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Power laws in stock market and fractal complexity of S&P500 and DAX

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Abstract

In this study the current research evidence about the role of power laws and fractals in stock market data is being reviewed. Then the fractal complexity of stock indexes in US (S&P 500) and Germany (DAX) is estimated. Daily closing prices from 1950 to 2017 are used for calculations. The results indicate a slightly downward trend in Hurst exponent during the investigated decades, meaning modest decrease in the spectral power-law exponent and increase in the fractal complexity of the indices. Possible links between the financial crises and changes in complexity are also discussed and questioned.
ITISE 2018
International Conference on Time Series and Forecasting
Proceedings of Papers
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Granada (Spain)
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Power laws in stock market and fractal
complexity of S&P500 and DAX
Anna Krakovsk´a
Institute of Measurement Science, Slovak Academy of Sciences,
ubravsk´a Cesta 9, 842 19 Bratislava, Slovak Republic
krakovska@savba.sk
Abstract. In this study the current research evidence about the role of
power laws and fractals in stock market data is being reviewed. Then
the fractal complexity of stock indexes in US (S&P 500) and Germany
(DAX) is estimated. Daily closing prices from 1950 to 2017 are used for
calculations.
The results indicate a slightly downward trend in Hurst exponent during
the investigated decades, meaning modest decrease in the spectral power-
law exponent and increase in the fractal complexity of the indices.
Possible links between the financial crises and changes in complexity are
also discussed and questioned.
Keywords: power law, fractal dimension, Hurst exponent, S&P 500,
DAX
1 Introduction
Complexity can be generated in a variety of ways. In chaos theory, irregular,
hardly predictable behaviour can arise from a few nonlinear differential equa-
tions. In agent based models, large number of individual objects governed by a
few simple rules are capable of generating remarkably complicated formations.
Moreover, complexity may be spatial or temporal - it may involve complicated
patterns that do not change over time, or may appear as a surprising time vary-
ing behaviour.
In terms of complexity, the stock market is a great place to look at. It is
composed of many decision-making individuals with similar motivations. The
market is open to the environment, and its dynamics has interesting features.
Moreover, the development of this remarkable system is well documented. In the
form of stock indexes and price series, we have decades of data to study.
However, if we see the stock market as a system of complex behaviour, then
we must call into question the standard random walk model with step sizes that
vary according to a normal distribution. However, questioning the normal or
log-normal distributions in economics, or in any real-world data, is not easy.
Gaussians seem to be all around us. Frequently mentioned examples are distri-
butions of height, weight, measurement errors. We even have the central limit
theorem that explains why normal distributions are so common. The theorem
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2 Power laws in stock market and fractal complexity of S&P500 and DAX
states that the normal distribution arises whenever a large number of indepen-
dent and identically distributed random variables with finite variances are added
together.
In economics, Gaussian distribution of stock returns and the correspond-
ing random walk model are the essence of the efficient market hypothesis [1].
According to this idea, the investors have all information about the expected
earnings of the stocks. They should always agree about the right price. The
prices can change a little bit after the companies release new estimates of the
future earnings. But no such things as bubbles and crises should happen.
Nevertheless, normal distribution may not be as typical as it has long been
thought of. Already in 1960 Benoit Mandelbrot pointed out that large move-
ments in prices are much more common than would be predicted from a normal
distribution [2]. As an alternative, he suggested the heavy tailed L´evy distribu-
tions. The importance of this result was under-valued by the economists until
the appearance of modern risk management methods around the 1990s. At that
time, the quantitative estimation of the heavy tailed distributions started to be
taken much more seriously. Mantegna and Stanley have shown that for the S&P
500 the central part of the distribution corresponds to the L´evy stable process
[3]. Scaling behaviour has been observed for time intervals spanning from min-
utes to weeks. The tails of the distribution deviated from that for a L´evy process
and were approximately exponential, ensuring - as one would expect for the price
returns - the finite variance of the distribution.
