## Power Law Distributions, 1/f Noise, Long-Memory Time Series

*30 Aug 2021 12:59*

Why do physicists care about power laws so much?

I'm probably not the best person to speak on behalf of our tribal obsessions (there was a long debate among the faculty at my thesis defense as to whether "this stuff is really physics"), but I'll do my best. There are two parts to this: power-law decay of correlations, and power-law size distributions. The link is tenuous, at best, but they tend to get run together in our heads, so I'll treat them both here.

The reason we care about power law correlations is that we're conditioned to think they're a sign of something interesting and complicated happening. The first step is to convince ourselves that in boring situations, we don't see power laws. This is fairly easy: there are pretty good and rather generic arguments which say that systems in thermodynamic equilibrium, i.e. boring ones, should have correlations which decay exponentially over space and time; the reciprocals of the decay rates are the correlation length and the correlation time, and say how big a typical fluctuation should be. This is roughly first-semester graduate statistical mechanics. (You can find those arguments in, say, volume one of Landau and Lifshitz's Statistical Physics.)

Second semester graduate stat. mech. is where those arguments break down ---
either for systems which are far from equilibrium
(e.g., turbulent flows), or in equilibrium but
very close to a critical point (e.g., the transition from a solid to liquid
phase, or from a non-magnetic phase to a magnetized
one). Phase transitions have fluctuations
which decay like power laws, and many non-equilibrium systems do too. (Again,
for phase transitions, Landau and Lifshitz has a good discussion.) If you're a
statistical physicist, phase transitions
and non-equilibrium processes *define* the
terms "complex" and "interesting" --- especially phase transitions, since we've
spent the last forty years or so developing a very successful theory of
critical phenomena. Accordingly, whenever we see power law correlations, we
assume there must be something complex and interesting going on to produce
them. (If this sounds like the fallacy of affirming the consequent, that's
because it is.) By a kind of transitivity, this makes power laws interesting
in themselves.

Since, as physicists, we're generally more comfortable working in the frequency domain than the time domain, we often transform the autocorrelation function into the Fourier spectrum. A power-law decay for the correlations as a function of time translates into a power-law decay of the spectrum as a function of frequency, so this is also called "1/f noise".

Similarly for power-law distributions. A simple use of the Einstein
fluctuation formula says that thermodynamic variables will have Gaussian
distributions with the equilibrium value as their mean. (The usual version of
this argument is *not* very precise.) We're also used to seeing
exponential distributions, as the probabilities of microscopic states. Other
distributions weird us out. Power-law distributions weird us out even more,
because they seem to say there's no typical scale or size for the variable,
whereas the exponential and the Gaussian cases both have natural scale
parameters. There is a connection here with fractals, which also lack typical
scales, but I don't feel up to going into that, and certainly a lot of the
power laws physicists get excited about have no obvious connection to any kind
of (approximate) fractal geometry. And there are lots of power law
distributions in all kinds of data, especially social data --- that's why
they're also called Pareto distributions, after the sociologist.

Physicists have devoted quite a bit of time over the last two decades to
seizing on what look like power-laws in various non-physical sets of data, and
trying to explain them in terms we're familiar with, especially phase
transitions. (Thus "self-organized criticality".) So
badly are we infatuated that there is now a huge, rapidly growing literature
devoted to "Tsallis statistics" or "non-extensive
thermodynamics", which is a recipe for modifying normal statistical
mechanics so that it produces power law distributions; and this, so far as I
can see, is its *only* good feature. (I will not attempt, here, to
support that sweeping negative verdict on the work of many people who have more
credentials and experience than I do.) This has not been one of our more
successful undertakings, though the basic motivation --- "let's see what we can
do!" --- is one I'm certainly in sympathy with.

There have been two problems with the efforts to explain all power laws
using the things statistical physicists know. One is that (to mangle Kipling)
there turn out to be nine and sixty ways of constructing power laws,
and *every single one of them is right*, in that it does indeed produce
a power law. Power laws turn out to result from a kind of central limit
theorem for multiplicative growth processes, an observation which apparently
dates back to Herbert Simon, and which has been rediscovered by a number of
physicists (for instance, Sornette). Reed and Hughes have established an even
more deflating explanation (see below). Now, just because these simple
mechanisms exist, doesn't mean they explain any particular case, but
it *does* mean that you can't legitimately argue "My favorite mechanism
produces a power law; there is a power law here; it is very unlikely there
would be a power law if my mechanism were not at work; therefore, it is
reasonable to believe my mechanism is at work here." (Deborah Mayo would say
that finding a power law does not constitute a severe test of your hypothesis.) You need to do
"differential diagnosis", by identifying other, *non*-power-law
consequences of your mechanism, which other possible explanations don't share.
This, we hardly ever do.

