Order book dynamics in liquid markets:
limit theorems and diffusion approximations
Rama CONT & Adrien de LARRARD
Columbia University, New York
&
Laboratoire de Probabilités et Modèles Aléatoires
CNRS  Université Pierre et Marie Curie (Paris VI)
Revised Feb 2012
We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jumpdiffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order flow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, selfexciting point processes and models of the ACDGARCH family.
Key words: limit order book, queueing systems, heavy traffic limit, functional central limit theorem, diffusion limit, highfrequency data, market microstructure, point process, limit order market
An increasing proportion of financial transactions in stocks, futures and other contracts take place in electronic markets where participants may submit limit orders (for buying or selling), market orders and order cancelations which are then centralized in a limit order book and executed according to precise time and price priority rules. The limit order book represents, at each point in time, the outstanding orders which are awaiting execution: it consists in queues at different price levels where these orders are arranged according to time of arrival. A limit new buy (resp. sell) order of size increases the size of the bid (resp. ask) queue by . Market orders are executed against limit orders at the best available price: a market order decreases of size the corresponding queue size by . Limit orders placed at the best available price are executed against market orders.
The availability of highfrequency data on limit order books has generated a lot of interest in statistical modeling of order book dynamics, motivated either by highfrequency trading applications or simply a better understanding of intraday price dynamics (see Cont (2011) for a recent survey). The challenge here is to develop statistical models which capture salient features of the data while allowing for some analytical and computational tractability.
Given the discrete nature of order submissions and precise priority rules for their execution, is quite natural to model a limit order book as a queueing system; early work in this direction dates back to Mendelson (1982). More recently, Cont, Stoikov and Talreja Cont et al. (2010b) have studied a Markovian queueing model of a limit order book, in which arrivals of market orders and limit orders at each price level are modeled as independent Poisson processes. Cont and de Larrard (2010) used this Markovian queueing approach to compute useful quantities (the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, conditional on the state of the order book) and relate the volatility of the price with statistical properties of the order flow.
However, the results obtained in such Markovian models rely on the fact that time intervals between orders are independent and exponentially distributed, orders are of the same size and that the order flow at the bid is independent from the order flow at the ask. Empirical studies on highfrequency data show these assumptions to be incorrect (Hasbrouck (2007), Bouchaud et al. (2002, 2008), Andersen et al. (2010)). Figure 1 compares the quantiles of the duration between order book events for CitiGroup stock on June 26, 2008 to those of an exponential distribution with the same mean, showing that the empirical distribution of durations is far from being exponential. Figure 9 shows the autocorrelation function of the inverse durations: the persistent positive value of this autocorrelation shows that durations may not be assumed to be independent. Finally, as shown in Figure 2 which displays the (positive or negative) changes in queue size induced by successive orders for CitiGroup shares, there is considerable heterogeneity in sizes and clustering in the timing of orders.
Other, more complex, statistical models for order book dynamics have been developed to take these properties into account (see Section id1). However, only models based on Poisson point processes such as Cont et al. (2010b), Cont and de Larrard (2010) have offered so far the analytical tractability necessary when it comes to studying quantities of interest such as durations or transition probabilities of the price, conditional on the state of the order book. It is therefore of interest to know whether the conclusions based on Markovian models are robust to a departure from these simplifying assumptions and, if not, how they must be modified in the presence of other distributional features and dependence in durations and order sizes.
The goal of this work is to show that it is indeed possible to restore analytical tractability without imposing restrictive assumptions on the order arrival process, by exploiting the separation of time scales involved in the problem. The existence of widely different time scales, from milliseconds to minutes, makes it possible to obtain meaningful results from an asymptotic analysis of order book dynamics using a diffusion approximation of the limit order book. We argue that this diffusion approximation provides relevant and computationally tractable approximations of the quantities of interest in liquid markets where order arrivals are frequent.
Regime  Time scale  Issues 

