**3. Results**

26 Neuroendocrinology and Behavior

**Figure 8.** Rat is on the platform after it found the hidden platform.

**Figure 9.** Total distance travelled is high. Time to find the platform is long.

Data were analyzed using SPSS (SPSS Statistical Software, SPSS Inc., Los Angeles, CA, USA, Ver. 15.0). A 2 (pinealectomy and control) X 4 (treatments: saline, diazepam, 1 μg/kg melatonin and 100 μg/kg melatonin) ANOVA analyses on data were performed with the last factor as a within subject or repeated design. Significant ANOVA results were also tested by the post test, namely the Tukey test which is assumed to be a strong test for comparison of groups that has equal variance and sample size. Values were considered statistically significant at p ≤ 0.05. Data are presented as MEAN SEM after back

**2.10. Statistical analyses**

transforming from ANOVA results.

#### **3.1. Anxiety measures**

#### *3.1.1. Open field measurements*

#### *<IP_ADDRESS>. Total distance travelled*

An interaction effect between the group and the treatment was significant on the total distance travelled on the open field, *F*(3, 36) = 6.15, *p* < 0.002, η2 = .34. This effect reflected the fact that in control condition subjects received 100 μg/kg melatonin (M = 699.65) and 1 μg/kg melatonin (M = 690.46) treatments travelled less distance than those received diazepam (M = 1400.04) and saline (M = 1214.95), whereas in the pinealectomy condition, the subjects received diazem (M = 643.75) travelled less distance than those received 100 μg/kg melatonin (M = 1070.22), 1 μg/kg melatonin (M = 914.38) and saline (M = 902.11) treatments.

**Figure 10.** The total distance travelled. Right striated bar represents the saline injection and black bar represents diazepam injections, cross striated bar represents 1 μg/kg melatonin injection and bricks striated bar represents 100 μg/kg melatonin injection for both control and pinealectomy groups. Data are presented as means (± S.E.M.). Different letters indicate the statistically different groups. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

#### *<IP_ADDRESS>. Time spent at the edge of the open field (Edge duration)*

An interaction effect between the group and the treatment was also significant, *F* (3, 36) = 5.38, *p* = .004, η2 = .31. Indicating that in control condition subjects received diazepam spent less time than the other treatments whereas, in pinealectomy condition the subjects were not significantly different from each other.

#### *<IP_ADDRESS>. Time spent at the center of the open field (Center duration)*

The interaction effect between the group and the treatment was also significant, *F*(3, 36) = 5.29, *p* < 0.004, η2 = .31. Indicating that in control condition subjects received diazepam spent

more time than the other treatments whereas, in pinealectomy condition the subjects were not significantly different from each other.

Intraamygdalar Melatonin Administration and Pinealectomy Affect Anxiety Like Behavior and Spatial Memory 29

saline was more mobile than the other treatments, whereas in pinealectomy condition the subjects who received 100 μg/kg melatonin treatment were more mobile than the other

**Figure 12.** The entrance frequency to the center of the open field. The bar explanations can be seen in

Figure 10. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

**Figure 13.** The mobility time . The bar explanations can be seen in Figure 10. (Reproduced with

received 100 μg/kg melatonin treatment were faster than the other treatments.

The interaction effect between the group and the treatment was significant, *F* (3, 36) = 6.52, *p* < 0.001, η2 = .35, indicating that in control condition subjects received saline and diazepam were faster than the other treatments, whereas in pinealectomy condition the subjects who

permission from Karakas et al.; published by Elsevier, 2011a.)

*<IP_ADDRESS>. Velocity*

treatments.

**Figure 11.** The time spent at the center of the open field. The bar explanations can be seen in Figure 10. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

#### *<IP_ADDRESS>. Entrance frequency to the edge of the open field (Edge frequency)*

The interaction effect between the group and the treatment was significant, *F*(3, 36) = 3.02, *p* < 0.04, η2 = .20, reflecting the fact that in control condition subjects received diazepam entered more frequently to the edge of the open field than the other treatments, whereas in pinealectomy condition the subjects who received saline treatment entered more frequently than the other treatments.

#### *<IP_ADDRESS>. Entrance frequency to the center of the open field (Center frequency)*

The interaction effect between the group and the treatment was significant, *F*(3, 36) = 3.02, *p* < 0.04, η2 = .20, reflecting the fact that in control condition subjects received diazepam entered more frequently to the center of the open field than the other treatments, whereas in pinealectomy condition the subjects who received saline treatment entered more frequently than the other treatments.

#### *<IP_ADDRESS>. Mobility*

The main effect of the group was significant, *F* (1, 36) = 6.89, *p* = .01, η2 = .16. Control group was more mobile than pinealectomy group. The main effect of the treatment was also significant, *F* (3, 36) = 6.73, *p* = .001, η2 = .36. The subjects who received saline were more mobile on the open field than the other subjects with each being not significantly different from each other.

In addition, the interaction effect between the group and the treatment was significant, *F*(3, 36) = 7.08, *p* < 0.001, η2 = .37. This reflected the fact that in control condition subjects received

saline was more mobile than the other treatments, whereas in pinealectomy condition the subjects who received 100 μg/kg melatonin treatment were more mobile than the other treatments.