Similarly for 1/f noise. Many different kinds of stochastic process, with no connection to critical phenomena, have power-law correlations. Econometricians and time-series analysts have studied them for quite a while, under the general heading of "long-memory" processes. You can get them from things as simple as a superposition of Gaussian autoregressive processes. (We have begun to awaken to this fact, under the heading of "fractional Brownian motion".)

The other problem with our efforts has been that a lot of the power-laws
we've been trying to explain are not, in fact, power-laws. I should perhaps
explain that statistical physicists are called that, not because we know a lot
of statistics, but because we study the large-scaled, aggregated effects of the
interactions of large numbers of particles, including, specifically, the
effects which show up as fluctuations and noise. In doing this we learn,
basically, nothing about drawing inferences from
empirical data, beyond what we may remember about curve fitting and
propagation of errors from our undergraduate lab courses. Some of us,
naturally, *do* know a lot of statistics, and
even teach it --- I might
mention Josef Honerkamp's superb Stochastic Dynamical Systems.
(Of course, that book is out of print and hardly ever cited...)

If I had, oh, let's say fifty dollars for every time I've seen a slide (or a
preprint) where one of us physicists makes a log-log plot of their data, and
then reports as the exponent of a new power law the slope they got from doing a
least-squares linear fit, I'd at least not grumble. If my colleagues had gone
to statistics textbooks and looked up how to estimate the parameters of a
Pareto distribution, I'd be a happier man. If any of them had actually tested
the hypothesis that they had a power law against alternatives like stretched
exponentials, or especially log-normals, I'd think
the millennium was at hand. (If you want to
know how to do these things, please
read this paper, whose merits are
entirely due to my co-authors.) The situation for 1/f noise is not so dire,
but there have been and still are plenty of abuses, starting with the fact that
simply taking the fast Fourier transform of the autocovariance function
does *not* give you a reliable estimate of the power
spectrum, *particularly* in the tails. (On that point, see, for
instance, Honerkamp.)

See also: Chaos and Dynamical Systems; Complex Networks; Self-Organized Criticality; Time Series; Tsallis Statistics

- Recommended, bigger picture:
- Michael Mitzenmacher, "A Brief History of Generative Models for
Power Law and Lognormal Distributions", Internet Mathematics
**1**(2003): 226--251 [PDF] - M. E. J. Newman, "Power laws, Pareto distributions and Zipf's law", cond-mat/0412004 [If you read one other thing on power laws, read this]
- Manfred Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise

- Recommended, more technical or more specialized:
- Robert J. Adler, Raise E. Feldman and Murad S. Taqqu (eds.), A Practical Guide to Heavy Tails [Presumes that you already know something about statistics and stochastic processes, so not suitable for beginners.]
- Barry C. Arnold, Pareto Distributions [Fine guide to the statistical literature, as it was in 1983; still valuable, though many things which were nasty computations then are easy now.]
- Ayan Bhattacharya, Bohan Chen, Remco van der Hofstad, Bert Zwart, "Consistency of the PLFit estimator for power-law data", arxiv:2002.06870 [That is, of the Clauset et al. 2009 estimator]
- Arijit Chakrabarty, "Effect of truncation on large deviations for heavy-tailed random vectors", arxiv:1107.2476
- Aaron Clauset, Maxwell Young, and Kristian Skrede Gleditsch, "Scale Invariance in the Severity of Terrorism", physics/0606007 [Surprising, but well-supported]
- F. Clementi, T. Di Matteo, M. Gallegati, "The Power-law Tail
Exponent of Income Distributions", Physica
A
**370**(2006): 49--53, physics/0603061 [An interesting way to improve the accuracy of Hill-type (tail-conditional maximum likelihood) estimates of the scaling parameter. Written with few concessions to those who are neither statisticians nor econometricians. Not directly suitable for determining the*range*of the scaling region. Income distribution is used only as an example.] - Andrew M. Edwards, Richard A. Phillips, Nicholas W. Watkins, Mervyn
P. Freeman, Eugene J. Murphy, Vsevolod Afanasyev, Sergey V. Buldyrev,
M. G. E. da Luz, E. P. Raposo, H. Eugene Stanley and Gandhimohan
M. Viswanathan, "Revisiting Lévy flight search patterns of wandering
albatrosses, bumblebees and
deer", Nature
**449**(2007): 1044--1048 - Paul Embrechts and Makoto Maejima, Selfsimilar Processes
- Robert P. Freckleton and William J. Sutherland [thanks to Nick Watkins for pointing these out]
- "Do in-hospital waiting lists show self-regulation?", Journal of the Royal Society of Medicine
**95**(2002): 164 - "Hospital waiting-lists (Communication arising): Do power laws imply self-regulation", Nature
**413**(2001): 382

