a practical guide to heavy tails statistical techniques and applications part 1 applications heavy tailed probability distributions in the world wide web m e crovella et al self similarity and heavy tails structural modelling of network traffic w willinger et al heavy tails in high frequency financial data u a muller et al stable paretian modelling in finance some empirical and theoretical aspects s mittnik et al risk management and quantile estimation f bassi et al part 2 time series analyzing stable time series r j adler et al inference for linear processes with stable noise m calder r a davis on estimating the intensity of long range dependence in finite and infinite variance time series m s taqqu v teverovsky why non linearities can ruin the heavy tailed modeller s day s i resnick periodogram estimates from heavy tailed data t mikosch bayesian inference for time series with infinite variance stable innovations n ravishanker z qiou part 3 heavy tail estimation hill bootstrap and jackknife estimators for heavy tails o v pictet et al characteristic function based estimation of stable distribution parameters s m kogan d b williams part 4 regression bootstrapping signs and permutations for regression with heavy tailed errors a robust resampling r lepage et al linear regression with stable disturbances j h mcculloch part 5 signal processing deviation from normality in statistical signal processing parameter estimation with alpha stable distributions p tsakalides c l nikias statistical modelling and receiver design for multi user communication networks g a tsihrintzis part 6 model structures subexponential distributions c m goldie c kluppelberg structure of stationary stable processes j rosinski tail behaviour of some shot noise processes g samorodnitsky part 7 numerical procedures numerical approximation of the symmetric stable distribution and density j h mcculloch table of the maximally skewed stable distributions j h mcculloch d b panton multivariate stable distributions approximation estimation simulation and identification j p nolan univariate stable distributions parametrizations and software j p nolan

power law statistics and universal scaling in the absence of criticality critical states are sometimes identified experimentally through power law statistics or universal scaling functions we show here that such features naturally emerge from networks in self sustained irregular regimes away from criticality in these regimes statistical physics theory of large interacting systems predict a regime where the nodes have independent and identically distributed dynamics we thus investigated the statistics of a system in which units are replaced by independent stochastic surrogates and found the same power law statistics indicating that these are not sufficient to establish criticality we rather suggest that these are universal features of large scale networks when considered macroscopically these results put caution on the interpretation of scaling laws found in nature

radial structure of the internet the structure of the internet at the autonomous system as level has been studied by the physics mathematics and computer science communities we extend this work to include features of the core and the periphery taking a radial perspective on as network structure new methods for plotting as data are described and they are used to analyse datasets that have been extended to contain edges missing from earlier collections the average distance from one vertex to the rest of the network is used as the baseline metric for investigating radial structure common vertex specific quantities are plotted against this metric to reveal distinctive characteristics of central and peripheral vertices two datasets are analysed

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

discrete approximations for complex kac moody groups abstract we construct a map from the classifying space of a discrete kac moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological kac moody group and prove that it is a homology equivalence at primes q different from p this generalizes a classical result of quillen friedlander and mislin for lie groups as an application we construct unstable adams operations for general kac moody groups compatible with the frobenius homomorphism our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete kac moody groups

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

on the fractal characterization of paretian poisson processes abstract paretian poisson processes are poisson processes which are defined on the positive half line have maximal points and are quantified by power law intensities paretian poisson processes are elemental in statistical physics and are the bedrock of a host of power law statistics ranging from pareto s law to anomalous diffusion in this paper we establish evenness based fractal characterizations of paretian poisson processes considering an array of socioeconomic evenness based measures of statistical heterogeneity we show that amongst the realm of poisson processes which are defined on the positive half line and have maximal points paretian poisson processes are the unique class of fractal processes exhibiting scale invariance the results established in this paper are diametric to previous results asserting that the scale invariance of poisson processes with respect to physical randomness based measures of statistical heterogeneity is characterized by exponential poissonian intensities

stability in a system subject to noise with regulated periodicity the stability of a simple dynamical system subject to multiplicative one side pulse noise with hidden periodicity is investigated both analytically and numerically the stability analysis is based on the exact result for the characteristic functional of the renewal pulse process the influence of the memory effects on the stability condition is analyzed for two cases i the dead time distorted poissonian process and ii the renewal process with pareto distribution we show that for fixed noise intensity the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity

understanding the determinants of volatility clustering in terms of stationary markovian processes abstract volatility is a key variable in the modeling of financial markets the most striking feature of volatility is that it is a long range correlated stochastic variable i e its autocorrelation function decays like a power law for large time lags in the present work we investigate the determinants of such feature starting from the empirical observation that the exponent of a certain stock s volatility is a linear function of the average correlation of such stock s volatility with all other volatilities we propose a simple approach consisting in diagonalizing the cross correlation matrix of volatilities and investigating whether or not the diagonalized volatilities still keep some of the original volatility stylized facts as a result the diagonalized volatilities result to share with the original volatilities either the power law decay of the probability density function and the power law decay of the autocorrelation function this would indicate that volatility clustering is already present in the diagonalized un correlated volatilities we therefore present a parsimonious univariate model based on a non linear langevin equation that well reproduces these two stylized facts of volatility the model helps us in understanding that the main source of volatility clustering once volatilities have been diagonalized is that the economic forces driving volatility can be modeled in terms of a smoluchowski potential with logarithmic tails

modeling statistics and kinetics of the natural aggregation structures and processes with the solution of generalized logistic equation abstract the generalized logistic equation is proposed to model kinetics and statistics of natural processes such as earthquakes forest fires floods landslides and many others this equation has the form d n a d a s 1 n a n a q a q 0 and a 0 is the size of an element of a structure and 0 the equation contains two exponents and q taking into account two important properties of elements of a system their fractal geometry and their ability to interact either to enhance or to damp the process of aggregation the function n a can be understood as an approximation to the number of elements the size of which is less than a the function d n a d a where n a is the general solution of this equation for q 1 is a product of an increasing bounded function and power law function with stretched exponential cut off the relation with tsallis non extensive statistics is demonstrated by solving the generalized logistic equation for q 0 in the case 0 q 1 the equation models super additive and the case q 1 it models sub additive structures the gutenberg richter g r formula results from interpretation of empirical data as a straight line in the area of stretched exponent with small the solution is applied for modeling distribution of foreshocks and aftershocks in the regions of napa valley 2014 and sumatra 2004 earthquakes fitting the observed data well both qualitatively and quantitatively

conditional subgradient optimization theory and applications abstract we generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients firstly we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization secondly we consider a particular choice of conditional subgradients obtained by projections which leads to an easily implementable modification of traditional subgradient optimization schemes to evaluate the subgradient projection method we consider its use in three applications uncapacitated facility location two person zero sum matrix games and multicommodity network flows computational experiments show that the subgradient projection method performs better than traditional subgradient optimization in some cases the difference is considerable these results suggest that our simply modification may improve subgradient optimization schemes significantly this finding is important as such schemes are very popular especially in the context of lagrangean relaxation

fully exponential laplace approximations to expectations and variances of nonpositive functions abstract tierney and kadane 1986 presented a simple second order approximation for posterior expectations of positive functions they used laplace s method for asymptotic evaluation of integrals in which the integrand is written as f exp nh and the function h is approximated by a quadratic the form in which they applied laplace s method however was fully exponential the integrand was written instead as exp nh log f this allowed first order approximations to be used in the numerator and denominator of a ratio of integrals to produce a second order expansion for the ratio other second order expansions hartigan 1965 johnson 1970 lindley 1961 1980 mosteller and wallace 1964 require computation of more derivatives of the log likelihood function in this article we extend the fully exponential method to apply to expectations and variances of nonpositive functions to obtain a second order approximation to an expectation e g we use the fully exponential method to approximate

frequency of frequencies distributions and size dependent exchangeable random partitions abstractmotivated by the fundamental problem of modeling the frequency of frequencies fof distribution this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions a cluster structure naturally arising from a completely random measure mixed poisson process allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size a distinct and motivated feature that makes it more flexible than a partition structure this allows it to model an entire fof distribution whose structural properties change as the population size varies a fof vector can be simulated by drawing an infinite number of poisson random variables or by a stick breaking construction with a finite random number of steps a generalized negative binomial process model is proposed to generate a cluster structure where in the prior the number of clusters is f

growing networks of overlapping communities with internal structure we introduce an intuitive model that describes both the emergence of community structure and the evolution of the internal structure of communities in growing social networks the model comprises two complementary mechanisms one mechanism accounts for the evolution of the internal link structure of a single community and the second mechanism coordinates the growth of multiple overlapping communities the first mechanism is based on the assumption that each node establishes links with its neighbors and introduces new nodes to the community at different rates we demonstrate that this simple mechanism gives rise to an effective maximal degree within communities this observation is related to the anthropological theory known as dunbar s number i e the empirical observation of a maximal number of ties which an average individual can sustain within its social groups the second mechanism is based on a recently proposed generalization of preferential attachment to community structure appropriately called structural preferential attachment spa the combination of these two mechanisms into a single model spa allows us to reproduce a number of the global statistics of real networks the distribution of community sizes of node memberships and of degrees the spa model also predicts a three qualitative regimes for the degree distribution within overlapping communities and b strong correlations between the number of communities to which a node belongs and its number of connections within each community we present empirical evidence that support our findings in real complex networks