Although heavy-tailed distributions are already accepted in finance, discus-
sions about their origins are still actively going on. For proponents of complex
systems, the heavy tails, implying occasional market crashes, and also the for-
mation of bubbles appear as a natural and inherent feature of the system. The
first artificial models that confirm this, were designed in the nineties [4]. In the
models - computer simulated markets - adaptive learning agents (traders) are
trying to predict price movements and buy or sell based on their forecasts. The
subsequent change in the price then gets fed back to the traders and is used
for further decision about buying or selling. The feedback makes the market a
specific dynamic system. Every emergence of obvious predictability and thus the
vision of profit opportunity is immediately revealed by a huge number of traders.
The massive attempt to use the new discovery immediately leads to canceling
or even reversing the pattern. But it is remarkable that in models (and also in
reality) sometimes the exact opposite occurs - market participants keep a certain
pattern alive. Usually it is a sustained rise in stock prices, known as a bubble.
Bubbles are accompanied by an unreasonable increase in the price-earning (P/E)
ratio of the shares concerned. The bubbles inevitably crash after some time, but
the height of peaks are hard to predict.
Excessive increase in P/E ratio can be caused by unrealistic expectations of
future earnings. However, it can also be triggered by herding behaviour. Nate
Silver emphasizes in his book [5] that, in the 1960s, only about 15 percent of
stocks were held by institutions rather than individuals. By 2007, the percentage
had risen to 68 percent. When a trader does not risk his own money, then it may
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Power laws in stock market and fractal complexity of S&P500 and DAX 3
be quite rational for him to stay with the herd when the market goes up. When
the trouble comes, the investor can blame the unpredictability of the market or,
at best, admit collective guilt.
Herding behaviour, along with fat tailed distributions, volatility persistence,
uncorrelated returns and other stylized facts were replicated by a plethora of
agent-based models to date (see e.g. [6] and references therein). What is inter-
esting is that the systems often converge to reality-like complex behavior, even
though they have started with very simple initial condition of huge number of
agents and a couple of trivial rules. Bak et al. have argued that under rather gen-
eral conditions the preferred state is the one for which there is no characteristic
spatial or temporal scale size [7]. Objects with no spatial scale are fractals (term
coined by Mandelbrot in 1970s), and fluctuations with no temporal scale are gov-
erned by power laws. For the above phenomenon, in 1980s Bak et al. introduced
the term self-organized criticality. In finance it immediately attracted increased
attention to the power-law process as to the next candidate for modeling the
distribution of returns.
Our variable of interest xhas a so called power-law distribution if
p(x)1/xα.
Power-law distributions are long-tailed, giving a relatively large probability
p(x) to extreme events. They diverge at zero, so there must be a minimum value
xmin >0 for which the power-law behaviour holds. If α > 1, then the median
can be computed as 21/(α1)xmin [8]. A power-law process has a well-defined
mean only if α > 2, and it has a finite variance only if α > 3.
Power laws are very common in nature. For example both the gravitational
and electrical force decrease inversely with the square of the distance from the
mass or charge and thus have no characteristic length scale (until the size of the
molecules). Let us also mention the Zipf’s law from 1929, valid for the frequency
of words in different languages, and for many other phenomena [9]. Another
similar law emerged in the 1930s from seismological research of Gutenberg and
Richter [10]. Power laws are also ubiquitous in economics. Just remember the
Pareto principle from 1896, concerning the income distribution [11].
It is unclear why power laws should be so common. They do not emerge
from the generalized central limit theorem as easily as the normal distribution.
A generalization of the central limit theorem due to Kolmogorov and Gnedenko
states that the sum of a number of random variables with power-law tails tends
to a stable distribution only for 1 < α 3 [12]. If α > 3 then the sum converges
to the Gaussian distribution. However, most identified power laws in nature are
below 3 and they are so common that, sometimes, it would seem that power
laws are ”the new normal”.
Self-organized criticality has been one of the first major attempt to explain
the abundance of power laws. Bak et al. declared discovery of a robust mech-
anism of spontaneous emergence of complexity from simple local interactions
- a promising source of ubiquitous complexity. However, the idea has finally
shown serious weaknesses and considerable efforts have been made to find new
candidates for universal mechanisms for generating power laws [13], [14].