- "Do in-hospital waiting lists show self-regulation?", Journal of the Royal Society of Medicine
- Alexander Gnedin, Ben Hansen, Jim Pitman, "Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws",
Probability Surveys
**4**(2007): 146--171, arxiv:math.PR/0701718 - Michel L. Goldstein, Steven A. Morris and Gary G. Yen, "Fitting to the Power-Law Distribution", cond-mat/0402322 [Pedestrian, but accurate, exposition in terms physicists and engineers are likely to understand. Insufficiently sourced to the statistical literature; e.g., their calculation of the maximum likelihood estimator was first published in 1952.]
- Timothy Graves, Robert B. Gramacy, Christian Franzke, Nicholas Watkins, "A brief history of long memory", arxiv:1406.6018
- Vladimir Hlasny, "Parametric representation of the top of income distributions: Options, historical evidence, and model selection",
Journal of Economic Surveys
**35**(2021): 1217--1256 - Josef Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis
- Yuji Ijiri and Herbert Simon, Skew Distributions and the Sizes of Business Firms [Collects Simon and co.'s pioneering papers on power laws and related distributions --- including "On a Class of Skew Distribution Functions", below --- as well as considering the limitations, alternatives, modifications to match data, statistical issues, the connection to Bose-Einstein statistics, the importance of going beyond just staring at distributional plots if you want to learn about mechanisms, etc., etc. This was all published in 1977...]
- A. James and M. J. Plank, "On fitting power laws to ecological data", arxiv:0712.0613
- Raya Khanin and Ernst Wit, "How Scale-Free Are Biological
Networks?",
Journal of Computational Biology
**13**(2006): 810--818 [Ans.: not very scale-free at all.] - Joel Keizer, Statistical Thermodynamics of Nonequilibrium Processes [Has a good discussion of critical fluctuations in chapter 8. Review: Molecular Fluctuations for Fun and Profit]
- Paul Krugman, The Self-Organizing Economy [Has a nice discussion of power-law size distributions in economics. Review]
- Michael LaBarbera, "Analyzing Body Size as a Factor in Ecology and
Evolution", Annual Review of Ecology and
Systematics
**20**(1989): 91--117 [Statistical problems in many studies of power-law scaling in biology, their effects on the conclusions of those studies (ranging from "wrong, but correctable" to "meaningless"), and how to do it right. JSTOR] - J. Laherrère and D. Sornette, "Stretched exponential
distributions in nature and economy: 'fat tails' with characteristic scales",
The European Physical Journal B
**2**(1998): 525--539 - L. D. Landau and E. M. Lifshitz, Statistical Physics [For the theory of fluctuations in statistical mechanics, and for critical phenomena in equilibrium]
- Adrián López García de Lomana, Qasim K. Beg, G. de Fabritiis and Jordi Villà-Freixa, "Statistical Analysis of Global Connectivity and Activity Distributions in Cellular Networks", arxiv:1004.3138
- R. Dean Malmgren, Daniel B. Stouffer, Adilson E. Motter, Luis A.N. Amaral, "A Poissonian explanation for heavy-tails in e-mail communication",
Proceedings of the
National Academy of Sciences (USA)
**105**(2008): 18153--18158, arxiv:0901.0585 - Elliott W. Montroll and Michael F. Shlesinger, "On 1/f noise and
other distributions with long
tails", Proceedings
of the National Academy of Sciences (USA)
**79**(1982): 3380--3383 - V. F. Pisarenko and D. Sornette, "New statistic for financial return distributions: power-law or exponential?", physics/0403075 [Actually, two new statistics: one converges to a constant if the distribution you're sampling from is an exponential, independent of the exponent, and the other converges to a constant if the distribution is a power law, independent of the power. They even have some indications of the sampling distributions, so you can at least gauge the statistical signifcance, i.e., the probability of deviations from the ideal value, even though the distribution really is of the appropriate type. I don't recall anything about the power of these statistics, however (i.e., the probability that a power law will look like an exponential, or vice-versa).]
- William J. Reed and Barry D. Hughes, "From Gene Families and Genera
to Incomes and Internet File Sizes: Why Power Laws are so Common in
Nature", Physical
Review E
**66**(2002): 067103 [This is, as I said, perhaps the most deflating possible explanation for power law size distributions. Imagine you have some set of piles, each of which grows, multiplicatively, at a constant rate. New piles are started at random times, with a constant probability per unit time. (This is a good model of my office.) Then, at any time, the age of the piles is exponentially distributed, and their size is an exponential function of their age; the two exponentials cancel and give you a power-law size distribution. The basic combination of exponential growth and random observation times turns out to work even if it's only the*mean*size of piles which grows exponentially.] - M. V. Simkin and V. P. Roychowdhury, "Re-inventing Willis", physics/0601192 [The comical, yet pathetic, history of the innumerable re-inventions of basic mechanisms which plague this area]
- Herbert Simon, "On a Class of Skew Distribution Functions",
Biometrika
**42**(1955): 425--440 [JSTOR] - Didier Sornette
- "Multiplicative Processes and Power Laws" cond-mat/9708231
= Physical Review E
**57**(1998): 4811--4813 - "Mechanism for Powerlaws without Self-Organization" cond-mat/0110426