**Figure 12.** The entrance frequency to the center of the open field. The bar explanations can be seen in Figure 10. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

**Figure 13.** The mobility time . The bar explanations can be seen in Figure 10. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

#### *<IP_ADDRESS>. Velocity*

28 Neuroendocrinology and Behavior

than the other treatments.

than the other treatments.

*<IP_ADDRESS>. Mobility*

from each other.

not significantly different from each other.

more time than the other treatments whereas, in pinealectomy condition the subjects were

**Figure 11.** The time spent at the center of the open field. The bar explanations can be seen in Figure 10.

The interaction effect between the group and the treatment was significant, *F*(3, 36) = 3.02, *p* < 0.04, η2 = .20, reflecting the fact that in control condition subjects received diazepam entered more frequently to the edge of the open field than the other treatments, whereas in pinealectomy condition the subjects who received saline treatment entered more frequently

The interaction effect between the group and the treatment was significant, *F*(3, 36) = 3.02, *p* < 0.04, η2 = .20, reflecting the fact that in control condition subjects received diazepam entered more frequently to the center of the open field than the other treatments, whereas in pinealectomy condition the subjects who received saline treatment entered more frequently

The main effect of the group was significant, *F* (1, 36) = 6.89, *p* = .01, η2 = .16. Control group was more mobile than pinealectomy group. The main effect of the treatment was also significant, *F* (3, 36) = 6.73, *p* = .001, η2 = .36. The subjects who received saline were more mobile on the open field than the other subjects with each being not significantly different

In addition, the interaction effect between the group and the treatment was significant, *F*(3, 36) = 7.08, *p* < 0.001, η2 = .37. This reflected the fact that in control condition subjects received

(Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

*<IP_ADDRESS>. Entrance frequency to the edge of the open field (Edge frequency)*

*<IP_ADDRESS>. Entrance frequency to the center of the open field (Center frequency)*

The interaction effect between the group and the treatment was significant, *F* (3, 36) = 6.52, *p* < 0.001, η2 = .35, indicating that in control condition subjects received saline and diazepam were faster than the other treatments, whereas in pinealectomy condition the subjects who received 100 μg/kg melatonin treatment were faster than the other treatments.

**Figure 14.** The velocity. The bar explanations can be seen in Figure 10. (Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

Intraamygdalar Melatonin Administration and Pinealectomy Affect Anxiety Like Behavior and Spatial Memory 31

**Figure 15.** The total distance travelled. Right striated bar represents the saline injection and black bar represents diazepam injections, cross striated bar represents 1 μg/kg melatonin injection and bricks striated bar represents 100 μg/kg melatonin injection for both control and pinealectomy groups. Data

are presented as means (± S.E.M.). Different letters indicate the statistically different groups.

**Figure 16.** The time spent in open arms (TSOA). The bar explanations can be seen in Figure 15.

The main effect of the treatment was significant, *F* (3, 39) = 6.56, *p* = .001, η2 = .34. The subjects who received 100 μg/kg melatonin treatment spent less time in closed arms than

(Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)

*<IP_ADDRESS>. Time spent in closed arms (Closed arm duration)*

those who received other treatments.

(Reproduced with permission from Karakas et al.; published by Elsevier, 2011a.)


**10. Enhanced electric properties in the artificial polymer multilayers**

Multilayers consisting of alternative ferroelectric P(VDF-TrFE) copolymer and relaxor P(VDF-TrFE-CFE) terpolymer with different periodicities in thickness were prepared. A superlatticelike structure is shown in the polymer multilayer as the periodic thickness is lower than a critical value. The dielectric constant of the multilayer with a small periodic thickness is two times higher than that of the P(VDF-TrFE) copolymer over a temperature range between 300 K and 350 K. The multilayer also shows a good ferroelectricity in the same temperature range. The enhanced electrical properties of the multilayers are due to the long-range ferroelectric coupling [28,29].

Organic ferroelectric polymers have recently attracted much attention for their potential applications in flexible electronic devices, such as display, solar cell, information storage, and so on [30–32]. However, compared with its inorganic counterpart, organic ferroelectric polymer has some drawbacks, e.g., the lower dielectric constant and electric polarization, which is an obstacle for their practical applications. It is well known that many artificial superlattices (SL) and multilayers (ML) based on perovskite ferroelectrics show some amazing properties [33–35].

High-quality ultrathin films of both ferroelectric P(VDF-TrFE) and relaxor ferroelectric P(VDF-TrFE-CFE) have been successfully fabricated using the LB technique, which provide precise control of the film thickness in molecular scale. In the present study, periodic multilayers composed of alternating ferroelectric P(VDF-TrFE, 70/30) layer and relaxor ferroelectric P(VDF-TrFE-CFE, 56.2/36.3/7.5) layer were fabricated.

