105 research outputs found
Anisotropic scaling of random grain model with application to network traffic
We obtain a complete description of anisotropic scaling limits of random
grain model on the plane with heavy tailed grain area distribution. The scaling
limits have either independent or completely dependent increments along one or
both coordinate axes and include stable, Gaussian and some `intermediate'
infinitely divisible random fields. Asymptotic form of the covariance function
of the random grain model is obtained. Application to superposed network
traffic is included
Projective stochastic equations and nonlinear long memory
A projective moving average is a Bernoulli shift
written as a backward martingale transform of the innovation sequence. We
introduce a new class of nonlinear stochastic equations for projective moving
averages, termed projective equations, involving a (nonlinear) kernel and a
linear combination of projections of on "intermediate" lagged innovation
subspaces with given coefficients . The class of such
equations include usual moving-average processes and the Volterra series of the
LARCH model. Solvability of projective equations is studied, including a nested
Volterra series representation of the solution . We show that under
natural conditions on , this solution exhibits
covariance and distributional long memory, with fractional Brownian motion as
the limit of the corresponding partial sums process
Scaling transition for nonlinear random fields with long-range dependence
We obtain a complete description of anisotropic scaling limits and the
existence of scaling transition for nonlinear functions (Appell polynomials) of
stationary linear random fields on with moving average
coefficients decaying at possibly different rate in the horizontal and vertical
direction. The paper extends recent results on scaling transition for linear
random fields in Puplinskait\.e and Surgailis (2016), Puplinskait\.e and
Surgailis (2015)
Scaling limits of nonlinear functions of random grain model, with application to Burgers' equation
We study scaling limits of nonlinear functions of random grain model
on with long-range dependence and marginal Poisson
distribution. Following Kaj et al (2007) we assume that the intensity of
the underlying Poisson process of grains increases together with the scaling
parameter as , for some . The
results are applicable to the Boolean model and exponential and rely on an
expansion of in Charlier polynomials and a generalization of Mehler's
formula. Application to solution of Burgers' equation with initial aggregated
random grain data is discussed
Moment bounds and central limit theorems for Gaussian subordinated arrays
A general moment bound for sums of products of Gaussian vector's functions
extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general
central limit theorem for triangular arrays of nonlinear functionals of
multidimensional non-stationary Gaussian sequences is proved. This theorem
extends the previous results of Breuer and Major (1981), Arcones (1994) and
others. A Berry-Esseen-type bound in the above-mentioned central limit theorem
is derived following Nourdin, Peccati and Podolskij (2011). Two applications of
the above results are discussed. The first one refers to the asymptotic
behavior of a roughness statistic for continuous-time Gaussian processes and
the second one is a central limit theorem satisfied by long memory locally
stationary process
A two-sample test for comparison of long memory parameters
We construct a two-sample test for comparison of long memory parameters based
on ratios of two rescaled variance (V/S) statistics studied in [Giraitis L.,
Leipus, R., Philippe, A., 2006. A test for stationarity versus trends and unit
roots for a wide class of dependent errors. Econometric Theory 21, 989--1029].
The two samples have the same length and can be mutually independent or
dependent. In the latter case, the test statistic is modified to make it
asymptotically free of the long-run correlation coefficient between the
samples. To diminish the sensitivity of the test on the choice of the bandwidth
parameter, an adaptive formula for the bandwidth parameter is derived using the
asymptotic expansion in [Abadir, K., Distaso, W., Giraitis, L., 2009. Two
estimators of the long-run variance: Beyond short memory. Journal of
Econometrics 150, 56--70]. A simulation study shows that the above choice of
bandwidth leads to a good size of our comparison test for most values of
fractional and ARMA parameters of the simulated series
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