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4 Power laws in stock market and fractal complexity of S&P500 and DAX
The attraction of power laws probably lies in the fact that processes with
power-law distribution are fractals. It means that they are self-similar or self
affine and hence scale-free. For example, graphs of share prices over minutes,
days, or weeks have a very similar overall form, displaying statistical self-affinity,
which means that a rescaled version of a small part of the graph has the same
statistical distribution as the larger part. Power-law distributions are accompa-
nying signs of fractals and vice versa. Evidence of inevitable ubiquity of fractals
would be impressive.
However, not just the search for a universal mechanism is unsuccessful. We
often fail even with a seemingly less ambitious goal of demonstrating that some
data follows a power law. The natural procedure is to take the histogram and plot
it on a log-log scale. If it looks linear then the estimate of the slope is considered
the exponent α. Unfortunately, the data may be well-fitted by a power law, and
another function might be an even better fit.
One of the latest studies on this topic comes from Broido and Clauset [15].
The authors compare the fitted power-law distributions to alternative heavy-
tailed distributions, like the log-normal or the stretched-exponential. The ap-
proximations are compared using a likelihood ratio test. The findings of the
authors are clear right from the title of the article - Scale-free networks are rare.
The provocative title has prompted an immediate response from Albert-L´aszl´o
Barab´asi, one of the advocates of ubiquitous scale-free networks [16]. He argues
that the fitting difficulties are not a reason to dismiss the idea of the scale-free
networks. Pure power law only emerges in idealized models. In real networks, we
have to admit additional effects. Then, the power law tends to coexist with some
corrective function, leading to power laws with exponential cutoffs, stretched ex-
ponentials, and so on.
The idea behind the Barab´asi–Albert networks driven by only growth and the
so called preferential attachment is that nodes with higher degree receive more
new links than nodes with lower degree [17]. Intuitively this makes sense. Web
pages with many incoming links are easier to find, so even more new Web pages
will link to them. In networks, this ”the rich get richer” principle of preferential
attachment leads to scale-free degree distributions. From this point of view the
stock market can be seen as a network of traded companies (nodes, vertices).
In the network, pairs of vertices are linked by edges, if the level of correlation
between the corresponding two price series is above a specified threshold.
Is the preferential attachment, resulting in power-law distributions, the right
model for the price returns? Or should we prefer one of many other proposed
models and mechanisms? It is not obvious how to decide.
Also, it is still not even clear whether it is appropriate to see the price series
and indexes as power-law processes and fractals. On the one hand, price returns
certainly have some kind of fat-tailed distributions, but verifying the specific
power-law distribution could prove to be be infeasible. On the other hand, we
have the remarkable indicators given by Mandelbrot: the apparent statistical
self-affinity in data and also the volatility persistence that is strongly supportive
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Power laws in stock market and fractal complexity of S&P500 and DAX 5
of a long memory property. These are typical features of fractal (power-law)
processes.
After this relatively extensive introductory review, the rest of the article is
organized as follows. In the next two sections we introduce the data sets and
the methods used for estimation of the fractal complexity. The fourth section
presents and discusses the results. Finally, the findings are summarized in Con-
clusion.
2 Methods
Although the idea of a power-law distribution in stock prices is still not justified,
use of fractal characteristics such as the Hurst exponent [18], has been quite
common for some time.
The Hurst exponent His usually referred to as a measure of persistence
but in fact it is also one of the fractal complexity measures. Hcan be derived
directly from the best-known fractal characteristic, which is the fractal dimension
D. Fractal dimension determines the irregularity of the signal. It tells us how
smooth or rough the trace of the graph is. Dcan take values between 1 and 2.
The more the graph fills the plane the closer it approaches the value of 2.
Let us also mention the possibility of spectral representation of the investi-
gated scale-free process:
S(f)1/fβ,
where S(f) is the power spectral density for the frequency components of the
signal. Processes with this power-law relationship are called 1/f noises.