Ultrahigh  s  Microstructure, 
frequency (UHF)  Latency  
High  s  Trade 
Frequency (HF)  execution  
“Daily”  s  Trading strategies, 
Option hedging 
Average no. of  Price changes  
orders in 10s  in 1 day  
Citigroup  4469  12499 
General Electric 
2356  7862 
General Motors 
1275  9016 
As shown in Table 1, most applications involve the behavior of prices over time scales an order of magnitude larger than the typical interevent duration: for example, in optimal trade execution the benchmark is the Volume weighted average price (VWAP) computed over a period which may range from 10 minutes to a day: over such time scales much of the microstructural details of the market are averaged out. Second, as noted in Table 2, in liquid equity markets the number of events affecting the state of the order book over such time scales is quite large, of the order of hundreds or thousands. The typical duration (resp. ) between limit orders (resp. market orders and cancelations) is typically (in seconds). These observations show that it is relevant to consider heavytraffic limits in which the rate of arrival of orders is large for studying the dynamics of order books in liquid markets.
In this limit, the complex dynamics of the discrete queueing system is approximated by a simpler system with a continuous state space, which can be either described by a system of ordinary differential equations (in the ’fluid limit’, where random fluctuations in queue size vanish) or a system of stochastic differential equations (in the ’diffusion limit’ where random fluctuations dominate) (Iglehart and Whitt (1970), Harrison and Nguyen (1993), Whitt (2002)). Intuitively, the fluid limit corresponds to the regime of law of large numbers, where random fluctuations average out and the limit is described by average queue size, whereas the diffusion limit corresponds to the regime of the central limit theorem, where fluctuations in queue size are asymptotically Gaussian. When order sizes or durations fail to have finite moments of first or second order, other scaling limits may intervene, involving Lévy processes (see Whitt (2002)) or fractional Brownian motion Araman and Glynn (2011). As shown by Dai and Nguyen (1994), there are also cases where such a ’heavy traffic limit’ may fail to exist. The relevance of each of these asymptotic regimes is, of course, not a matter of ‘taste’ but an empirical question which depends on the behavior of highfrequency order flow in these markets.
Using empirical data on US stocks, we argue that for most liquid stocks, while the rate of arrival of market orders and limit orders is large, the imbalance between limit orders, which increase queue size, and market orders and cancels, which decrease queue size, is an order of magnitude smaller: over, say, a 10 minute interval, one observes an imbalance ranging from 1 to 10 % of order flow. In other words, over a time scale of several minutes, a large number of events occur, but the bid/ask imbalance accumulating over the same interval is of order . In this regime, random fluctuations in queue sizes cannot be ignored and it is relevant to consider the diffusion limit of the limit order book.
In this paper we study the behavior of a limit order book in this diffusion limit: we prove a functional central limit theorem for the joint dynamics of the bid and ask queues when the intensity of orders becomes large, and use it to derive an analytically tractable jumpdiffusion approximation. More precisely, we show that under a wide range of assumptions, which are shown to be plausible for empirical data on liquid US stocks, the intraday dynamics of the limit order book behaves like as a planar Brownian motion in the interior of the positive orthant, and jumps to the interior of the orthant at each hitting time of the boundary.
This jumpdiffusion approximation allows various quantities of interest to be computed analytically: we obtain analytical expressions for various quantities such as the probability that the price will increase at the next price change, and the distribution of the duration between price changes, conditional on the state of the order book.
Our results extend previous analysis of heavy traffic limits for such auction processes (Kruk (2003), Bayraktar et al. (2006), Cont and de Larrard (2010)) to a setting which is relevant and useful for quantitative modeling of limit order books and provide a foundation for recently proposed diffusion models for order book dynamics Avellaneda et al. (2011).
Outline. The paper is organized as follows. Section id1 describes a general framework for the dynamics of a limit order book; various examples of models studied in the literature are shown to fall within this modeling framework (Section id1). Section id1 reviews some statistical properties of high frequency order flow in limit order markets: these properties highlight the complex nature of the order flow and motivate the statistical assumptions used to derive the diffusion limit. Section id1 contains our main result: Theorem 2 shows that, in a limit order market where orders arrive at high frequency, the bid and ask queues behaves like a Markov process in the positive quadrant which diffuses inside the quadrant and jumps to the interior each time it hits the boundary. We provide a complete description of this process, and use it to derive, in Section 1, a simple jumpdiffusion approximation for the joint dynamics of bid and ask queues, which is easier to study and simulate than the initial queueing system.
In particular, we show that in this asymptotic regime the price process is characterized as a piecewise constant process whose transition times correspond to hitting times of the axes by a two dimensional Brownian motion in the positive orthant (Proposition 1). This result allows to study analytically various quantities of interest, such as the distribution of the duration between price moves and the probability of an increase in the price: this is discussed in Section id1.
Empirical studies of limit order markets suggest that the major component of the order flow occurs at the (best) bid and ask price levels (see e.g. Biais et al. (1995)). All electronic trading venues also allow to place limit orders pegged to the best available price (National Best Bid Offer, or NBBO); market makers used these pegged orders to liquidate their inventories. Furthermore, studies on the price impact of order book events show that the net effect of orders on the bid and ask queue sizes is the main factor driving price variations (Cont et al. (2010a)). These observations, together with the fact that queue sizes at the best bid and ask of the consolidated order book are more easily obtainable (from records on trades and quotes) than information on deeper levels of the order book, motivate a reducedform modeling approach in which we represent the state of the limit order book by