Figure 14 shows the XRD patterns of the multilayers with different periodicities. Two diffrac‐ tion peaks were observed in the multilayer structure of (15/15)2, i.e., 2θ = 18.57 and 19.97, which are assigned to the typical (110,200) reflection of the P(VDF-TrFE-CFE) and (110,200) reflection of P(VDF-TrFE), respectively. Compared with the pure P(VDF-TrFE-CFE) and P(VDF-TrFE), there is no shift in the (110,200) peaks for the (15/15)2 multilayer, suggesting that both the terpolymer and copolymer components in the multilayer keep their original phase structure. When the periodic thickness is decreased to five transfer layers, a shift is observed in both the diffraction peaks. Simultaneously, the intensity of the diffraction peak associated with the pure terpolymers becomes weaker.

or in the bulk polymer films. Various transport mechanisms are observed in the P(VDF-TrFE)

Schottky emission and Frenkel–Poole emission are found to be the dominant transport mechanism in the temperature range from 100 K to 200 K and the range from 200 K to 260 K, respectively. Space-charge-limited current conduction is the main transport mechanism as the

**10. Enhanced electric properties in the artificial polymer multilayers**

Multilayers consisting of alternative ferroelectric P(VDF-TrFE) copolymer and relaxor P(VDF-TrFE-CFE) terpolymer with different periodicities in thickness were prepared. A superlatticelike structure is shown in the polymer multilayer as the periodic thickness is lower than a critical value. The dielectric constant of the multilayer with a small periodic thickness is two times higher than that of the P(VDF-TrFE) copolymer over a temperature range between 300 K and 350 K. The multilayer also shows a good ferroelectricity in the same temperature range. The enhanced electrical properties of the multilayers are due to the long-range ferroelectric

Organic ferroelectric polymers have recently attracted much attention for their potential applications in flexible electronic devices, such as display, solar cell, information storage, and so on [30–32]. However, compared with its inorganic counterpart, organic ferroelectric polymer has some drawbacks, e.g., the lower dielectric constant and electric polarization, which is an obstacle for their practical applications. It is well known that many artificial superlattices (SL) and multilayers (ML) based on perovskite ferroelectrics show some amazing

High-quality ultrathin films of both ferroelectric P(VDF-TrFE) and relaxor ferroelectric P(VDF-TrFE-CFE) have been successfully fabricated using the LB technique, which provide precise control of the film thickness in molecular scale. In the present study, periodic multilayers composed of alternating ferroelectric P(VDF-TrFE, 70/30) layer and relaxor ferroelectric

Figure 14 shows the XRD patterns of the multilayers with different periodicities. Two diffrac‐ tion peaks were observed in the multilayer structure of (15/15)2, i.e., 2θ = 18.57 and 19.97, which are assigned to the typical (110,200) reflection of the P(VDF-TrFE-CFE) and (110,200) reflection of P(VDF-TrFE), respectively. Compared with the pure P(VDF-TrFE-CFE) and P(VDF-TrFE), there is no shift in the (110,200) peaks for the (15/15)2 multilayer, suggesting that both the terpolymer and copolymer components in the multilayer keep their original phase structure. When the periodic thickness is decreased to five transfer layers, a shift is observed in both the diffraction peaks. Simultaneously, the intensity of the diffraction peak associated with the pure

films, which are influenced by both temperature and applied voltage.

temperature is higher than 260 K.

164 Ferroelectric Materials – Synthesis and Characterization

coupling [28,29].

properties [33–35].

terpolymers becomes weaker.

P(VDF-TrFE-CFE, 56.2/36.3/7.5) layer were fabricated.

**Figure 14.** The XRD patterns of the multilayers with various periodicities. The inset shows the XRD patterns of copoly‐ mer and terpolymer

The temperature dependence of the dielectric constant of the multilayer structures on the heating process is shown in Fig. 15. Dielectric constant calculated with a series capacitor model of the individual copolymer and terpolymer is also shown in Fig. 15. It can be seen that the temperature dependence of the dielectric constant for the multilayer with the transfer number m larger than 10 is analogous to that of the series model, which shows a platform between 300 K and 360 K. This suggests that these multilayers with thick interlayer are just a combination of individual copolymer and terpolymer components.

**Figure 15.** The dielectric constant of multilayers with different periodicities and the calculated dielectric constant using series model as a function of temperature

The *P-E* hysteresis loops for the multilayers as well as the pure copolymer and terpolymer films are shown in Fig. 16. The (1/1)30 multilayer structure displays a good ferroelectricity with a remanent polarization (*P*r) of ~ 5.2 μC/cm2 , which is just a little smaller than that of pure P(VDF-TrFE) thin films. The temperature dependence of the *P*r of the multilayers with various periodic thicknesses is summarized in the inset of Fig. 3.

**Figure 16.** *P-E* hysteresis loops of the multilayers with different periodicities, pure copolymer and terpolymer films at 1 kHz. The inset shows *Pr* of (1/1)30 multilayer as a function of temperature

The piezoelectric properties of the copolymer P(VDF-TrFE) and terpolymer P(VDF-TrFE-CFE) multilayers are presented in Fig. 17.

**Figure 17.** The piezoelectric response curves of multilayers with different periodicities. The inset compares the piezo‐ electric response curves of the (1/1)30 multilayer, P(VDF-TrFE) and P(VDF-TrFE-CFE) homogenous films

The piezoelectricity of the samples was measured by piezoresponse force microscopy (PFM). To increase the degree of accuracy and allow meaningful comparison of the piezoresponses of the samples, measurements were made at ten different locations for all samples and then averaged. The piezoelectric coefficient *d33* for various multilayer structures are listed in Fig. 19. The *d33* of the pure copolymer and terpolymer are 41 pC/N and 52 pC/N, respectively, which is close to the reported value for the copolymer (49 pC/N) and equal to that observed for the terpolymer. Interestingly, an enhanced piezoelectric response of 81.6 pC/N is observed for the (1/1)30 multilayers, which is nearly 57 % larger than the pure terpolymer. As the periodic thickness increases, the piezoelectric response of the multilayers decreases dramatically.

a remanent polarization (*P*r) of ~ 5.2 μC/cm2

166 Ferroelectric Materials – Synthesis and Characterization

periodic thicknesses is summarized in the inset of Fig. 3.