- "Multiplicative Processes and Power Laws" cond-mat/9708231
= Physical Review E
- Stilian A. Stoev, George Michailidis, and Murad S. Taqqu, "Estimating heavy-tail exponents through max self-similarity", math.ST/0609163
- Bruce J. West and Bill Deering, The Lure of Modern Science:
Fractal Thinking [Despite the
*painful*title, this is actually a very good book. I disagree with some of the more philosophical positions they take, but on the actual science and math they're quite sound.] - Wei Biao Wu, Yinxiao Huang and Wei Zheng, "Covariances estimation
for long-memory
processes", Advances
in Applied Probability
**42**(2010): 137--157 [How big are the errors in your covariance estimates?] - Damian H. Zanette, "Zipf's law and the creation of musical
context", cs.CL/0406015 [This
*sounds*bizarre, and I'd not have bothered to even note it if I didn't know Zanette's work in other areas, which shows him to be a good and careful scientist. And this is actually an interesting and meaningful little paper, which has something non-trivial to say about music. It's worth noting, perhaps, that the distribution he actually ends up fitting isn't a pure power law, but a modification inspired by Simon's paper. Thanks to John Burke for prodding me to actually read it.]

- Not altogether recommended (without being actively dis-recommended either):
- R. Alexander Bentley, Paul Ormerod, Michael Batty, "An evolutionary model of long tailed distributions in the social sciences", arxiv:0903.2533 [This is a minor modification of the classical Yule/Simon mechanism for random growth, with the main advantage being that (with the right parameter tweaking) it allows for more turn-over of which values are most common. Unsurprisingly, this is done by adding extra parameters, and so the family of distributions is more flexible. But they use bad statistical procedures, and the finding that the estimated power law exponent grows as the amount of data held in the tail shrinks is simply explained: the tails aren't power laws.]

- Recommended, of a not entirely serious character:
- Mason Porter's Power Law Shop

- Modesty forbids me to recommend:
- Aaron Clauset, CRS and M. E. J. Newman, "Power-law distributions in
empirical data", SIAM Review
**51**(2009): 661--703, arxiv:0706.1062 [with commentary by Aaron and myself]

- Pride compels me to recommend:
- Georg M. Goerg, "Lambert W random variables: A new family of generalized skewed distributions with applications to risk estimation",
Annals of Applied
Statistics
**5**(2011): 2197--2230, arxiv:0912.4554 [By my student, but entirely independent work]