on the bias of traceroute sampling or power law degree distributions in regular graphs understanding the graph structure of the internet is a crucial step for building accurate network models and designing efficient algorithms for internet applications yet obtaining this graph structure can be a surprisingly difficult task as edges cannot be explicitly queried for instance empirical studies of the network of internet protocol ip addresses typically rely on indirect methods like traceroute to build what are approximately single source all destinations shortest path trees these trees only sample a fraction of the network s edges and a paper by lakhina et al 2003 found empirically that the resulting sample is intrinsically biased further in simulations they observed that the degree distribution under traceroute sampling exhibits a power law even when the underlying degree distribution is poisson in this article we study the bias of traceroute sampling mathematically and for a very general class of underlying degree distributions explicitly calculate the distribution that will be observed as example applications of our machinery we prove that traceroute sampling finds power law degree distributions in both regular and poisson distributed random graphs thus our work puts the observations of lakhina et al on a rigorous footing and extends them to nearly arbitrary degree distributions

power law distribution of degree degree distance a better representation of the scale free property of complex networks whether real world complex networks are scale free or not has long been controversial recently in broido and clauset a d broido a clauset nat commun 10 1017 2019 it was claimed that the degree distributions of real world networks are rarely power law under statistical tests here we attempt to address this issue by defining a fundamental property possessed by each link the degree degree distance the distribution of which also shows signs of being power law by our empirical study surprisingly although full range statistical tests show that degree distributions are not often power law in real world networks we find that in more than half of the cases the degree degree distance distributions can still be described by power laws to explain these findings we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted two way selection process the model does not always produce solid power law distributions but predicts that the degree degree distance distribution exhibits stronger power law behavior than the degree distribution of a finite size network especially when the network is dense we test the strength of our model and its predictive power by examining how real world networks evolve into an overly dense stage and how the corresponding distributions change we propose that being scale free is a property of a complex network that should be determined by its underlying mechanism e g preferential attachment rather than by apparent distribution statistics of finite size we thus conclude that the degree degree distance distribution better represents the scale free property of a complex network

dynamical generalized hurst exponent as a tool to monitor unstable periods in financial time series we investigate the use of the hurst exponent dynamically computed over a weighted moving time window to evaluate the level of stability instability of financial firms financial firms bailed out as a consequence of the 2007 2008 credit crisis show a neat increase with time of the generalized hurst exponent in the period preceding the unfolding of the crisis conversely firms belonging to other market sectors which suffered the least throughout the crisis show opposite behaviors we find that the multifractality of the bailed out firms increase at the crisis suggesting that the multi fractal properties of the time series are changing these findings suggest the possibility of using the scaling behavior as a tool to track the level of stability of a firm in this paper we introduce a method to compute the generalized hurst exponent which assigns larger weights to more recent events with respect to older ones in this way large fluctuations in the remote past are less likely to influence the recent past we also investigate the scaling associated with the tails of the log returns distributions and compare this scaling with the scaling associated with the hurst exponent observing that the processes underlying the price dynamics of these firms are truly multi scaling

estimating heavy tail exponents through max self similarity in this paper a novel approach to the problem of estimating the heavy tail exponent 0 of a distribution is proposed it is based on the fact that block maxima of size m scale at a rate m1 for independent as well as for a number of dependent data this scaling rate can be captured well by the max spectrum plot of the data that leads to regression based estimators for consistency and asymptotic normality of these estimators is established for independent data under mild conditions on the behavior of the tail of the distribution the proposed estimators have an important computational advantage over existing methods namely they can be calculated and updated sequentially in an on line fashion without having to store the entire data set practical issues on the automatic selection of tuning parameters for the estimators and corresponding confidence intervals are also addressed extensive numerical simulations show that the proposed method is competitive for both small and large sample sizes robust to contaminants and continues to work under the presence of substantial amount of dependence the proposed estimators are used to illustrate the close connection between long range dependence and heavy tails over an internet traffic trace