When we have self-affine time series, some of the properties as D, decay of au-
tocorrelation or power spectrum, persistence, etc. are elegantly connected. Above
all, the next relation between the fractal dimension D, decay of autocorrelation
γ, Hurst exponent, and spectral decay βholds:
D=2 + γ
2= 2 H=5β
2(1)
for 1 β3 [19], [20].
The consequence of (1) is that if you estimate one of the characteristics, you
also get the others. For example, in this study, we estimated Dand used it to
get the estimate of the Hurst exponent and spectral decay.
For computation of D, we chose the Higuchi method, introduced in [21].
The method stood out from our earlier comparison of four different estimation
techniques [22].
The Higuchi’s method scans fluctuations of the signal by investigating the
defined length of the curve for different magnifications of the time axis of the
signal. We take time-series X(1), X(2), ..., X (N),and make klagged time series
that start from m-th place (m= 1,2, ..., k) with gap of the size k:
Xm
k=X(m), X(m+k), ..., X m+Nm
kk
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6 Power laws in stock market and fractal complexity of S&P500 and DAX
Then Higuchi defines a length of the curve of Xm
kas follows:
Lm
k=
[Nm)
k]
X
i=1
|X(m+ik)X(m+ (i1)k|N1
Nm
kk2
When the length of the curve is calculated (and normalized) for every mand k,
we get an L(k) as the mean of all lengths Lm
k. If L(k)kD, then the curve is
a fractal with dimension D.
To estimate the Higuchi dimension, we used a Matlab code shared by J.
Monge- ´
Alvarez [23].
Real data, however, can only be investigated over a restricted range of scales.
We have to determine how high should the value kgo. The recommendation is
to compute the estimates for increasing kand use the value where the estimates
reach a plateau. For the indexes analyzed in this study, the optimal choice was
found to be k= 4.
After estimating the Higuchi fractal dimension D, we use Eq. (1) to get the
corresponding Hurst exponent: H= 2 D.
When we estimate Hor other fractal characteristics from the financial series,
we actually work with a hypothesis that the given series is a power-law process
and thus a fractal.
Finding H > 0.5 indicates that the increments of the process are positively
correlated, meaning that a high value in the series will probably be followed by
another high value (persistence). Variation increases at a faster rate than what
is expected for the Gaussian case, so that large jumps are possible, and the series
may travel a larger distance than a random walk would imply.
On the other hand, H < 0.5 means negative correlation and long-term ten-
dency to switch between high and low values (anti-persistence).
A value of H= 0.5 means that the investigated process has independent
increments (e.g. ordinary Brownian motion), or the absolute values of the auto-
correlations decay exponentially quickly to zero, unlike the above-mentioned case
of typically power-law decay. In stock returns data, H= 0.5 can be considered
to support the efficient market hypothesis.
3 Data
In this study, the daily closing values of S&P 500 and DAX indexes are examined.
The S&P 500 is an American stock market index based on 500 large compa-
nies listed on the New York Stock Exchange or the Nasdaq Stock Market. Daily
closing values (17110) of S&P 500 from January 3, 1950 to December 29, 2017
were downloaded from https://finance.yahoo.com.
The DAX is a German stock market index consisting of the 30 major com-
panies trading on the Frankfurt Stock Exchange. Daily closing values (14578)
of DAX from January 4, 1960 to December 29, 2017 were downloaded from
https://stooq.com.
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Power laws in stock market and fractal complexity of S&P500 and DAX 7
For the computation of the fractal characteristics, the daily closing values of
the indexes were used. For the plot of a histogram and the corresponding fit of a
normal density function, ordinary returns expressed as a percentage were used.
4 Results and Discussion
Fig. 1 shows the empirical distribution of the returns of S&P 500 for years 1950–
2017. The estimated σis 0.9608. We can see, that the distribution has fatter tails
than predicted by the Gaussian. Events that should theoretically be extremely
rare, such as price changes by more than 3 percent (above 3σ), are actually quite
common. The result suggests that the market does not follow a random walk.