- To read:
- Eduardo G. Altmann and Holger Kantz, "Recurrence time analysis, long-term correlations, and extreme events", physics/0503056
- J. A. D. Aston, "Modeling macroeconomic time series via heavy tailed distributions", math.ST/0702844
- Stefan Aulbach and Michael Falk, "Testing for a generalized Pareto
process", Electronic
Journal of Statistics
**6**(2012): 1779--1802 - Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger, "Infinite variance stable limits for sums of dependent random variables", arxiv:0906.2717
- Michael Batty, "Rank Clocks", Nature
**444**(2006): 592--596 - Marco Bee, Massimo Riccaboni and Stefano Schiavo, "Pareto versus lognormal: A maximum entropy test", Physical
Review E
**84**(2011); 026104 - Jan Beran, Bikramjit Das, Dieter Schell, "On robust tail index estimation for linear long-memory processes", Journal of Time Series Analysis
**33**(2012): 406--423 - Patrice Bertail, Stéphan Clémençon, and
Jessica Tressou, "Regenerative block-bootstrap confidence intervals for tail
and extremal indexes", Electronic Journal of Statistics
**7**(2013): 1224--1248 - P. Besbeas and B. J. T. Morgan, "Improved estimation of the stable
laws", Statistics
and Computing
**18**(2008): 219--231 - Eric Beutner, Henryk Zähle, "Continuous mapping approach to the asymptotics of U- and V-statistics", arxiv:1203.1112
- Danny Bickson, Carlos Guestrin, "Linear Characteristic Graphical Models: Representation, Inference and Applications", arxiv:1008.5325 [Graphical models with heavy-tailed latent variables]
- Thierry Bochud and Damien Challet, "Optimal approximations of power-laws with exponentials", physics/0605149 ["We propose an explicit recursive method to approximate a power-law with a finite sum of weighted exponentials. Applications to moving averages with long memory are discussed in relationship with stochastic volatility models." The last part sounds like a rediscovery of Granger.]
- Stéphane Boucheron and Maud Thomas, "Tail index estimation, concentration and adaptivity", arxiv:1503.05077
- Christian Brownlees, Emilien Joly, Gabor Lugosi, "Empirical risk minimization for heavy-tailed losses", arxiv:1406.2462
- Laurent E. Calvet and Adlai J. Fisher, Multifractal Volatility: Theory, Forecasting, and Pricing [Thanks to Prof. Calvet for bringing this to my attention]
- Anna Carbone and Giuliano Castelli, "Scaling Properties of Long-Range Correlated Noisy Signals," cond-mat/0303465
- Alexandra Carpentier, Arlene K.H. Kim, "Adaptive and minimax optimal estimation of the tail coefficient", arxiv:1309.2585
- C. Cattuto, V. Loreto and V. D. P. Servedio, "A Yule-Simon process with memory", cond-mat/0608672 [Memo to self: compare this to the auto-correlated Yule-Simon process in Ijiri and Simon's book.]
- Arijit Chakrabarty, "Central Limit Theorem and Large Deviations for truncated heavy-tailed random vectors", arxiv:1003.2159
- Arijit Chakrabarty, Gennady Samorodnitsky, "Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?", arxiv:1001.3218
- Anirban Chakraborti, Marco Patriarca, "A Variational Principle for Pareto's power law", cond-mat/0605325
- Ali Chaouche and Jean-Noel Bacro, "Statistical Inference for
the Generalized Pareto Distribution: Maximum Likelihood Revisited", Communications in Statistics: Theory and Methods
**35**(2006): 785--802 - F. Clementi, M. Gallegati, "Pareto's Law of Income Distribution: Evidence for Germany, the United Kingdom, and the United States", physics/0504217
- Cline, heavy-tailed noise, 1983 (?)
- B. Conrad and M. Mitzenmacher, "Power Laws for Monkeys Typing
Randomly: The Case of Unequal Probabilities", IEEE Transactions on
Information Theory
**50**(2004): 1403--1414 - J. Danielsson, L. de Haan, L. Peng and C. G. de Vries, "Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation",
Journal of Multivariate Analysis
**76**(2001): 236--248 - Bikramjit Das and Siddney I. Resnick, "QQ plots, Random sets and data from a heavy tailed distribution", math.PR/0702551
- Anirban Dasgupta, John Hopcroft, Jon Kleinberg and Mark Sandler, "On Learning Mixtures of Heavy-Tailed Distributions"
- Nima Dehghani, Nicholas G. Hatsopoulos, Zach D. Haga, Rebecca A. Parker, Bradley Greger, Eric Halgren, Sydney S. Cash, Alain Destexhe, "Avalanche analysis from multi-electrode ensemble recordings in cat, monkey and human cerebral cortex during wakefulness and sleep", arxiv:1203.0738 [Ummm, we explain why you can't use $R^2$ that way in the paper you cite...]
- T. Di Matteo, T. Aste and M. Gallegati, "Innovation flow through social networks: Productivity distribution", physics/0406091 [Those look an awful lot like log-normals to me.]
- Paul Doukhan, George Oppenheim and Murad S. Taqqu (eds.), Theory and Applications of Long-Range Dependence
- Rick Durrett and Jason Schweinsberg, "Power laws for family sizes
in a duplication model", math.PR/0406216
Annals of Probability
**33**(2005): 2094--2126 - R. Fox and M. S. Taqqu
- "Noncentral Limit Thorems for Quadratic Forms in Random
Variables Having Long-Range Dependence," Annals of Probability
**13**(1985) 428--446 - "Central Limit Theorems for Quadratic Forms in Random
Variables Having Long-Range Dependence," Probability Theory and Related
Fields
**74**(1987): 213--240

- "Noncentral Limit Thorems for Quadratic Forms in Random
Variables Having Long-Range Dependence," Annals of Probability
- G. Frenkel, E. Katzav, M. Schwartz and N. Sochen, "Distribution of
Anomalous Exponents of Natural Images", Physical Review
Letters
**97**(2006): 103902 - U. Frisch and D. Sornette, "Extreme Deviations and Applications",
J. Phys. I France
**7**(1997): 1155--1171 - Akihiro Fujihara, Toshiya Ohtsuki and Hiroshi Yamamoto
- "Power-law tails in nonstationary stochastic processes with
asymmetrically multiplicative interactions", Physical Review
E
**70**(2004): 031106 = cond-mat/0506785 - "Similarity and Probability Distribution Functions in Many-body Stochastic Processes with Multiplicative Interactions", cond-mat/0508615