1 kHz. The inset shows *Pr* of (1/1)30 multilayer as a function of temperature

multilayers are presented in Fig. 17.

, which is just a little smaller than that of pure

P(VDF-TrFE) thin films. The temperature dependence of the *P*r of the multilayers with various

**Figure 16.** *P-E* hysteresis loops of the multilayers with different periodicities, pure copolymer and terpolymer films at

The piezoelectric properties of the copolymer P(VDF-TrFE) and terpolymer P(VDF-TrFE-CFE)

**Figure 17.** The piezoelectric response curves of multilayers with different periodicities. The inset compares the piezo‐

The piezoelectricity of the samples was measured by piezoresponse force microscopy (PFM). To increase the degree of accuracy and allow meaningful comparison of the piezoresponses of the samples, measurements were made at ten different locations for all samples and then

electric response curves of the (1/1)30 multilayer, P(VDF-TrFE) and P(VDF-TrFE-CFE) homogenous films

In summary, the multilayers composed of alternating P(VDF-TrFE) copolymer and P(VDF-TrFE-CFE) terpolymer layers have been prepared, and their crystal structure and dielectric and ferroelectric properties have been studied. The multilayers with a periodic thickness of ~3 nm shows a superlattice-like crystal structure, high dielectric constant, good ferroelectricity, and piezoelectricity over a wide temperature range from 300 K to 350 K. The long-range ferroelectric coupling is considered to be dominant for multilayers with a smaller periodic interlayer.


**2. Motivation**

Models with interdependent waiting times are used to describe electron transfer [63], the firing of a single neuron [64], interhuman communication [65], and the modeling of earthquakes [66–69]. An excellent example of a process with correlated inter-event times that we will describe in this manuscript is tick-by-tick transaction price data from the stock market [70]. These data are very convenient to use, as they are of high quality and easily accessible in large amounts.

Firstly, let us recall two basic stylized facts observed in the majority of stock markets [71].

The latter is considered to be reminiscent of the volatility clustering phenomenon.

Of course, these are not the only or the most significant stylized facts, but these two do not directly depend on the log-return distribution. The list should also contain the broad distribution of log-returns [72]; multi-fractality [73,74]; universal scaling of the distribution of times between large jumps [75,76]; and the slow, power-law decay of the correlation between these times. We will further discuss the latter in this manuscript. Usually, the CTRW models used to describe high-frequency stock market data consider waiting times Δ*tn* as inter-transaction times, and process increments Δ*xn* as logarithmic returns between consecutive transactions. Taking into account the so-called bid-ask bounce phenomenon allows CTRW processes to reproduce the first stylized fact of short-term negative autocorrelation [58,77,78]. In this type of models, waiting times Δ*tn* are i.i.d. variables and only the dependence between Δ*xn* and Δ*xn*−1 is considered. Unfortunately, models considering only this type of dependencies turned out to be unable to describe the time ACF of absolute values of price changes [60]. Technically, it is possible to obtain a CTRW model reproducing both stylized facts, but it requires a power-law waiting-time distribution *ψ*(Δ*t*). However, this solution is not satisfying as we can obtain waiting-time distribution directly from the empirical data of inter-transaction times. It turns out that this distribution is far from a power-law one [58]. These results suggest that the source of the second stylized fact is not in the distributions of increments *h*(Δ*x*) and waiting times *ψ*(Δ*t*), but in the dependence between consecutive Δ*x* and Δ*t* values.

Let us start with an empirical analysis of the step ACF of series Δ*tn* and |Δ*xn*|. We observe approximately power-law memories in waiting times and absolute values of price changes; see Figure 2a. For a lag (in the number of steps) - 3, the autocorrelation of |Δ*xn*| is higher than the autocorrelation of Δ*tn*, but for a lag > 3 it is otherwise. This result suggests that in the limit of long times, the dependence between waiting times may be more critical than dependence between price changes. To verify this hypothesis we perform a shuffling test. We compare the time ACF of price changes' absolute values for four samples of time series. The first one is the original time series of tick-by-tick transaction data. The second time series keeps the price changes Δ*xn* in the original order but shuffles the order of waiting times Δ*tn*. This way, we obtained a time series keeping all dependencies between price changes Δ*xn*, but without any dependencies between waiting times Δ*tn*. In the third time series, we kept the original waiting times Δ*tn* but shuffled the price changes Δ*xn*. In the last, fourth time series, both Δ*tn* and Δ*xn* were shuffled. Let us emphasise that all four time series have the same, unchanged distributions *ψ*(Δ*tn*) and *h*(Δ*xn*). The results are shown in Figure 2b. As expected, we observe the slow, almost power-law decay of the time ACF for the first empirical time series. Surprisingly, removing dependencies between waiting times does not change the time ACF in the limit of *t* → 0, but significantly increases its slope of decay in the long-term. On the other hand, removing dependencies between price changes decreases the time ACF, dividing it by an almost constant factor but does not change the slope of the decay. The removal of all dependencies still leads to a positive time ACF, resulting from the non-exponential empirical distribution of waiting times.

