100 research outputs found
The Defect of Random Hyperspherical Harmonics
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the
unit -sphere (). We investigate the distribution of their defect
i.e., the difference between the measure of positive and negative regions.
Marinucci and Wigman studied the two-dimensional case giving the asymptotic
variance (Marinucci and Wigman 2011) and a Central Limit Theorem (Marinucci and
Wigman 2014), both in the high-energy limit. Our main results concern
asymptotics for the defect variance and quantitative CLTs in Wasserstein
distance, in any dimension. The proofs are based on Wiener-It\^o chaos
expansions for the defect, a careful use of asymptotic results for all order
moments of Gegenbauer polynomials and Stein-Malliavin approximation techniques
by Nourdin and Peccati. Our argument requires some novel technical results of
independent interest that involve integrals of the product of three
hyperspherical harmonics.Comment: Accepted for publication in Journal of Theoretical Probabilit
On L\'evy's Brownian motion indexed by the elements of compact groups
We investigate positive definiteness of the Brownian kernel
K(x,y)=1/2(d(x,x_0) + d(y,x_0) - d(x,y)) on a compact group G and in particular
for G=SO(n).Comment: Accepted for publication. 10 page
Representation of Gaussian Isotropic Spin Random Fields
We develop a technique for the construction of random fields on algebraic
structures. We deal with two general situations: random fields on homogeneous
spaces of a compact group and in the spin-line bundles of the 2-sphere. In
particular, every spin Gaussian isotropic field can be obtained with this
construction. Our construction extends P. L\'evy's original idea for the
spherical Brownian Motion.Comment: 27 pages. Accepted for publication on Stoch. Processes App
Large Deviation asymptotics for the exit from a domain of the bridge of a general Diffusion
We provide Large Deviation estimates for the bridge of a -dimensional
general diffusion process as the conditioning time tends to and apply these
results to the evaluation of the asymptotics of its exit time probabilities. We
are motivated by applications to numerical simulation, especially in connection
with stochastic volatility models.Comment: 15 pages, 2 figure
Nodal Statistics of Planar Random Waves
We consider Berry's random planar wave model (1977) for a positive Laplace
eigenvalue , both in the real and complex case, and prove limit theorems
for the nodal statistics associated with a smooth compact domain, in the
high-energy limit (). Our main result is that both the nodal
length (real case) and the number of nodal intersections (complex case) verify
a Central Limit Theorem, which is in sharp contrast with the non-Gaussian
behaviour observed for real and complex arithmetic random waves on the flat
-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings
can be naturally reformulated in terms of the nodal statistics of a single
random wave restricted to a compact domain diverging to the whole plane. As
such, they can be fruitfully combined with the recent results by Canzani and
Hanin (2016), in order to show that, at any point of isotropic scaling and for
energy levels diverging sufficently fast, the nodal length of any Gaussian
pullback monochromatic wave verifies a central limit theorem with the same
scaling as Berry's model. As a remarkable byproduct of our analysis, we
rigorously confirm the asymptotic behaviour for the variances of the nodal
length and of the number of nodal intersections of isotropic random waves, as
derived in Berry (2002).Comment: Preliminary version. 51 page
On the excursion area of perturbed Gaussian fields
We investigate Lipschitz-Killing curvatures for excursion sets of random
fields on under small spatial-invariant random perturbations. An
expansion formula for mean curvatures is derived when the magnitude of the
perturbation vanishes, which recovers the Gaussian Kinematic Formula at the
limit by contiguity of the model. We develop an asymptotic study of the
perturbed excursion area behaviour that leads to a quantitative non-Gaussian
limit theorem, in Wasserstein distance, for fixed small perturbations and
growing domain. When letting both the perturbation vanish and the domain grow,
a standard Central Limit Theorem follows. Taking advantage of these results, we
propose an estimator for the perturbation which turns out to be asymptotically
normal and unbiased, allowing to make inference through sparse information on
the field
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