- "Power-law tails in nonstationary stochastic processes with
asymmetrically multiplicative interactions", Physical Review
E
- Akihiro Fujihara, Satoshi Tanimoto, Toshiya Ohtsuki, Hiroshi Yamamoto, "Log-normal distribution in growing systems with weighted multiplicative interactions", cond-mat/0511625
- Yoshi Fujiwara, Corrado Di Guilmi, Hideaki Aoyama, Mauro Gallegati, Wataru Souma, "Do Pareto-Zipf and Gibrat laws hold true? An analysis with European Firms", cond-mat/0310061
- Xavier Gabaix, "Power Laws in Economics and Finance" [PDF preprint]
- Michael Golosovsky and Sorin Solomon, "Stochastic Dynamical Model
of a Growing Citation Network Based on a Self-Exciting Point
Process", Physical
Review Letters
**109**(2012): 098701 - M. Ivette Gomes, M. Isabel Fraga Alves, Paulo Araujo Santos,
"PORT Hill and Moment Estimators for Heavy-Tailed Models",
Communications in Statistics:
Simulation and Computation
**37**(2008): 1281--1306 - Alexander Gnedin, Ben Hansen, Jim Pitman, "Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws", math.PR/0701718
- J. A. Gubner, "Theorems and Fallacies in the Theory of
Long-Range-Dependent Processes", IEEE Transactions on
Information Theory
**51**(2005): 1234--1239 - Alexandra Guerrero and Leonard A. Smith, "A maximum likelihood
estimator for long-range persistence", Physica
A
**355**(2005): 619--632 - Rudolf Hanel and Stefan Thurner, "On the Derivation of power-law distributions within standard statistical mechanics", cond-mat/0412016
- Bruce M. Hill and Michael Woodroofe, "Stronger Forms of Zipf's
Law", Journal of the American Statistical
Association
**70**(1975): 212--219 [Deriving Zipf's law from Bose-Einstein statistics. JSTOR] - Byoung Hee Hong, Kyoung Eun Lee, Jae Woo Lee, "Power Law in Firms Bankruptcy", physics/0701302
- Y. Hosoya
- "The quasi-likelihood approach to statistical inference on
multiple time-series with long-range dependence," Journal of
Econometrics
**73**(1996): 217--236 - "A limit theory for long-range dependence and statistical
inference on related models," Annals of Statistics
**25**(1997): 105--137

