8,807 research outputs found
Super Generalized Central Limit Theorem: Limit distributions for sums of non-identical random variables with power-laws
In nature or societies, the power-law is present ubiquitously, and then it is
important to investigate the mathematical characteristics of power-laws in the
recent era of big data. In this paper we prove the superposition of
non-identical stochastic processes with power-laws converges in density to a
unique stable distribution. This property can be used to explain the
universality of stable laws such that the sums of the logarithmic return of
non-identical stock price fluctuations follow stable distributions.Comment: 4pages,1figur
Position space formulation for Dirac fermions on honeycomb lattice
We study how to construct Dirac fermion defined on the honeycomb lattice in
position space. Starting from the nearest neighbor interaction in tight binding
model, we show that the Hamiltonian is constructed by kinetic term and second
derivative term of three flavor Dirac fermions in which one flavor has a mass
of cutoff order and the other flavors are massless. In this formulation the
structure of the Dirac point is simplified so that its uniqueness can be easily
shown even if we consider the next-nearest neighbor interaction. We also show
the chiral symmetry at finite lattice spacing, which protects the masslessness
of the Dirac fermion, and discuss the analogy with the staggered fermion
formulation.Comment: 19 pages, 7 figure
Nonparametric Neural Network Estimation of Lyapunov Exponents and a Direct Test for Chaos
This paper derives the asymptotic distribution of the nonparametric neural network estimator of the Lyapunov exponent in a noisy system. Positivity of the Lyapunov exponent is an operational definition of chaos. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. A simulation study to evaluate small sample performance is reported. We also apply our procedures to daily stock return data. In most cases, the hypothesis of chaos in the stock return series is rejected at the 1% level with an exception in some higher power transformed absolute returns.Artificial neural networks, nonlinear dynamics, nonlinear time series, nonparametric regression, sieve estimation
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