-10 -5 0 5 10
100
101
102
103
104
histogram and a normal distribution fit
histogram
normal fit
Fig. 1. Histogram for the daily returns of the S&P 500 during the period 1950–2017
compared to the Gaussian normal distribution. In order to amplify the heavy tails of
the distributions, semilogarithmic plot is used.
Fig. 2 shows estimates of Hurst exponent for the S&P 500 during the period
1950–2017. Each value of His estimated from 250 previous daily closing values
of the index. Most of the time, the H > 0.5 indicates significantly persistent
behaviour. However, Hshows a slightly downward trend over the years and
during the last two decades it oscillates quite close to the value of 0.5 typical for
the ordinary random walk. Similar decreasing pattern has been observed when
high frequency (1-minute) data were used for S&P 500 index for the period
1983–2009 [24]. The similarity of the results obtained from daily and minute
data supports the possible presence of scale-free processes.
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8 Power laws in stock market and fractal complexity of S&P500 and DAX
1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Hurst exponent
0
500
1000
1500
2000
2500
S&P 500
Hurst exponent
S&P 500
Fig. 2. Hurst exponent for the S&P 500 during the period 1950–2017. Each value of
His estimated from the previous year of daily closing values.
Fig. 3 shows estimates of Hurst exponent for the German DAX during the
period 1960–2017. Again, each value of His estimated from 250 previous daily
closing values of the index. Also in this case the Hurst exponent shows a moderate
downward trend over the investigated years.
Fig. 3. Hurst exponent for the DAX during the period 1960–2017. Each value of His
estimated from the previous year of daily closing values.
In case of S&P 500, the highest values of Hwere found for the years 1971–
1972. Left part of the Fig. 5 shows a closer look at these years. The index is
relatively smooth and persistent. The estimation of the Higuchi dimension is
D1.3, so based on Eq. (1), H0.7 and the signal resembles 1/f noise with
an exponent β2.4.
The lowest values of Hwere found for the turbulent years of 2008–2009.
Right part of the Fig. 6 shows S&P 500 and Hfor these two years. The index
here is mostly anti-persistent and it looks more rough and space-filling than
in the previous case of 1971–1972. The corresponding estimates of the fractal
complexity measures are D1.6, H0.4, β1.8.
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Power laws in stock market and fractal complexity of S&P500 and DAX 9
Can we learn more from the course of the Hurst exponent?
The most important question is, of course, whether Hcan help to predict the
coming crash. By stock market crash we understand panic selling and double-
digit percentage losses in a stock market index over a period of few days. Al-
though long-term market predictions are undoubtedly illusory, some authors
believe that limited short-term trend predictions are possible [25], [26].
To discuss this topic, let us look closer at the Hurst exponent’s course over
some specific 2-year periods.
1960-01-01 1961-01-01 1962-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
50
55
60
65
70
75
S&P 500
Hurst exponent
S&P 500
1962-01-01 1963-01-01 1964-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
50
55
60
65
70
75
80
S&P 500
Hurst exponent
S&P 500
Fig. 4. Hurst exponent (blue) and the S&P 500 (red) during the period 1960–1961 (on
the left) and during 1962–1963 (on the right). Each value of His estimated from 50
previous days.
1971-01-01 1972-01-01 1973-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
90
95
100
105
110
115
120
S&P 500
Hurst exponent
S&P 500
1987-01-01 1988-01-01 1989-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
220
240
260
280
300
320
340
S&P 500
Hurst exponent
S&P 500
Fig. 5. Hurst exponent (blue) and the S&P 500 (red) during the period 1971–1972 (on
the left) and during 1987–1988 (on the right). Each value of His estimated from 50
previous days.