**Figure 2.** Figures 2 and 3 were prepared using transaction data for KGHM (one of the most liquid Polish stocks) from period of January 2013 to July 2017. Both figures are on a log-log scale. (**a**) The plot of normalized empirical step ACF of Δ*t* and |Δ*x*|. Both functions decay like a power-law. For lag = 1, the autocorrelation of |Δ*x*| is higher. However, it decays faster, and for long times the memory in waiting times is stronger. (**b**) The plot of the normalized time ACF of |Δ*x*| for four time series. The presented lines are for empirical data (thick black), empirical price changes, and intra-daily shuffled waiting times (dotted red); intra-daily shuffled price changes and empirical waiting times (dash-dotted green); and intra-daily independently shuffled price changes and waiting times (thin blue). Considering only empirical dependencies of waiting times reproduces the ACF, which decays with almost the same slope as the empirical one.

The empirical observations presented above convinced us that it is necessary to consider long-range dependencies between waiting times within CTRW to reproduce the slowly decaying ACF of price changes' absolute values observed in the financial data.

Please note that in Figure 2, we analyzed the step ACF for lags up to 100 and the time ACF for times up to 1000 s. The procedure used to estimate the time ACF was presented in [58] and is a modification of the classical slotting technique introduced in [79]. Such limits were chosen due to the length of trading sessions (around 8 h or 1000 trades). Unfortunately, these limits are not long enough to detect power-law dependencies. The only way to increase these limits is by joining all sessions into one sequence. In this procedure, we merge the end of one session with the beginning of the following one (we omit overnight price changes). These two periods of the sessions are different, as we observe intraday activity in financial data [80]. The session begins with short inter-transaction times and a high standard deviation of price changes. Usually, up to the middle of the session, average inter-trade times increase, and the standard deviation of price changes decreases. The situation reverts again close to the end of the session. This phenomenon is called the *lunch effect* [81]. We use the canonical method to remove intraday non-stationarity by dividing each waiting time by the corresponding average waiting time, depending on the time that has elapsed since the beginning of the session for each day of the week separately [82,83]. The comparison of the step ACFs of waiting times for non-stationarized and stationarized

data is presented in Figure 3a. As a result of this procedure, we obtain the power-law decay over four orders of magnitude of lag. In Figure 3b, we present the time ACF of price changes' absolute values for stationarized data, which also exhibit power-law decay over four orders of magnitude of time lag. It is now reasonable to ask what the relationship is between the decay exponents of these autocorrelations. Fortunately, the model studied in this paper gives a strict answer to this question.

**Figure 3.** All intraday data (waiting times and corresponding price changes) are joined into one data set. (**a**) The plot shows the normalized step ACF of Δ*t* for non-stationary and stationarized cases. The stationarizing procedure is described in the main text. (**b**) The plot of the normalized time ACF of |Δ*x*| with stationarized waiting times. Both stationarized autocorrealations decay like a power-law with similar slope.

#### **3. Process of Waiting Times**

Let us now focus on the sequence of inter-transaction times Δ*t*1, Δ*t*2, ... , Δ*tn*, .... We are now looking for the point process to describe this series, which will be suitable for use in CTRW. For this reason, we need analytically solvable models. Moreover, we would like to use the empirical distribution of inter-event times *ψ*(Δ*tn*) and observe the power-law step ACF, as shown in Figure 3a. Even these two simple conditions exclude ACD models and Hawkes processes from our considerations. We are not interested in ACD models, as the power-law ACF can be obtained only within the fractional extension. In the Hawkes process, both the waiting time distribution and autocorrelation depend on the memory kernel [15,84]. Therefore, they cannot be set independently. As the Hawkes process is defined solely by its kernel, both waiting time distribution and autocorrelation depend on it. Thus, it would be difficult (if it is possible at all) to reproduce both empirical WTD and ACF at the same time. This feature of the Hawkes process hampers its use in the description of empirical data.

As the solution to our search, we propose a simple point process in which waiting times Δ*t* are repeated. In a very general sense, our proposition can be interpreted as a discretized version of CTRW, adapted to the role of the point process. Let us briefly

describe this analogy. Within the canonical CTRW, values of the process are represented by a spatial variable, and the time is continuous. The spatial variable remains constant for a given period of continuous waiting time. Now, we define the point process by the series of waiting times. Here, the number of repetitions *νi* of the same value of waiting time is the analog of waiting time in the canonical CTRW. The exemplary realization of such an adapted process of waiting times is shown in Figure 4.

We require the waiting times Δ*tn* (values of the process in the discrete subordinated time *n*) to come from the distribution *ψ*(Δ*tn*) (Δ*tn* > 0), with a finite mean Δ*t*. We define *νi* as the number of repetitions of the same waiting times (drawn independently for each series of repetitions). Let *νi* be the i.i.d. random variables with the distribution *ω*(*νi*). In general, it can be any distribution, but to recreate the power-law step ACF of waiting times we will focus on a fat-tailed distribution with a finite first moment *ν*. In particular, we use the zeta distribution with parameter *ρ*

$$
\omega\_{\rho}(k) = k^{-\rho} / \zeta(\rho); \quad \zeta(\rho) = \sum\_{i=1}^{\infty} i^{-\rho}, \; \rho > 1,\tag{1}
$$

where *ζ*(*ρ*) is Riemann's zeta function. Its expected value is equal to *ω* = *ζ*(*ρ*−1) *ζ*(*ρ*) for *ρ* > 2 and the variance is finite for *ρ* > 3. The cumulative distribution function is given by *Hk*,*ρ ζ*(*ρ*), where *Hk*,*ρ* = ∑*ki*=1 *i*−*ρ* is the generalized harmonic number. Let us introduce Ω(*k*) = ∑∞*i*=*k ω*(*i*) as a sojourn probability. We have Ω(*k*) = 1 − *Hk*−1,*ρ ζ*(*ρ*) for the zeta distribution.

