Long-range memory test by the burst and inter-burst duration distribution
LLong-range memory test by the burst andinter-burst duration distribution
Vygintas GontisInstitute of Theoretical Physics and Astronomy, Vilnius University
Abstract
It is empirically established that order flow in the financial markets ispositively auto-correlated and can serve as an example of a social systemwith long-range memory. Nevertheless, widely used long-range memoryestimators give varying values of the Hurst exponent. We propose the burstand inter-burst duration statistical analysis as one more test of long-rangememory and implement it with the limit order book data comparing itwith other widely used estimators. This method gives a more reliableevaluation of the Hurst exponent independent of the stock in considerationor time definition used. Results strengthen the expectation that burst andinter-burst duration analysis can serve as a better method to investigatethe property of long-range memory.
The long-range memory in natural and social systems ranges from the levelof rivers to the financial markets. The vast amount of empirical data andobserved power-law statistical properties of volatility and trading activity inthe financial markets are still among the most mysterious features attractingthe permanent attention of researchers [1–5]. From our point of view, thedefinition of long-range memory based on the self-similarity and power-lawstatistical properties are ambiguous as Markov processes can exhibit all theseproperties, including slowly decaying autocorrelation [6–9]. Econometricianstend to conclude that the statistical analysis, in general, cannot be expectedto provide a definite answer concerning the presence or absence of long-rangememory in asset price return [10–12]. Nevertheless, it is widely accepted that thevolatility of prices exhibits long-range memory properties. Thus the alternativemodels, such as FIGARCH, FIEGARCH, LM-ARCH and ARFIMA, includingfractional Brownian noise (fBn), have been proposed for the volatility in thefinancial markets [1, 13–18].Earlier we have proposed an agent-based model, macroscopic dynamics ofwhich can be reduced to a set of stochastic differential equations (SDEs) able1 a r X i v : . [ q -f i n . S T ] M a y o precisely reproduce empirical probability density function (PDF) and powerspectral density (PSD) of absolute return [19, 20] as well as scaling behavior ofvolatility return intervals [21]. Later, investigating empirical PDF of burst andinter-burst duration compared with the model properties we have explained theso-called long-range memory in the financial markets by ordinary non-linear SDEsrepresenting multifractal stochastic processes with non-stationary increments[22, 23]. The proposed description is a realistic alternative to the modelingincorporating fractional Brownian motion (fBm) and might be applicable inthe modeling of other social systems, where models of opinion or populationsdynamics lead to the macroscopic description by the non-linear SDEs [24].From our perspective, there is a fundamental problem empirically establishingwhich of the possible alternatives, fBm or stochastic processes with non-stationaryincrements, is most well-suited to describe natural and social systems exhibitingpower-law statistical properties and self-similarity. Our first idea was to employthe dependence of first passage time PDF on Hurst parameter H for the fBm[22, 23, 25, 26] in the empirical analysis of long-range memory properties for thevolatility in the financial markets.As was shown in the empirical analysis of order flow in the financial markets,there is strong evidence of the persistence of the order signs [27–29]. Authorsemployed the main statistical methods to evaluate the Hurst exponent of theorder flow time series and have found that, in most cases H (cid:39) . . Here weinvestigate burst and inter-burst duration statistical properties of order dis-balance time series seeking to confirm or reject the long-range memory in theorder flow.The rest of this paper is organized as follows. First, we describe methodsused to evaluate the property of long-range memory. Second, we describe useddata sources of limit order books (LOB) and define order dis-balance time series.Further, we apply our methods to the data and demonstrate the advantages ofburst and inter-burst duration analysis. In the concluding part, we discuss theresults and summarize findings. The most widespread definition of long-range memory is based on the power-lawautocovariance γ ( τ ) ∼ τ − α L ( τ ) (1)with divergent integral in the limit τ → ∞ , where < α < and L ( x ) is a slowlyvarying function at infinity. This power-law behavior of autocovariance is relatedto the self-similarity and other power-law statistical properties. Thus, the Hurstexponent H , parameter of self-similarity, is also essential in long-range memorytheory, where H = 1 − α/ . For