from X, we define A/sub i//sup N/(x) as the length of the shortest substring, starting at x/sub i/, that does not appear as a contiguous substring of (x/sub i-N/,x/sub i-N+1/,...,x/sub i-1/). We consider stationary, ergodic processes that have a discrete (finite or infinite) alphabet and also satisfy the Doeblin condition (Kontoyiannis and Suhov, 1994).