The right part of Fig. 6 shows estimates of Hurst exponent for the S&P
500 around the financial crisis in September 2008. In the first half of 2008, the
stock market was at its peak with H > 0.5 suggesting a slight persistence in
the data. In such situation, the short-term predictability of the market can be
increased, traders may become confident about their strategies and the liquidity,
but also the vulnerability of the market grows. What seems to be interesting is
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10 Power laws in stock market and fractal complexity of S&P500 and DAX
1989-01-01 1990-01-01 1991-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
260
280
300
320
340
360
380
S&P 500
Hurst exponent
S&P 500
2008-01-01 2009-01-01 2010-01-01
0
0.2
0.4
0.6
0.8
1
Hurst exponent
700
800
900
1000
1100
1200
1300
1400
S&P 500
Hurst exponent
S&P 500
Fig. 6. Hurst exponent (blue) and the S&P 500 (red) during the period 1989–1990 (on
the left) and during 2008–2009 (on the right). Each value of His estimated from 50
previous days.
that well before the crash Hbegan to fall sharply. Unusually low values of the
exponent (H < 0.3), shortly before the crash, could be interpreted as the onset
of strong anti-persistence and therefore extreme nervousness in the market. Do
we see a useful prognostic pattern here? Unfortunately, analysis of other similar
situations does not confirm this.
Let us see, for example, the strong market decline in 1987. On Black Monday
(October 19, 1987), the S&P 500 dropped more than 20%. The crash came
after years of strong economic optimism and unprecedented market increase.
Estimates of the Hurst exponent for period 1987–1988 can be seen on the right
graph of Fig. 5. Similarly, as in 2008, we can see the stock market at its peak
with H > 0.5 suggesting persistence in data. However, this time, no warning
sign of nervousness in the form of a falling Hurst exponent is present.
In other situations, on the contrary, the market looks nervous in the sense
that it is anti-persistent, but it still rises. See for example Fig. 4, with periods
of rising market, accompanied by Hwell below 0.5. Another significant example
is the rapid market growth in 2017 with S&P 500 booming and anti-persistent
at the same time.
5 Conclusion
In this study, we estimated fractal complexity characteristics from daily closing
values of US and German stock market indexes. Estimates were mostly found in
the following ranges:
1.31.6 for the fractal dimension D,
0.40.7 for the Hurst exponent H,
1.82.4 for the spectral decay β.
In case of S&P 500, the results showed change from the strongly persistent
behaviour until 1970s to mostly anti-persistent behaviour of the last two decades.
Then we looked for patterns that are potentially useful for predicting forth-
coming danger of stock collapses. However, the prognostic value of the measures
of fractal complexity was not confirmed.
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Power laws in stock market and fractal complexity of S&P500 and DAX 11
We also discussed the role of the power-law processes in stock market mod-
eling. If we accepted power-law processes as inevitable consequences of the evo-
lution of the market as a large complex system, then we would have to accept
that:
The market is very susceptible to any disturbance. Minor shocks can lead to
fluctuations of all large sizes, just as it is with earthquakes and avalanches.
Catastrophes can occur for no obvious reason, either due to the internal
dynamics of the system or triggered by sudden external events like terrorist
attacks. But collapses are fundamentally due to the unstable position and
they are unavoidable.
It is not possible to predict the long-term course of the market.
It is not possible to get rid of the fluctuations through some sort of economic
regulations.
Nevertheless, the most robust and efficient state of the economy could be the
one with occasional fluctuations of extreme sizes and durations. It is probably
not the best imaginable state, but it might be the best achievable state.
The questions that we continue to ask are: Are power laws and fractals typ-
ical for the stock market dynamics? If so, is there a universal mechanism that
inevitably leads to these power laws?
An overview of current literature shows that we do not have definitive an-
swers. It seems that there are many different ways that fractals and power laws
can arise, and that the long sought universal mechanism may not exist. And we
still can not rule out the possibility that ubiquitous power laws (fractals) are
just mere idealization. As we mentioned above, many published claims for the
existence of power laws are almost surely wrong. Maybe what we see are mostly
just heavy-tailed distributions. They still indicate a very different phenomenon
from Gaussian or exponential, and force us to look at the rare events in a new
way. However, there is no doubt that they are not nearly as exciting as power
laws and fractals.
Acknowledgments. This work was supported by the Slovak Grant Agency for
Science (grants no. 2/0011/16) and by the Slovak Research and Development
Agency (grant no. APVV-15-0295). I would also like to thank Hana Krakovsk´a
for her help with the computational part of the paper.
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