the short-memory Markov processes H = 1 / and the positively correlated long-memory process . < H < . This definitionof long-range memory is ambiguous, when one deals with the real-time seriesfinite in length, as Markov processes in some regions of variables can generatepower-law statistical properties as well.2he Hurst exponent is a universal parameter of self-similarity. It defines therelation of the standard deviation to the time t , t H L ( t ) , where H = 1 / only inthe cases when stochastic increments are not correlated. Yet another equivalentdefinition of the long-range memory is used in terms of the spectral density forlow frequencies S ( f ) = f − H L ( f ) , (2)where f is the frequency, and L ( f ) is a slowly varying function in the limit f → . Notice that this last condition has the same constrain related to therequirement of time series infinite in time.There are many different methods to evaluate H from empirical time series.Authors seek to combine different methods to increase the reliability of results.In this paper we will use three well known Hurst exponent estimators describedbellow in comparison with our estimation from the exponent of burst andinter-burst duration PDF, see for example [22–24].The first method. Estimation of the spectral density S ( f ) using the peri-odogram method S ( f ) = 12 πt n | j = n (cid:88) j =1 X j e it j f | , (3)where t j is the time of event – occurrence of value change in the time series X j and t n is the length of time series. We evaluate the exponent − H from thepower-law (2) expected for the low values of the frequency.The second method is the rescaled range (R/S) method [30]. R/S is awidely used and described method, in summary, one can divide a time series X , X , ..., X N with equal N − time steps into m subseries each of length n, ( m · n = N ) , mean e j and standard deviation S j . Then define cumulativedeviations from the mean x k,j , the range R j , and the value of rescaled range ( R/S ) n x k,j = i = k (cid:88) i =1 ( X i,j − e j ) ,R j = max k ( x k,j ) − min k ( x k,j ) , (4) ( R/S ) n = (cid:80) mj =1 R j /S j m . We repeat the procedure of ( R/S ) n calculation for the consecutive n , dividingthe series of data into disjoint intervals and finding the mean value ( R/S ) n foreach value of n . Finally, we calculate the Hurst exponent H as the slope in thelog-log graph of lg( R/S ) n versus lg n .The third method is the Multifractal Detrended Fluctuation Analysis (MF-DFA) [31] as a generalized version of Detrended Fluctuation Analysis (DFA) [32].First, we integrate the initial time series of length N , denote it X t , and divideinto m boxes of equal length n . In each box j , a least-squares line is fit to thedata giving us Y n ( k, j ) coordinates of the straight line segments. Then calculate3ariances of deviation from the each trend line F ( n, j ) = n (cid:80) mk =1 ( X n ( k, j ) − Y n ( k, j )) . Average overall segments to obtain the q-th order fluctuation function F q ( n ) = { m m (cid:88) j =1 (cid:2) F ( n, j ) (cid:3) q } q . (5)Finally we calculate the generalized Hurst exponent H q as the slope in the log-loggraph of lg F q ( n ) versus lg n . We use financial data as the best available time series of a social systemswith expected real long-range memory property. High-frequency, easy to uselimit order book data for all NASDAQ traded stocks is provided by LimitOrder Book System LOBSTER [33]. The limit order book (LOB) data thatLOBSTER reconstructs originates from NASDAQ’s Historical TotalView-ITCHfiles (http://nasdaqtrader.com). We have selected only five stocks as LOBSTERprovides AAPL, AMZN, GOOG, INTLC, MSFT as free access sample files ofdata for these stocks.LOBSTER generates two files: message.csv and orderbook.csv for eachselected trading day and ticker (stock). The message.csv file contains the full listof events causing an update of LOB in the requested price range. We includeorders up to the 10 levels of prices for this research. All events are time-stampedin seconds with a precision of at least milliseconds and up to the nanosecondsdepending on the selected period. Both files provide exact information aboutthe instantaneous state of LOB needed to define order dis-balance time series.Any event j changing the LOB state has a time record t j and 10 values of buyvolumes v + k ( t j ) as well as sell volumes v − k ( t j ) , which are retrieved from LOBdata at any moment. We define order dis-balance time series X ( t j ) as follows X ( t j ) = (cid:80) k =1 ( v + k ( t j ) − v − k ( t j )) (cid:80) k =1 ( v + k ( t j ) + v − k ( t j )) . (6)It follows from the definition that order dis-balance time series is bounded andfluctuates in the interval − ≤ X ( t j ) ≤ . It is reasonable to consider that sodefined order dis-balance is a measure of traders opinion about the price andthis empirical time series is an example of opinion dynamics in the social system.For the application of statistical methods, we have to combine time seriesprepared from daily data into series up to the few months. Thus we drop smallparts of daily series X ( t j ) at the very beginning of the day and at the endto ensure that daily series start and end with the value X ( t j ) (cid:39) . Then wecombine these successive daily intervals into continuing order dis-balance timeseries of the needed length.For this research, we use two types of time series: when the time is measuredin seconds as retrieved from data and when the time is measured in ticks ofevents. The event time changes by one tick when any change of order book4