the bid price and the ask price

the size of the bid queue representing the outstanding limit buy orders at the bid, and

the size of the ask queue representing the outstanding limit sell orders at the ask
Figure 3 summarizes this representation.
If the stock is traded in several venues, the quantities and represent the best bids and offers in the consolidated order book, obtained by aggregating over all (visible) trading venues. At every time , (resp. ) corresponds to all visible orders available at the bid price (resp. ) across all exchanges.
The state of the order book is modified by order book events: limit orders (at the bid or ask), market orders and cancelations (see Cont et al. (2010b, a), Smith et al. (2003)). A limit buy (resp. sell) order of size increases the size of the bid (resp. ask) queue by , while a market buy (resp. sell) order decreases the corresponding queue size by . Cancelation of orders in a given queue reduces the queue size by . Given that we are interested in the queue sizes at the best bid/ask levels, market orders and cancelations have the same effect on the queue sizes .
The bid and ask prices are multiples of the tick size . When either the bid or ask queue is depleted by market orders and cancelations, the price moves up or down to the next level of the order book. The price processes are thus piecewise constant processes whose transitions correspond to hitting times of the axes by the process .
If the order book contains no ‘gaps’ (empty levels), these price increments are equal to one tick:

when the bid queue is depleted, the (bid) price decreases by one tick.

when the ask queue is depleted, the (ask) price increases by one tick.
If there are gaps in the order book, this results in ’jumps’ (i.e. variations of more than one tick) in the price dynamics. We will ignore this feature in what follows but it is not hard to generalize our results to include it.
The quantity is the bidask spread, which may be one or several ticks. As shown in Table 3, for liquid stocks the bidask spread is equal to one tick for more than of observations.
Bidask spread  1 tick  2 tick  3 tick 

Citigroup 
98.82  1.18  0 
General Electric 
98.80  1.18  0.02 
General Motors 
98.71  1.15  0.14 
When either the bid or ask queue is depleted, the bidask spread widens immediately to more than one tick. Once the spread has increased, a flow of limit sell (resp. buy) orders quickly fills the gap and the spread reduces again to one tick. When a limit order is placed inside the spread, all the limit orders pegged to the NBBO price move in less than a millisecond to the price level corresponding to this new order. Once this happens, both the bid price and the ask price have increased (resp. decreased) by one tick.
The histograms in Figure 4 show that this ’closing’ of the spread takes place very quickly: as shown in Figure 4 (left) the lifetime of a spread larger than one tick is of the order of a couple of milliseconds, which is negligible compared to the lifetime of a spread equal to one tick (Figure 4 , right). In our model we assume that the second step occurs infinitely fast: once the bidask spread widens, it returns immediately to one tick. For the example of Dow Jones stocks (Figure 4 ), this is a reasonable assumption since the widening of the spread lasts only a few milliseconds. This simply means that we are not trying to describe/model how the orders flow inside the bidask spread at the millisecond time scale and, when we describe the state of the order book after a price change we have in mind the state of the order book after the bidask spread has returned to one tick.
Under this assumption, each time one of the queues is depleted, both the bid queue and the ask queues move to a new position and the bidask spread remains equal to one tick after the price change. Thus, under our assumptions the bidask spread is equal to one tick, i.e. , resulting in a further reduction of dimension in the model.
Once either the bid or the ask queue are depleted, the bid and ask queues assume new values. Instead of keeping track of arrival, cancelation and execution of orders at all price levels (as in Cont et al. (2010b), Smith et al. (2003)), we treat the queue sizes after a price change as a stationary sequence of random variables whose distribution represents the depth of the order book in a statistical sense. More specifically, we model the size of the bid and ask queues after a price increase by a stationary sequence of random variables with values in . Similarly, the size of the bid and ask queues after a price decrease is modeled by a stationary sequence of random variables with values in . The sequences and summarize the interaction of the queues at the best bid/ask levels with the rest of the order book, viewed here as a ’reservoir’ of limit orders.
The variables (resp. ) have a common distribution which represents the depth of the order book after a price increase (resp. decrease): Figure 5 shows the (joint) empirical distribution of bid and ask queue sizes after a price move for Citigroup stock on June 26th 2008.
The simplest specification could be to take , to be IID sequences; this approach, used in Cont and de Larrard (2010), turns out to be good enough for many purposes. But this IID assumption is not necessary; in the next section we will see more general specifications which allow for serial dependence.
In summary, state of the limit order book is thus described by a continuoustime process which takes values in the discrete state space , with piecewise constant sample paths whose transitions correspond to the order book events. Denoting by (resp. ) the event times at the ask (resp. the bid), (resp. ) the corresponding change in ask (resp. bid) queue size, and the number of price changes in , the above assumptions translate into the following dynamics for :