- "The quasi-likelihood approach to statistical inference on
multiple time-series with long-range dependence," Journal of
Econometrics
- Henrik Hult and Gennady Samorodnitsky, "Large deviations for point processes based on stationary sequences with heavy tails", Journal of Applied Probability
**47**(2010): 1--40 - Takashi Ichinomiya, "Power-law distribution in Japanese racetrack betting", physics/0602165
- Sanja Janicevic, Lasse Laurson, Knut Jorgen Maloy, Stephane Santucci, and Mikko J. Alava, "Interevent Correlations from Avalanches Hiding Below the Detection Threshold", Physical Review Letters
**117**(2016): 230601 - Milton Jara, Tomasz Komorowski and Stefano Olla, "Limit theorems for additive functionals of a Markov chain", arxiv:0809.0177 [Convergence to alpha-stable distributions]
- Predrag R. Jelenkovic, Jian Tan, "Modulated Branching Processes, Origins of Power Laws and Queueing Duality", 0709.4297
- Junghyo Jo, Jean-Yves Fortin, M. Y. Choi, "Weibull-type limiting distribution for replicative systems", Physical Review E
**83**(2011): 031123, arxiv:1103.3038 - Taisei Kaizoji, "Power laws and market crashes", physics/0603138
- Imen Kammoun, Vernoique Billat and Jean-Marc Bardet, "A new stochastic process to model Heart Rate series during exhaustive run and an estimator of its fractality parameter", arxiv:0803.3675 [Includes statistical criticism of the common, but deeply unsatisfying, "detrended fluctuation analysis" method of estimating the Hurst exponent.]
- B. Kaulakys and J. Ruseckas, "Stochastic nonlinear
differential equation generating 1/f noise", Physical Review E
**70**(2004): 020101 = cond-mat.0408507 - K. Kiyani, S. C. Chapman and B. Hnat, "A method for extracting the scaling exponents of a self-affine, non-Gaussian process from a finite length timeseries", physics/0607238
- K. H. Kiyani, S. C. Chapman, N. W. Watkins, "Pseudo-nonstationarity in the scaling exponents of finite interval time series", Physical
Review E
**79**(2009): 036109, arxiv:0808.2036 - Henry Lam, Jose Blanchet, Damian Burch, Martin Z. Bazant, "Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities",
Journal of Theoretical Probability
**24**(2011): 895--927, arxiv:1103.4306 - Francois M. Longin, "The Asymptotic Distribution of Extreme Stock
Market Returns", The Journal of Business
**69**(1996): 383--408 [JSTOR] - Fotis Loukissas, "Precise Large Deviations for Long-Tailed Distributions", Journal of Theoretical Probability
**25**(2012): 913--924 - Bruce D. Malamud, James D. A. Millington and George L. W. Perry,
"Characterizing wildfire regimes in the United States", Proceedings of the
National Academy of Sciences (USA)
**102**(2005): 4694--4699 - Y. Malevergne, V.F. Pisarenko, D. Sornette, "Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?", physics/0305089
- Alon Manor and Nadav M. Shnerb, "Multiplicative Noise and Second Order Phase Transitions", Physical
Review Letters
**103**(2009): 030601 - Natalia Markovich, Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice
- Matteo Marsili, "On the concentration of large deviations for fat tailed distributions", arxiv:1201.2817
- Yosef E. Maruvka, David A. Kessler, Nadav M. Shnerb, "The Birth-Death-Mutation process: a new paradigm for fat tailed distributions", arxiv:1011.4110 [I suspected from the abstract that this was Yet Another Rediscovery of the Yule-Simon mechanism. However, after actually looking through the paper (prompted by Dr. Shnerb), I see that they are in fact doing something more, and that I was just wrong. I still need to read it properly, however, before deciding what I think about the actual proposal.]
- Joseph L. McCauley, Gemunu H. Gunaratne, Kevin E. Bassler, "Hurst Exponents, Markov Processes, and Fractional Brownian motion", cond-mat/0609671
- Richard Metzler, "Comment on 'Power-law correlations in the
southern-oscillation-index fluctuations characterizing El
Nino'", Physical
Review E
**67**(2003): 018201 - Salvatore Miccich`, "Modeling long-range memory with stationary Markovian processes", Physical
Review E
**79**(2009): 031116, arxiv:arxiv:0806.0722 - Thomas Mikosch, Sidney Resnick, Holger Rootzén, and Alwin Stegeman. "Is Network Traffic Appriximated by Stable Lévy Motion or Fractional Brownian Motion?", Annals of Applied Probability
**12**(2002): 23--68 - Thomas Mikosch, Olivier Wintenberger, "Precise large deviations for dependent regularly varying sequences", arxiv:1206.1395
- Edoardo Milotti, "Model-based fit procedure for power-law-like spectra", physics/0510011
- Mariusz Mirek, "Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps", Probability Theory and Related Fields
**151**(2011): 705--734 - Elliott W. Montroll and Michael Shlesinger, "Maximum entropy
formalism, fractals, scaling phenomena and 1/f noise: A tale of tails",
Journal of Statistical Physics
**32**(1983): 209--230 - Eric Moulines, Francois Roueff, Murad S. Taqqu, "A Wavelet Whittle estimator of the memory parameter of a non-stationary Gaussian time series", math/0601070
- Newton J. Moura Jr. and Marcelo B. Ribeiro, "Zipf Law for Brazilian Cities", physics/0511216
- J. F. Muzy, E. Bacry and A. Kozhemyak, "Extreme values and fat
tails of multifractal
fluctuations", Physical Review
E