000 4500 5000 5500 6000 6500 7000 - - ( s ) X ( t ) AAPL 4000 4500 5000 5500 6000 6500 7000 - - ( s ) X ( t ) AMZN4000 4500 5000 5500 6000 6500 7000 - - ( s ) X ( t ) INTC 4000 4500 5000 5500 6000 6500 7000 - - ( s ) X ( t ) MSFT
Figure 1: Examples of order dis-balance real-time series. The excerpts ofthe second hour of order dis-balance real-time series for four stocks: AAPL,AMZN, INTC, MSFT traded on the June 21 of 2012 and reconstructed from theLOBSTER sample data.state appears. In Fig 1 we demonstrate the one trading hour (the second in thetrading day) excerpt of order dis-balance real-time series. One can observe thatthe range of fluctuations is varying for different stocks. The intensity of orderflow ν measured, for example, in the number of events per hour, is varying aswell. Values ν for the stocks AAPL, AMZN, GOOG, INTC, MSFT are {60752,49903, 25868, 87163, 106258}. The most innovative part of this contribution is related to the burst and inter-burst duration statistical analysis. The scientific motivation to work on theempirical analysis of burst duration statistics comes from the need for criteria todiscriminate between real long-range memory processes such as fBm and spuriouslong-range memory as modeled by the non-linear SDEs [22–24]. The definition ofburst and inter-burst duration is given in Fig 2, for more details, see [22, 24]. Inthis contribution, we use the only notation T for burst and inter-burst durationas differences are not relevant here.The idea of how to discriminate is based on the PDF of burst and inter-burstduration T defined for the fBm [22, 23, 25, 26] P ( T ) ∼ T H − . (7)Power-law (7) is defined by the Hurst exponent H and any deviation from the5igure 2: Definition of burst and inter-burst duration. Excerpt from a generictime series X ( t ) . Three threshold, h x , passage events, t i , are shown. Thus burstduration can be defined as t − t , and inter-burst duration can be defined as t − t .exponent / or H = 1 / should indicate the presence of correlations in thenoise increments and long-range memory. Markov processes should always giveus at least a part of a burst duration PDF as power-law with exponent / . Inthe previous work, we have shown that non-linear SDEs, being a macroscopicdescription of many agent-based systems or birth-death processes, generatetime series with power-law / [20, 21, 23, 24, 34]. The non-linear SDEs are inthe background of the stochastic models of trading activity and volatility inthe financial markets and explain power-law statistical and long-range memoryproperties, including power-spectral density and auto-correlation [20, 35–37].This type of spurious long-range memory is a very realistic alternative to themodeling incorporating fBm. The empirical analysis of burs and inter-burstduration in the order dis-balance time series should give us an answer aboutwhether the origin of long-range memory in the order flow is spurious or real. We consider the order dis-balance time series retrieved from LOBSTER dataas described in the previous section and compare the statistical properties ofreal-time and event time series. First of all, the complexity of these financialtime series is revealed in the power spectral density (PSD) S ( f ) calculated fromthe periodogram (3). In Fig 3 we demonstrate averaged PSD of real-time seriescalculated for the four stocks: AAPL, AMZN, GOOG, INTC. The average ofspectra here is calculated from the 48 daily series in the period from June 26 toAugust 31 of 2012.Note that the PSD is fractured and one can define at least two power-law S ( f ) ∼ /f β exponents: β for the lower frequencies and β for the higher.From first glance, this property looks similar to the empirical PSD of the returnvolatility in the financial markets [20, 21]. In Table 1 we present values of6 - - f ( Hz ) S ( f ) AAPL 10 - - f ( Hz ) S ( f ) AMZN10 - - f ( Hz ) S ( f ) GOOG 10 - - f ( Hz ) S ( f ) INTC
Figure 3: Power spectral density of order dis-balance real-time series. Thespectra are calculated by periodogram (3) for the four stocks: AAPL, AMZN,GOOG, INTC and is averaged over 48 trading days. Straight lines fit the PSDgiven on the log-log scale.Table 1: Power-law exponents of the power spectral density for the five stocksorder dis-balance time series.