**Figure 4.** The example realization of the process of waiting times, the values of which correspond to the waiting times Δ*tn* of the point process used in the primary CTRW process. Process values Δ*t*1, Δ*t*2, ... , Δ*tn* come from the values Δ*t*1, Δ*t*2, ... , Δ*tk* repeated *ν*1, *ν*2, ... , *νk* times, respectively. Number of repetitions *νi* are drawn from the distribution *ω*(*νi*). In the example above: *ν*1 = 1, *ν*2 = 3, *ν*3 = 2, . . . and Δ*t*1 = Δ*t*1, Δ*t*2 = Δ*t*3 = Δ*t*4 = Δ*t*2, Δ*t*5 = Δ*t*6 = Δ*t*3, . . ..

We define a soft propagator of the process of times P(Δ*t*; *n*|Δ*t*0, 0), which is the conditional probability density that the waiting time, which was initially (at *n* = 0) in the origin value (Δ*t* = Δ*t*0), is equal to Δ*t* after *n* steps. The soft propagator can be expressed by

$$\mathcal{P}(\Delta t; n | \Delta t\_0, 0) = \delta(\Delta t - \Delta t\_0) \Omega^{\text{first}}(n) + [1 - \Omega^{\text{first}}(n)] \psi(\Delta t),\tag{2}$$

where Ωfirst(*n*) is the sojourn probability obtained from *ω*first(*n*), which is the stationarized distribution of the repetition of the first waiting time:

$$\begin{split} \omega^{\text{first}}(n) &= \frac{\sum\_{n'=1} \omega(n+n')}{\sum\_{n''=0} \sum\_{n'=1} \omega(n''+n')} = \frac{\sum\_{n'=1} \omega(n+n')}{\sum\_{n=1} n\omega(n)} = \frac{\sum\_{n'=n+1} \omega(n')}{\langle \omega \rangle}, \\ \Omega^{\text{first}}(n) &= \frac{\sum\_{i=n} \sum\_{n'=i+1} \omega(n')}{\langle \omega \rangle} = \frac{\sum\_{i=1} i\omega(i+n)}{\langle \omega \rangle} = \frac{\langle \omega \rangle - n\Omega(n+1) - \sum\_{i=1}^{n} i\omega(i)}{\langle \omega \rangle}. \end{split} \tag{3}$$

The first term of the right-hand side of Equation (2) is the probability that the process value will stay constant (equal Δ*t*0) after *n* jumps. The second term indicates that there will be a process value jump with probability 1 − Ωfirst(*n*), so new process values will be completely independent, drawn from the distribution *ψ*(Δ*t*).

Restricting ourselves to *ω*(*n*) in the form of the zeta distribution, we can obtain

$$
\Omega^{\text{first}}(n) = 1 - \frac{n}{\langle \omega \rangle} + \frac{nH\_{n,\rho}}{\zeta(\rho - 1)} - \frac{H\_{n,\rho - 1}}{\zeta(\rho - 1)},\tag{4}
$$

and hence the propagator given by Equation (2). The step autocovariance of waiting times Δ*tn* can be expressed as

$$\text{cov}(n) = \left< \Delta t\_i \Delta t\_{i+n} \right> - \left< \Delta t\_i \right> \left< \Delta t\_{i+n} \right> = \left< \Delta t\_i \Delta t\_{i+n} \right> - \left< \Delta t \right>^2,\tag{5}$$

where symbol ... means taking the average. Note that Δ*ti*+*n* = Δ*ti* with probability *p* = Ωfirst(*n*). With probability 1 − *p*, the Δ*ti* is independent. This leads to

$$\text{cov}(n) = p\left<\Delta t^2\right> + \left(1 - p\right)\left<\Delta t\right>^2 - \left<\Delta t\right>^2 = \sigma\_{\Delta t}^2 p = \sigma\_{\Delta t}^2 \Omega^{\text{first}}(n). \tag{6}$$

We are interested in the asymptotic form of autocorrelation for *n* 1. We can use following approximation (Theorem 12.21 from [85])

$$
\zeta(\rho) - H\_{n,\rho} \approx \frac{n^{1-\rho}}{\rho - 1}.\tag{7}
$$

Finally, we obtain the normalized step ACF

$$\text{corr}(n) = \frac{\text{cov}(n)}{\text{cov}(0)} \approx \frac{n^{-(\rho - 2)}}{\zeta(\rho - 1)(\rho - 2)(\rho - 1)}.\tag{8}$$

The step ACF of waiting times decays like a power-law and the decay exponent is *ρ* − 2. It is worth emphasizing that even considering only *ρ* > 2, required for the existence of a finite average number of repetitions, we can obtain any value of the decay exponent.

#### **4. The Primary Process**

Now we are ready to define the primary CTRW process with repeating waiting times. This process is characterized by two key properties:

Note that we do not assume any dependence within the series of consecutive changes of the process value Δ*x*1, Δ*x*2, ... , Δ*xn*. We do not make any further assumptions about the shape of distributions *h*(Δ*x*). The memory in this process is present only in the sequence of waiting times.