If an order or cancelation of size arrives on the ask side at

if , the order can be satisfied without changing the price;

if , the ask queue is depleted, the price increases by one ’tick’ of size , and the queue sizes take new values ,
(1) 

If an order or cancelation of size arrives on the bid side at

if , the order can be satisfied without changing the price;

if , the bid queue gets depleted, the price decreases by one ’tick’ of size and the queue sizes take new values :
(2) 
As in the case of reflected processes arising in queueing networks, the process may be constructed from the net order flow process
where (resp. ) is the number of events (i.e. orders or cancelations) occurring at the bid (resp. the ask) during . is analogous to the ’net input’ process in queuing systems Whitt (2002): (resp. ) represents the cumulative sum of all orders and cancelations at the bid (resp. the ask) between and .
which takes values in the positive orthant, may be constructed from by reinitializing its value to a a new position inside the positive orthant according to the rules (section2.I3.i1)–(section2.I3.i2) each time one of the queues is depleted: every time attempts to exit the positive orthant, it jumps to a a new position inside the orthant, taken from the sequence .
This construction may be done path by path, as follows:
Definition 1
Let be a rightcontinuous function with left limits (i.e. a cadlag function), and two sequences with values in . There exists a unique cadlag function such that

For , let where
is the first exit time of from the positive orthant.

if if .

For :
for , where 
if and otherwise.
The path is obtained by ”regulating” the path with the sequences : in between two exit times, the increments of follow those of and each time the process attempts to exit the positive orthant by crossing the axis (resp. the axis), it jumps to a a new position inside the orthant, taken from the sequence (resp. from the sequence ).
Unlike the more familiar case of a continuous reflection at the boundary, which arises in heavytraffic limits of multiclass queueing systems (see Harrison (1978), Harrison and Nguyen (1993), Whitt (2002), Ramanan and Reiman (2003) for examples), this construction introduces a discontinuity by pushing the process into the interior of the positive orthant each time it attempts to exit from the axes.
To study the continuity properties of this map, we endow with Skorokhod’s topology Billingsley (1968), Lindvall (1973) and the set with the topology induced by ’cylindrical’ seminorms, defined as follows: for a sequence in
is then endowed with the corresponding product topology.
Theorem 1
Let be sequences in which do not have any accumulation point on the axes. If is such that
(3) 
Then the map
(4) 
is continuous at .
Proof: see Section id1 in the Appendix.
This construction may be applied to any cadlag stochastic process: given a cadlag process with values in and (random) sequences and with values in , the process is a cadlag process with values in .
It is easy to see that the order book process may be constructed by this procedure:
Lemma 1
where