Exponents AAPL AMZN GOOG INTC MSFT β .
46 0 .
65 0 .
58 0 .
75 0 . β .
46 0 .
35 0 .
31 0 .
14 0 . β + β ) / .
46 0 .
50 0 .
45 0 .
45 0 . H .
73 0 .
75 0 .
73 0 .
73 0 . exponents evaluated by the log-log linear fit of PSDs for the five stocks, seefitting lines given in Fig 3.The straightforward interpretation of PSD and its exponents by the relation(2) becomes complicated, when we have fractured PSD with different values of β for the various stocks and frequencies. The fluctuations in order flow intensity,probably, are important and contribute to the behavior of PSD. In Table 1 wepresent the formal estimation of H = (1 + β ) / using ( β + β ) / instead of β . Fortunately, this estimation looks reasonable, comparing it with the othermethods given below. Considerable fluctuations in the slopes of PSDs for thevarious stocks and time scales implies the need to evaluate Hurst exponent usingother methods.The second, R/S method to evaluate Hurst exponent gives more definiteresults than PSD. We chose the same time steps τ = 200 s in the implementation7able 2: Hurst parameter H evaluated by R/S and MDFA methods for the fivestocks order dis-balance real and event time series. Exponents AAPL AMZN GOOG INTC MSFT H , τ = 200 s .
64 0 .
74 0 .
69 0 .
78 0 . H , τ = 500 ticks .
59 0 .
66 0 . H , τ = 2000 ticks .
78 0 . H (2) , τ = 200 s .
59 0 .
57 0 .
55 0 .
64 0 . H (2) , τ = 500 ticks .
60 0 .
59 0 . H ( , τ = 2000 ticks .
64 0 . H ( q ) Real time 1 2 3 4 5 60.560.580.600.620.64 q H ( q ) Event time
Figure 4: Generalized Hurst exponent of order dis-balance real and event timeseries. The generalized Hurst exponent H ( q ) is calculated using the third method(5) for the five stocks: AAPL, AMZN, GOOG, INTC, and MSFT. Here theequal time steps are: τ = 200 s for the real-time series and for the event timeseries τ = 500 ticks for the AAPL, AMZN, GOOG stocks, and τ = 2000 ticks for the INTC, MSFT stocks.of R/S method for the order dis-balance real-time series. In the case of the eventtime series we used τ = 500 ticks for the AAPL, AMZN, GOOG stocks and τ = 2000 ticks for the INTC and MSFT stocks. We provide the list of evaluated H values in Table 2.Though the values of H in Table 2 are scattered, these numbers are compatiblewith the estimation by PSD, other order flow empirical researches [27–29,38] andconfirm the presence of long-range memory. Values of H for the event time serieslook slightly lower than for the real-time, but in both cases, they are scatteredaround H = 0 . and confirm findings of other researches.The multifractal detrended fluctuation analysis is one more essential methodproviding us the information whether time series are multifractal or mono-fractal. This method applies to the non-stationary time series and can be easilyimplemented for the analysis of order dis-balance real and event time series. InFig 4 we provide results of our calculations described here as the third method.These results confirm that the order dis-balance real and event time series,at least for the stocks considered, are mono-fractal as functions H ( q ) are slowlyvarying and almost linear. For both time series H (2) values are in the interval8 H ( q ) Real time 1 2 3 4 5 60.30.40.50.60.70.8 q H ( q ) Event time
Figure 5: The comparison of generalized Hurst exponent of absolute return andorder dis-balance time series. The generalized Hurst exponent H ( q ) is calculatedby the third method (5) for the absolute return and order dis-balance real andevent time series of stocks AAPL. Here we used the three different time stepvalues for the absolute return and one for the order dis-balance time series. Inthe real-time sub-figure: (blue) absolute return τ = 10 s ; (green) absolute return τ = 60 s ; (red) absolute return τ = 200 s ; (black) order dis-balance τ = 200 s .In the event time sub-figure: (blue) absolute return τ = 200 ticks ; (green)absolute return τ = 500 ticks ; (red) absolute return τ = 2000 ticks ; (black)order dis-balance τ = 2000 ticks . . ≤ H (2) ≤ . , see Table 2, and are lower than the values we get by the R/Smethod. Though the Hurst exponent values for the order dis-balance real andevent time series of the investigated stocks are scattered, we have to concludethat R/S and MDFA methods give comparable results and confirm the presenceof the long-range memory in the order dis-balance time series.Many research papers have been devoted to the analyses of multifractalityin the financial data, including absolute return time series [31, 39, 40]. Thequestion which properties of the financial time series contribute to the multifractalbehavior the most is still open. We compare the generalized Hurst exponent H ( q ) evaluated for the absolute return and orders dis-balance real and eventtime series of the stock AAPL in Fig 5. The multifractal behavior is evident inthe absolute return real-time series and is very weak for the order dis-balancereal time series. The event time series exhibit almost constant H ( q ) for bothabsolute return and order dis-balance empirical LOB data for the stock AAPL.Finally, we present our results of the burst and inter-burst duration analysis.From the R/S and MDFA results, we expect that H (cid:39) . and the PDF of T should have, at least in some region, a power-law part with the exponent γ = 2 − H (cid:39) . . We will denote the exponent of another power-law PDF partby γ plotting both lines in the PDF of T log-log plot. Thus in both followingfigures the line γ = 1 . is shown to guide the eye after the power-law we arelooking for. In Fig 6 we demonstrate inter-burst duration, T , PDFs for the threedifferent threshold values of order dis-balance real time series of four stocks:AAPL, AMZN, GOOG, INTC.In the middle part of empirical PDFs, calculated as histograms of T from9 .001 0.100 10 100010 - - - - burst duration ( s ) P D F AAPL, threshold ={
0, 0.3, 0.6 } - - burst duration ( s ) P D F AMZN, threshold ={
0, 0.3, 0.6 } - - - - burst duration ( s ) P D F GOOG, threshold ={
0, 0.3, 0.6 } - - - - burst duration ( s ) P D F INTC, threshold ={
0, 0.1, 0.2 } Figure 6: Inter-burst duration PDF for the order dis-balance real-time series.Sub-figures show PDFs of the following stocks: AAPL, AMZN, GOOG, INTC.Values of thresholds are given in the plot labels of sub-figures. The plots are redfor the lowest thresholds, green for the higher and blue for the highest. Straightlines, power-laws with exponents γ = 0 . and γ = 1 . guide the eye.the time series having up to 25 mln. points, can be fitted very well by thepower-law T − . . In the region of lower T values PDFs can be approximated bythe power-law with exponent γ = 0 . and in the region of the highest valuesone can observe exponential like cut-of as is expected for the fBm or Markovprocesses [25, 26, 41]. These results confirm that order dis-balance real timeseries, at least for the stocks considered, have long-range memory characterizedby the Hurst exponent approximately equal to . .Seeking to find whether observed long-range memory is related to the prop-erties of order flow intensity, we investigate event time series. In Fig 7 wedemonstrate inter-burst duration, T , PDFs for the three different thresholdvalues of order dis-balance event time series for the same stocks as in Fig 6. Inthis case only one power-law with the same exponent γ = 1 . is present.This result confirms long-range memory for the order dis-balance event timeseries with approximately the same Hurst exponent equal to . , and is evenstronger than in the case of real-time series as power-law of PDF spans in theinterval of T values up to the five orders. At the first glance, the defined H is very stable, being the same for the all considered 5 stocks, including MSFTand both types of time series. To quantify this stability, we make the numericalleast-square log-log fit of T PDF in the observed region of power-law. Numericalresults are given in Table 3. Numerical values of H are scattered around thevalues . in the narrow region compared to the other methods of H evaluation,10 - - - - Inter - burst duration ( ticks ) P D F AAPL, threshold ={
0, 0.3, 0.6 }
10 100 1000 10 - - - - Inter - burst duration ( ticks ) P D F AMZN, threshold ={
0, 0.3, 0.6 }
10 100 1000 10 - - - - Inter - burst duration ( ticks ) P D F GOOG, threshold ={
0, 0.3, 0.6 }
10 100 1000 10 - - - - Inter - burst duration ( ticks ) P D F INTC, threshold ={
0, 0.2, 0.4 } Figure 7: Inter-burst duration PDF for the order dis-balance event time series.Sub-figures show PDFs of the following stocks: AAPL, AMZN, GOOG, INTC.Values of thresholds are given in the plot labels of sub-figures. The plots are redfor the lowest thresholds, green for the higher values and blue for the highest.Straight lines, power-laws with exponents γ = 1 . , guides the eye.Table 3: Hurst parameter H evaluated by the burst and inter-burst durationmethod for the five stocks order dis-balance real and event time series. Exponents AAPL AMZN GOOG INTC MSFT H , real time .