Let us start the analysis of the properties of this process with the following observation. As the changes Δ*xn* are independent, the changes above any given threshold occur independently. Knowing the result in (8), we can calculate the autocorrelation of the series of inter-occurrence times between changes above or below any threshold. The details of

the derivation are presented in Appendix B. It turns out that we also obtain power-law decay with the exponent −(*ρ* − 2), the same as in (8).

Moreover, we managed to obtain the soft propagator of the primary CTRW process and the characteristics derived from it. The details of calculations can be found in Appendix A. Here we present selected results, namely, the first two moments and the time autocorrelation of changes, in the limit of long times (*t* → ∞). We consider analytical terms (*t*, *t*2, *t*3, . . .) and the most significant power-law term when *ρ* is non-integer.

Using results from Appendix A, the first moment of the process for *t* → ∞ can be approximated as

$$m\_1(t) = \mathcal{L}^{-1} \left[ -i \frac{\partial \varGamma(k; s)}{\partial k} \Big|\_{k=0} \right](t) \approx \frac{\mu\_1}{\langle \varDelta t \rangle} t + \mu\_1 \frac{a \{ \psi \}}{\Gamma(4-\rho)} t^{3-\rho}, \quad \rho \in (2; 4), \tag{9}$$

where L−1[·](*t*) is the inverse Laplace transform, *P*˜(*k*;*s*) is the propagator of the process in the Fourier–Laplace domain, Γ(·) is Euler's gamma function, and *α*{*ψ*} is a complex functional of *ψ*, which has to be calculated separately for each *ψ*. The most important term is typical, linear behavior, but we observe an additional power-law term. The second moment can be written in the form

$$\begin{split} m\_2(t) &= \mathcal{L}^{-1} \left[ -\frac{\partial^2 \tilde{P}(k;s)}{\partial k^2} \Big|\_{k=0} \right](t) \\ &\approx \mu\_1^2 \left( \frac{t}{\langle \Delta t \rangle} \right)^2 + \sigma\_x^2 \frac{t}{\langle \Delta t \rangle} + \mu\_1^2 \beta \{\Psi \} \frac{t}{\langle \Delta t \rangle} + \mu\_1^2 \frac{\gamma \{\Psi \}}{\Gamma(5-\rho)} t^{4-\rho}, \quad \rho \in (2;5), \end{split} \tag{10}$$

where *β*{*ψ*}, *γ*{*ψ*} are complex functionals of *ψ*, which have to be calculated separately for each *ψ*. From the first two moments of the process, we calculate the process variance (still considering only analytical and the most important power-law term)

$$
\sigma^2(t) = m\_2(t) - m\_1^2(t) \approx \left(\sigma\_x^2 + \mu\_1^2 \beta \{\psi\}\right) \frac{t}{\langle \Delta t \rangle} + \mu\_1^2 \frac{\gamma \{\psi\}}{\Gamma(5-\rho)} t^{4-\rho}, \quad \rho \in (2;5). \tag{11}
$$

It is worth mentioning that for variance the power-law term from the second moment is more important than the power-law term from the first moment. We can observe normal diffusion for *ρ* > 3. However, there is superdiffusion in the case of *ρ* ∈ (2; 3). We obtain ballistic diffusion in the limit *ρ* → 2.

Having the first two moments, one can calculate velocity ACF, which is equivalent to normalized ACF of changes for fixed sampling for the stationary process

$$\mathbb{C}(t) = \frac{1}{2} \frac{\partial^2 m\_2(t)}{\partial t^2} - \left(\frac{\partial m\_1(t)}{\partial t}\right)^2 \Rightarrow \mathbb{C}(t) \approx \mu\_1^2 \frac{1}{\Gamma(3-\rho)} \kappa \{\psi\} t^{2-\rho},\tag{12}$$

where *κ*{*ψ*} = *γ*{*ψ*} 2 − 2*α*{*ψ*} Δ*t* , for *ρ* ∈ (2; 4). In the limit of *t* → ∞ and *μ*1 = 0 we observe a power-law decay of ACF with the exponent *ρ* − 2. In the case of *μ*1 = 0, it can be proven that this exponent is *ρ* − 1, so the decay is faster (A5).

It is crucial to emphasize that in Equations (9)–(12) for *ρ* exceeding the mentioned range, there is still a power-law term with the same dependence on *μ*1 and the same time exponent. However, the dependence of the amplitude on *ρ* takes a different, more complex form.

#### **5. Empirical Results**

We use the constructed process to investigate the role of correlated inter-trade times in the volatility clustering effect. We consider this process as a toy model, describing high-frequency financial data. The value of the process represents the logarithm of the stock price. We can treat transactions as events that change the price. Therefore, the inter-transaction times correspond to waiting times in our model. The jumps represent the difference in the logarithmic prices of consecutive transactions, which are logarithmic returns [52].