is the net order flow at the bid and the ask,

is the sequence of queue sizes after a price increase, and

is the sequence of queue sizes after a price decrease.
One can thus build a statistical model for the limit order book by specifying the joint law of and of the regulating sequences . This approach simplifies the study of the (asymptotic) properties of .
Example 1 (IID reinitializations)
The simplest case is the case where the queue length after each price change is independent from the history of the order book, as in Cont and de Larrard (2010). and are then IID sequences with values in . Figure 5 shows an example of such a distribution for a liquid stock.
The law of the process is then entirely determined by the law of the net order flow and the distributions of , : it can be constructed from the concatenation of the laws of for (where we define ).
Example 2 (Pegged limit orders)
Most electronic trading platforms allow to place limit orders which are pegged to the best quote: if the best quote moves to a new price level, a pegged limit order moves along with it to the new price level. The presence of pegged orders leads to positive autocorrelation and dependence in the queue size before/after a price change. The queue size after a price change may be modeled as

if the price has increased, and

if the price has decreased
where are IID sequences. Empirically, one observes a correlation of between the queue lengths before and after a price change, which suggests an order magnitude for the fraction of pegged orders.
As in the previous example, the law of of the process is determined by the law of the net order flow , the coefficients and the distributions of ,: it can be constructed from the concatenation of the laws of for .
More generally, one could consider other extensions where the queue size after a price move may depend in a (nonlinear) way on the queue size before the price move and a random term representing the inflow of new orders after the th price change:
(5) 
The results given below hold for this general specification although the examples 1 and 2 above are sufficiently general for most applications.
The framework described in Section id1 is quite general: it allows a wide class of specifications for the order flow process,and contains as special cases various models proposed in the literature. Each model involves a specification for the (random) sequences , and or, equivalently, , and where (resp. ) are the durations between order book events on the ask (resp. the bid) side.
Cont and de Larrard (2010) study a stylized model of a limit order market in which market orders, limit orders and cancelations arrive at independent and exponential times with corresponding rates , and , the process becomes a Markov process. If we assume additionally that all orders have the same size, the dynamics of the reduced limit order book is described by:

The sequence is a sequence of independent random variables with exponential distribution with parameter

The sequence is a sequence of independent random variables with exponential distribution with parameter

The sequence is a sequence of independent random variables with

The sequence is a sequence of independent random variables with

All these sequences are independent.
It is readily verified that this model is a special case of the framework of Section id1: may be constructed as in Definition 1, where the unconstrained process is now a compound Poisson process.
Empirical studies of order durations highlight the dependence in the sequence of order durations. This feature, which is not captured in models based on Poisson processes, may be adequately represented by a multidimensional selfexciting point process Andersen et al. (2010), Hautsch (2004), in which the arrival rate of an order of type is represented as a stochastic process whose value depends on the recent history of the order flow: each new order increases the rate of arrival for subsequent orders of the same type (selfexciting property) and may also affect the rate of arrival of other order types (mutually exciting property):
Here measures the impact of events of type on the rate of arrival of subsequent events of type : as each event of type occurs, increases by . In between events, decays exponentially at rate . Maximum likelihood estimation of this model on TAQ data Andersen et al. (2010) shows evidence of selfexciting and mutually exciting features in order flow: the coefficients are all significantly different from zero and positive, with for .
Models based on Poisson process fail to capture serial dependence in the sequence of durations, which manifests itself in the form of clustering of order book events. One approach for incorporating serial dependence in event durations is to represent the duration between transactions and as
where is a sequence of independent positive random variables with common distribution and and the conditional duration is modeled as a function of past history of the process:
Engle and Russell’s Autoregressive Conditional Duration model Engle and Russell (1998) propose an ARMA representation for :
where and are positive constants. The ACDGARCH model Ghysels and Jasiak (1998) combine this model with a GARCH model for the returns. Engle (2000) proposes a GARCHtype model with random durations where the volatility of a price change may depend on the previous durations. Variants and extensions are discussed in Hautsch (2004). Such models, like ARMA or GARCH models defined on fixed time intervals, have likelihood functions which are numerically computable. Although these references focus on transaction data, the framework can be adapted to model the durations and between order book events with the ACD framework Hautsch (2004).
Another way of specifying a stochastic model for the order flow in a limit order market is to use an ’agentbased’ formulation where agent types are characterized in terms of the statistical properties of the order flow they generate. Consider for example a market with three types of traders:

impatient traders who only submit market orders:

patient traders who use only limit orders: this is the case for example of traders who place stop loss orders or engage in strategies such as meanreversion arbitrage or pairs trading which are only profitable with limit orders.

