635 research outputs found
On low-sampling-rate Kramers-Moyal coefficients
We analyze the impact of the sampling interval on the estimation of
Kramers-Moyal coefficients. We obtain the finite-time expressions of these
coefficients for several standard processes. We also analyze extreme situations
such as the independence and no-fluctuation limits that constitute useful
references. Our results aim at aiding the proper extraction of information in
data-driven analysis.Comment: 9 pages, 4 figure
Extreme statistics for time series: Distribution of the maximum relative to the initial value
The extreme statistics of time signals is studied when the maximum is
measured from the initial value. In the case of independent, identically
distributed (iid) variables, we classify the limiting distribution of the
maximum according to the properties of the parent distribution from which the
variables are drawn. Then we turn to correlated periodic Gaussian signals with
a 1/f^alpha power spectrum and study the distribution of the maximum relative
height with respect to the initial height (MRH_I). The exact MRH_I distribution
is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random
acceleration), and alpha=infinity (single sinusoidal mode). For other,
intermediate values of alpha, the distribution is determined from simulations.
We find that the MRH_I distribution is markedly different from the previously
studied distribution of the maximum height relative to the average height for
all alpha. The two main distinguishing features of the MRH_I distribution are
the much larger weight for small relative heights and the divergence at zero
height for alpha>3. We also demonstrate that the boundary conditions affect the
shape of the distribution by presenting exact results for some non-periodic
boundary conditions. Finally, we show that, for signals arising from
time-translationally invariant distributions, the density of near extreme
states is the same as the MRH_I distribution. This is used in developing a
scaling theory for the threshold singularities of the two distributions.Comment: 29 pages, 4 figure
A theory for long-memory in supply and demand
Recent empirical studies have demonstrated long-memory in the signs of orders
to buy or sell in financial markets [2, 19]. We show how this can be caused by
delays in market clearing. Under the common practice of order splitting, large
orders are broken up into pieces and executed incrementally. If the size of
such large orders is power law distributed, this gives rise to power law
decaying autocorrelations in the signs of executed orders. More specifically,
we show that if the cumulative distribution of large orders of volume v is
proportional to v to the power -alpha and the size of executed orders is
constant, the autocorrelation of order signs as a function of the lag tau is
asymptotically proportional to tau to the power -(alpha - 1). This is a
long-memory process when alpha < 2. With a few caveats, this gives a good match
to the data. A version of the model also shows long-memory fluctuations in
order execution rates, which may be relevant for explaining the long-memory of
price diffusion rates.Comment: 12 pages, 7 figure
Extreme times for volatility processes
We present a detailed study on the mean first-passage time of volatility
processes. We analyze the theoretical expressions based on the most common
stochastic volatility models along with empirical results extracted from daily
data of major financial indices. We find in all these data sets a very similar
behavior that is far from being that of a simple Wiener process. It seems
necessary to include a framework like the one provided by stochastic volatility
models with a reverting force driving volatility toward its normal level to
take into account memory and clustering effects in volatility dynamics. We also
detect in data a very different behavior in the mean first-passage time
depending whether the level is higher or lower than the normal level of
volatility. For this reason, we discuss asymptotic approximations and confront
them to empirical results with a good agreement, specially with the ExpOU
model.Comment: 10, 6 colored figure
Non-characteristic Half-lives in Radioactive Decay
Half-lives of radionuclides span more than 50 orders of magnitude. We
characterize the probability distribution of this broad-range data set at the
same time that explore a method for fitting power-laws and testing
goodness-of-fit. It is found that the procedure proposed recently by Clauset et
al. [SIAM Rev. 51, 661 (2009)] does not perform well as it rejects the
power-law hypothesis even for power-law synthetic data. In contrast, we
establish the existence of a power-law exponent with a value around 1.1 for the
half-life density, which can be explained by the sharp relationship between
decay rate and released energy, for different disintegration types. For the
case of alpha emission, this relationship constitutes an original mechanism of
power-law generation
Pareto versus lognormal: a maximum entropy test
It is commonly found that distributions that seem to be lognormal over a broad range change to a power-law (Pareto) distribution for the last few percentiles. The distributions of many physical, natural, and social events (earthquake size, species abundance, income and wealth, as well as file, city, and firm sizes) display this structure. We present a test for the occurrence of power-law tails in statistical distributions based on maximum entropy. This methodology allows one to identify the true data-generating processes even in the case when it is neither lognormal nor Pareto. The maximum entropy approach is then compared with other widely used methods and applied to different levels of aggregation of complex systems. Our results provide support for the theory that distributions with lognormal body and Pareto tail can be generated as mixtures of lognormally distributed units
Modeling long-range memory with stationary Markovian processes
In this paper we give explicit examples of power-law correlated stationary
Markovian processes y(t) where the stationary pdf shows tails which are
gaussian or exponential. These processes are obtained by simply performing a
coordinate transformation of a specific power-law correlated additive process
x(t), already known in the literature, whose pdf shows power-law tails 1/x^a.
We give analytical and numerical evidence that although the new processes (i)
are Markovian and (ii) have gaussian or exponential tails their autocorrelation
function still shows a power-law decay =1/T^b where b grows with a
with a law which is compatible with b=a/2-c, where c is a numerical constant.
When a<2(1+c) the process y(t), although Markovian, is long-range correlated.
Our results help in clarifying that even in the context of Markovian processes
long-range dependencies are not necessarily associated to the occurrence of
extreme events. Moreover, our results can be relevant in the modeling of
complex systems with long memory. In fact, we provide simple processes
associated to Langevin equations thus showing that long-memory effects can be
modeled in the context of continuous time stationary Markovian processes.Comment: 5 figure
The Bivariate Normal Copula
We collect well known and less known facts about the bivariate normal
distribution and translate them into copula language. In addition, we prove a
very general formula for the bivariate normal copula, we compute Gini's gamma,
and we provide improved bounds and approximations on the diagonal.Comment: 24 page
Upper bounds for number of removed edges in the Erased Configuration Model
Models for generating simple graphs are important in the study of real-world
complex networks. A well established example of such a model is the erased
configuration model, where each node receives a number of half-edges that are
connected to half-edges of other nodes at random, and then self-loops are
removed and multiple edges are concatenated to make the graph simple. Although
asymptotic results for many properties of this model, such as the limiting
degree distribution, are known, the exact speed of convergence in terms of the
graph sizes remains an open question. We provide a first answer by analyzing
the size dependence of the average number of removed edges in the erased
configuration model. By combining known upper bounds with a Tauberian Theorem
we obtain upper bounds for the number of removed edges, in terms of the size of
the graph. Remarkably, when the degree distribution follows a power-law, we
observe three scaling regimes, depending on the power law exponent. Our results
provide a strong theoretical basis for evaluating finite-size effects in
networks
Intermediate Tail Dependence: A Review and Some New Results
The concept of intermediate tail dependence is useful if one wants to
quantify the degree of positive dependence in the tails when there is no strong
evidence of presence of the usual tail dependence. We first review existing
studies on intermediate tail dependence, and then we report new results to
supplement the review. Intermediate tail dependence for elliptical, extreme
value and Archimedean copulas are reviewed and further studied, respectively.
For Archimedean copulas, we not only consider the frailty model but also the
recently studied scale mixture model; for the latter, conditions leading to
upper intermediate tail dependence are presented, and it provides a useful way
to simulate copulas with desirable intermediate tail dependence structures.Comment: 25 pages, 1 figur
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