The CTRW formalism allows us to obtain the autocorrelation of price returns. Moreover, the same formalism can be used to obtain the nonlinear ACF of absolute increments. This can be achieved by using different jump distributions *h*(Δ*x*). To model the process of price changes in time, we should use the symmetric distribution *h*(Δ*x*), as the empirical distribution of returns is symmetrical. As a result, we obtain the vanishing mean *μ*1 = 0 and the quickly decaying ACF of returns. To derive the nonlinear ACF of absolute returns, we define the new CTRW process, and by calculating its linear ACF, we obtain the nonlinear ACF of price increments. Following [60], if as *h*(Δ*x*) we use only the positive half of the previous distribution multiplied by 2, we deal with the case of non-zero drift and obtain an artificial, monotonically increasing process. As *μ*1 = 0, we obtain the slow power-law decay of the autocorrelation of absolute returns, as in the empirical results presented as a solid black line in Figure 2b.

Since we assumed only one type of memory in our model, introduced by the distribution *ω*(*ν*), we cannot expect that the model will be able to reproduce exact values of the empirical nonlinear ACF of the absolute returns. The model, however, should be able to reproduce its slope (as in Figure 2b, in which the green dash-dotted line reproduces the slope of the solid black line). The theoretical slope is obtained analytically and is equal 2 − *ρ*. It is worth emphasizing that the slope does not depend on the distribution of price changes *h*(Δ*x*) or waiting times *ψ*(Δ*t*) and is fully determined by the single parameter *ρ*, characterizing the distribution *ω*(*ν*). This fact significantly simplifies the comparison with the empirical data, as we are required to estimate only one parameter *ρ*. On the other hand, the assumption of repeated waiting time is a technical method introducing memory. We cannot expect to observe such a phenomenon in the empirical time series. The parameter *ρ* is a measure of the memory present in the sequence of consecutive waiting times. Therefore, we estimate this parameter using the slope of the step ACF of waiting times, which is equal to 2 − *ρ* in the model. It is a surprising and potentially essential fact that the exponent of the decay of the nonlinear time ACF is the same as in the step ACF of waiting times. This result motivates us to compare these two values for empirical financial data. Of course, in the empirical data we also observe a long-term positive step ACF of |Δ*x*|, which was not included in our model. Therefore, we can expect that the slope of time ACF of |Δ*x*| should be slightly higher than the slope of the step ACF of Δ*t*. Since a long-term nonlinear autocorrelation is usually interpreted as a reminiscence of the volatility clustering phenomenon, it is interesting to check what part of the observed volatility clustering effect can be explained only by memory between inter-trade times. We present the results for the five most traded stocks from the Warsaw Stock Exchange in Table 1 (ordered by the number of transactions), with the average inter-trade time not being greater than 30 s.

**Table 1.** Table with fitted slopes of the empirical stationarized step ACF of waiting times and the time ACF of price changes' absolute values for the five most liquid stocks from the WSE. The time ACF slopes are close to the corresponding step ACF slopes. The analysis was performed on the tick-by-tick market data from the public domain database [70]. The data covers the period from 3 January 2013 to 14 July 2017. For instance, the data set for KGHM contains 3,096,625 transactions.

We see that our model can estimate the slope of time ACF with an accuracy of around 10%. Moreover, our model can successfully reproduce the power-law decay of the autocorrelation of inter-occurrence times between changes below or above any given threshold reported in [75,76]. Please note that the decay exponent predicted by our model −(*ρ* − 2), with empirical values presented in the Table 1, is close to 0.31, as reported in [76].

### **6. Conclusions**

We introduced a new continuous-time random walk (CTRW) model with long-term memory within a sequence of waiting times. We use a simple model of repeating waiting times instead of commonly-used point processes such as the ACD and the Hawkes process. Despite its simplicity, our model of repeating waiting times has a few valuable properties. It is stationary, can be treated analytically, and the distribution of waiting times and memory in its series can be set independently.

As we observe many phenomena with dependencies between waiting times, possible applications of this family of CTRW models go beyond the exemplary application presented here.

However, in this manuscript, we applied the proposed model to describe highfrequency financial time series. We asked ourselves which commonly known properties of the financial time series can be reproduced by the long-term memory introduced in our model, only by means of the repeating waiting times. We have to emphasize that part of these properties, known as stylized facts, depend on the waiting time distribution *ψ*(Δ*t*) and price change distribution *h*(Δ*x*). As we are not trying to study the general ability of continuous-time random walk to describe the high-frequency financial time series, we have not studied the broad distribution of log-returns [72], multi-fractality [73,74], or universal scaling of the distribution of times between large jumps [75,76]. We have analyzed the decay of the nonlinear time autocorrelation function of log-returns and the decay of the step autocorrelation function of times between large jumps. Although we considered only memory in a sequence of waiting times, we managed to show that long-term dependencies in waiting times are crucial in explaining the volatility clustering effect and results in the power-law decay of both measures mentioned above.

Our results indicate that the dependence between consecutive price changes is not the primary carrier of long-range memory in the volatility clustering phenomenon. To verify these results, we conducted another simulation. We prepared autocorrelated series of waiting times according to the Fourier filtering method (for example, described in [86]). Similarly, as in our model, both the slopes of the step ACF of WTs and the time ACF of absolute returns were the same. This verification confirms our conclusion and indicates that it is general, independently of the origin of the autocorrelation between inter-trade intervals.

**Author Contributions:** Data curation, J.K.; formal analysis, J.K. and T.G.; investigation, J.K.; methodology, T.G.; software, J.K.; supervision, T.G; validation, T.G.; visualization, J.K.; writing—original draft, J.K. and T.G.; writing—review and editing, J.K. and T.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Acknowledgments:** We want to thank Ryszard Kutner and Tomasz Raducha for their helpful remarks and comments on the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.