72 0 .
74 0 .
75 0 .
70 0 . γ .
28 1 .
26 1 .
25 1 .
30 1 . H , event time .
73 0 .
72 0 .
71 0 .
68 0 . γ .
27 1 .
28 1 .
29 1 .
32 1 . see table 2. The average value of H for all stocks calculated from the real-timeseries is . and calculated from the event time series . . These findings servein a favor for the proposed method to evaluate Hurst exponent from the burstand inter-burst duration analysis. From our point of view, this advantage of themethod is related to the dependence of the evaluated H only on the correlationsof the noise increments and low sensitivity to the other non-linear origins ofpower-law statistical properties. Power-law statistical properties are the characteristic feature of social systems.The financial markets providing us with a vast amount of empirical LOB data11xhibit such power-law statistical properties as well [42]. Here we investigatedburst and inter-burst duration statistical properties in the order dis-balancetime series seeking to develop the test for the presence of long-range memory.Our previous analysis of burst and inter-burst time statistical properties in thetime series of absolute return and trading activity has shown that the observedproperty of long-range property might be spurious [22, 23]. Thus the search forreal long-range memory property with correlated stochastic increments in thelimit order flow has become of greater interest.Here we demonstrate, that order dis-balance time series have many pecu-liarities related to the stock considered. The order flow intensity, the range oforder dis-balance fluctuations, the exponents of PSD, and other measures oflong-range memory are varying for different stocks. PSD with two power-lawexponents reveals the complexity of these time series. Though the values of β , fluctuate considerably from stock to stock, values of ( β + β ) / give a muchbetter estimate of H .We implemented the second, widely used rescaled range method to evaluate H for the real and event time series. Though the defined values of H are scattered,they are comparable for the real and event time series and with the estimatefrom the PSD. Fluctuations of Hurst parameter around H = 0 . are comparablewith defined in order arrivals studied by Lillo, Mike and Farmer [27, 38] showingstatistically significant variations of the estimated values of H for different stocks.Despite considerable variations of Hurst parameter the R/S method confirmsthe long-range memory property.The third MF-DFA method applicable to the non-stationary time seriesgives us additional information on whether the order dis-balance time seriesare mono-fractal or multifractal. This information is essential for comparingstatistical properties observed in the order dis-balance and absolute return timeseries [37, 39]. The MF-DFA implemented for the LOB data confirms thatorder dis-balance real and event time series are mono-fractal for the consideredstocks. The observed dependence of H ( q ) on q is low and can be comperedwith the H ( q ) variations from stock to stock. For the absolute return real-timeseries extracted from the AAPL LOB data, Fig 5, H ( q ) exhibits much strongernon-linear dependence. The detailed analysis of multifractal spectra is itselfeffort consuming task [40]. Here we made very preliminary multifractal testingof order dis-balance time series concluding mono-fractal behavior. Even for theabsolute return, multifractal behavior disappeared when we switched to theevent time series, see Fig 5. Values H (2) defined by MF-DFA are slightly lowerthan H defined by the R/S method and confirm that widely used long-rangememory estimators give varying results.We investigated statistical properties of burst and inter-burst duration inorder dis-balance real and event time series seeking to develop one more test oflong-range memory. The big choice of LOB data available and high expectationthat the order flow has a real long-range memory property attracted our attentionto this type of social system. The results of this empirical analysis are givenin Fig 6, Fig 7, and Table 3. All calculated histograms of inter-burst durationfor the five stocks investigated can be well-fitted in the interval of three orders12or the real-time series and in the interval of four orders for the event timeseries by the same power-law with exponent γ = 1 . . Though the more precisenumerical evaluation of the exponents γ , see results in Table 3, are slightlydispersed, corresponding values of Hurst exponent are in a very narrow regionaround . . From our point of view, these results confirm that the limit orderflow exhibits real long-range memory, and with high probability, this propertymay be independent of the stock and time definition.Our expectation that burst and inter-burst duration analysis can give anadditional value in the estimation of long-range memory property is based onthe previous study of the first passage time problem in birth-death processesand the duration PDF in-variance regarding non-linear transformations of thetime series [22, 24, 34, 41, 43]. This empirical analysis of LOB data strengthensour expectation providing evidence that the definition of Hurst exponent usingburst and inter-burst duration analysis can be more reliable in comparison withother widely used methods. 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