**73**(2006): 066114 = cond-mat/0509357 ["problem of the estimation of extreme event occurrence probability for data drawn from some multifractal process. We also study the heavy (power-law) tail behavior of probability density function associated with such data. We show that because of strong correlations, standard extreme value approach is not valid and classical tail exponent estimators should be interpreted cautiously"] - Fuyuo Nagayama, "Wealth Inequality Among the Forbes 400 and U.S. Households Overall" [PDF preprint via Federal Reserve Bank of Philadelphia]
- Marko Obradović, Milan Jovanović, Bojana Milošević, "Goodness-of-Fit Tests for Pareto Distribution Based on a Characterization and their Asymptotics", arxiv:1310.5510
- Richard Perline, "Strong, Weak and False Inverse Power Laws",
Statistical
Science
**20**(2005): 68--88 - Sergei Petrovskii, Alla Mashanova, and Vincent A. A. Jansen, "Variation in individual walking behavior creates the impression of a Lévy flight", Proceedings of
the National Academy of Sciences (USA)
**108**(2011): 8704--8707 - William Rea, Les Oxley, Marco Reale and Jennifer Brown, "Estimators for Long Range Dependence: An Empirical Study", arxiv:0901.0762
- S. Redner, "Random multiplicative processes: An elementary
tutorial", American Journal of Physics
**58**(1990): 267--273 - Sidney I. Resnick, Heavy-Tail Phenomena: Probabilistic and Statistical Modeling
- Sidney Resnick and Catalin Starica, "Tail Index Estimation for
Dependent Data", The Annals of Applied
Probability
**8**(1998): 1156--1183 - Massimo Riccaboni, Fabio Pammolli, Sergey V. Buldyrev, Linda Ponta,
H. Eugene Stanley , "The Size Variance Relationship of Business Firm Growth
Rates", arxiv:0904.1404
= Proceedings of the National Academy of Sciences
(USA)
**105**(2008): 19595--19600 - C. Y. Robert, "Automatic Declustering of Rare Events", Biometrika
**100**(2013): 587--606 - Alexander Roitershtein, "One-dimensional linear recursions with
Markov-dependent
coefficients", math/0409335
= Annals of
Applied Probability
**17**(2007): 572--608 [To summarize the abstract, suppose S(n) = A(n) + B(n)*S(n-1), where A(n) and B(n) are Markov sequences. Then "the distribution tail of its stationary solution has a power law decay." This sounds like Simon's argument made more general.] - Holger Rootzen, M. Ross Leadbetter and Laurens de Haan, "On the
distribution of tail array sums for strongly mixing stationary
sequences", Annals of Applied Probability
**8**(1998): 868--885 - Gordon J Ross, Tim Jones, "Understanding the Heavy Tailed Dynamics in Human Behavior", arxiv:1505.01547
- Gennady Samorodnitsky and Murad S. Taqqu, Stable Non-Gaussian Random Processes
- Francois G. Schmitt and Yongxiang Huang, Stochastic Analysis of Scaling Time Series: From Turbulence Theory to Applications
- David J. Schwab, Ilya Nemenman, Pankaj Mehta, "Zipf's law and criticality in multivariate data without fine-tuning", arxiv:1310.0448
- D. Sornette and V. F. Pisarenko, "Properties of a simple bilinear stochastic model: estimation and predictability", physics/0703217
- Attilio L. Stella, Fulvio Baldovin, "Anomalous scaling due to correlations: Limit theorems and self-similar processes", arxiv:0909.0906
- Stilian A Stoev, George Michailidis, "On the Estimation of the Heavy-Tail Exponent in Time Series using the Max-Spectrum", arxiv:1005.4329
- Stilian A. Stoev and Murad S. Taqqu, "Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights",
Methodology and
Computing in Applied Probability
**9**(2007): 55--87 - Sarah Touati, Mark Naylor, and Ian G. Main, "Origin and Nonuniversality of the Earthquake Interevent Time Distribution", Physical Review Letters
**102**(2009): 168501 - Ciprian Tudor and Frederi Viens, "Variations and estimators for the selfsimilarity order through Malliavin calculus", arxiv:0709.3896
- Caglar Tuncay, "A universal model for languages and cities, and their lifetimes", physics/0703144
- Marta Tyran-Kaminska, "Convergence to Lévy stable processes under strong mixing conditions", arxiv:0907.1185
- Sergio Venturini, Francesca Dominici, Giovanni Parmigiani, "Gamma
shape mixtures for heavy-tailed distributions", Annals of Applied
Statistics
**2**(2008): 756--776 = arxiv:0807.4663 - Yogesh Virkar, Aaron Clauset, "Power-law distributions in binned empirical data", arxiv:1208.3524
- Nicholas Watkins, "Mandelbrot's 1/f fractional renewal models of 1963-67: The non-ergodic missing link between change points and long range dependence", Advances in Time Series Analysis and Forecasting ITISE 2016, arxiv:1603.00738
- Rafal Weron
- "Estimating long range dependence: finite sample properties and confidence intervals," cond-mat/0103510
- "Measuring long-range dependence in electricity prices," cond-mat/0103621

- T. S. T. Wong and W. K. Li, "A note on the estimation of extreme value distributions using maximum product of spacings", math.ST/0702830
- Wei Biao Wu, Xiaofeng Shao, "Invariance principles for fractionally integrated nonlinear processes", math.PR/0608223
- Seokhoon Yun, "The Extremal Index of a Higher-Order
Stationary Markov Chain", The Annals of Applied Probability
**8**(1998): 408--437 - Damian H. Zanette, "Zipf's law and city sizes: A short tutorial review on multiplicative processes in urban growth", arxiv:0704.3170
- Ting Zhang, Hwai-Chung Ho, Martin Wendler, Wei Biao Wu, "Block Sampling under Strong Dependence", arxiv:1312.5807
- Qiuye Zhao and Mitch Marcus, "Long-tail Distributions and Unsupervised Learning of Morphology" [PDF. Replaces Zipf distribution over words with a log-normal. Doesn't test whether that's a better fit, but claims to give nice results in other tasks.]