other traders who use both limit and market orders; we will assume these traders submit a proportion of their orders as limit orders and as market orders, where .
Denote by (resp. ) the proportion of orders generated by impatient (resp. patient) traders:
Assume that the sequence of duration between consecutive orders is a stationary ergodic sequence of random variables with , that each trader has an equal chance of being a buyer or a seller and that the type of trader (buyer or seller) is independent from the past:
Trader generates an order of size , where is an IID sequence with:
As described in Section id1, the sequence of order book events –the order flow– is characterized by the sequences and of durations between orders and the sequences of order sizes and . In this section we illustrate the statistical properties of these sequences using highfrequency quotes and trades for liquid US stocks –CitiGroup, General Electric, General Motors– on June 26th, 2008.
Empirical studies Bouchaud et al. (2002, 2008), Gopikrishnan et al. (2000), Maslov and Mills (2001) have shown that order sizes are highly heterogeneous and exhibit heavytailed distributions, with Paretotype tails:
with tail exponent between 2 and 3, which corresponds to a series with finite variance but infinite moments of order . The tail exponent is difficult to estimate precisely, but the Hill estimator Resnick (2006) can be used to measure the heaviness of the tails. Table 4 gives the Hill estimator of the tail coefficient of order sizes for our samples. This estimator is larger than for both the bid and the ask; this means that the sequence of order sizes have a finite moment of order two.
Bid side  Ask side  

Citigroup  [0.42, 0.46]  [0.29, 0.32] 
General Electric 
[0.42, 0.45]  [0.41, 0.46] 
General Motors 
[0.36, 0.42]  [0.44, 0.51] 
The sequences of order sizes and exhibit insignificant autocorrelation, as observed on Figure 6. However, they are far from being independent: the series of squared order sizes and are positively correlated, as revealed by their autocorrelation functions (displayed in Figure 7).
Finally, the sequences and may be negatively correlated. This stems from the fact that a buyer can simultaneously use market orders on the ask side (which correspond to negative values of and limit orders on the bid side (which correspond to positive values of ); the same argument holds for sellers (see Section id1).
These properties of the sequence may be modeled using a bivariate ARCH process:
and are positive coefficients satisfying
(6) 
As shown by Bougerol and Picard (1992), under the assumption (6), the sequences of order sizes and is then a well defined, stationary sequence of random variables with finite secondorder moments, satisfying the properties enumerated above.
The timing of order book events is describe by the sequence of durations at the bid and at the ask. These sequences have zero autocorrelation (see Figure 8) but are not independence sequences: for example, as shown in Figure 9, the sequence of inverse durations and is strongly correlated in each case.
Figure 10 represents the empirical distribution functions and in logarithmic scale. Both empirical distributions exhibit thin, exponentialtype tails (which implies in particular that and have finite expectation).
At very high frequency, the limit order book is described by a twodimensional piecewise constant process , whose evolution is determined by the flow of orders. The complex nature of this order flow –heterogeneity and serial dependence in order sizes, dependence between orders coming at the ask and at the bid– described in section id1, makes it difficult to describe in an analytically tractable manner which would allow the quantities of interest to be computed either in closed form or numerically in real time applications. However, if one is interested in the evolution of the order book over time scales much larger than the interval between individual order book events, the (coarsegrained) dynamics of the queue sizes may be described in terms of a simpler process , called the heavy traffic approximation of . In this limit, the complex dynamics of the discrete queueing system is approximated by a simpler system with a continuous state space, which can be either described by a system of ordinary differential equations (in the ’fluid limit’, where random fluctuations in queue sizes vanish) or a system of stochastic differential equations (in the ’diffusion limit’ where random fluctuations dominate). This idea has been widely used in queueing theory to obtain useful analytical insights into the dynamics of queueing systems Harrison and Nguyen (1993), Iglehart and Whitt (1970), Whitt (2002).
We argue in this section that the heavy traffic limit is highly relevant for the study of limit order books in liquid markets, and that the correct scaling limit for the liquid stocks examined in our data sets is the ”diffusion” limit. This heavy traffic limit is then derived in Section id1 and described in Section sec.Markovianapproximation.
One way of viewing the heavy traffic limit is to view the limit order book at a lower time resolution, by grouping together events in batches of size . Since the interevent durations are finite, this is equivalent to rescaling time by . The impact, on the net order flow, of a batch of events at the ask is
where is the sequence of order sizes at the ask and . Under appropriate assumptions (see next section), this sum behaves approximately as a Gaussian random variable for large :
(7) 
Two regimes are possible, depending on the behavior of the ratio as grows:

If as , the correct approximation is given by the fluid limit, which describe the (deterministic) behavior of the average queue size.

If , the rescaled queue sizes behave like a diffusion process.
The fluid limit corresponds to the regime of law of large numbers, where random fluctuations average out and the limit is described by average queue size, whereas the diffusion limit corresponds to the regime of the (functional) central limit theorem, where fluctuations in queue size are asymptotically Gaussian.
Figure 11 displays the histogram of the ratio for stocks in the Dow Jones index, where for each stock is chosen to represent the average number of order book events in a 10 second interval (typically ). This ratio is shown to be rather small at such intraday time scales, showing that the diffusion approximation, rather than the fluid limit, is the relevant approximation to use here.
Indeed, bid and ask queue sizes exhibit a diffusiontype behavior at such intraday time scales: Figure 12 shows the path of the net order flow process
(8) 
sampled every second for CitiGroup stocks on a typical trading day. In this example, for which the average time between consecutive orders is second, we observe that the process behaves like a diffusion in the orthant with negative drift: the randomness of queue sizes does not average out at this time scale.
We will now show that this is a general result: under mild assumptions on the order flow process, we will show that the (rescaled) queue size process
(9) 
converges in distribution to a Markov process in the positive orthant, whose features we will describe in terms of the statistical properties of the order flow.
Consider now a sequence of processes, where represents the dynamics of the bid and ask queues in the limit order book at a time resolution corresponding to events (see discussion above). The dynamics of is characterized by the sequence of order sizes , durations between orders and the fact that, at each price change

if the price has increased, and

if the price has decreased,
where is an IID sequence with distribution , and is an IID sequence with distribution . Note that this specification includes Examples 1 and 2 as special cases.
We make the following assumptions, which allow for an analytical study of the heavy traffic limit and are sufficiently general to accommodate high frequency data sets of trades and quotes such as the ones described in Section id1:
Assumption 1
is a stationary array of positive random variables whose common distribution has a continuous density and satisfies
(resp. ) represents the arrival rate of orders at the ask (resp. the bid).
Assumption 2
is a stationary, uniformly mixing array of random variables satisfying
(10) 
The assumption of uniform mixing (Billingsley 1968, Ch. 4) implies that the partial sums of order sizes verify a central limit theorem, but allows for various types of serial dependence in order sizes. The scaling assumptions on the first two moments corresponds to the properties of the empirical data discussed in Section id1. Under Assumption 2, one can define
(11) 
may be interpreted as a measure of ‘correlation’ between event sizes at the bid and event sizes at the ask.
These assumptions hold for the examples of Section id1. In the case of the Hawkes model, Assumption 1 was shown to hold in Bacry et al. (2010). Also, these assumptions are quite plausible for high frequnecy quotes for liquid US stocks since, as argued in Section id1:

The tail index of order sizes is larger than two, so the sequences and have a finite second moment.

The sequence of order sizes is uncorrelated i.e. has statistically insignificant autocorrelation. Therefore the sum of autocorrelations of order sizes is finite (zero, in fact).

The sequence of interevent durations has a finite empirical mean and is not autocorrelated.
These empirical observations support the plausibility of Assumptions 1 and 2 for the data sets examined.
Assumption 2 has an intuitive interpretation: if orders are grouped in batches of orders, then Assumption 2 amounts to stating that the variance of batch sizes should scale linear with . This assumption can be checked empirically, using a variance ratio test for example: Figure 13 shows that this linear relation is indeed verifies for the data sets examined in Section id1.
The following scaling assumption states that, when grouping orders in batches of orders, a good proportion of batches should have a size (otherwise their impact will vanish in the limit when becomes large):
Assumption 3
There exist probability distributions on the interior of the positive orthant, such that
Assumption 4
and
Finally, we add the following condition for the initial value of the queue sizes:
(12) 
The following theorem, whose proof is given in the Appendix, describes the joint dynamics of the bid and ask queues in this heavy traffic limit: