scholarly journals Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes

Bernoulli ◽  
2007 ◽  
Vol 13 (3) ◽  
pp. 712-753 ◽  
Author(s):  
Arnaud Begyn
Keyword(s):  
Asymptotic Expansion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Gaussian Processes ◽  
Quadratic Variations

Central limit theorem for wave-functionals of Gaussian processes

1999 ◽  
Vol 31 (01) ◽  
pp. 158-177 ◽  
Author(s):  
Vladimir Piterbarg ◽  
Igor Rychlik
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Gaussian Processes ◽  
Sample Paths

In this paper a central limit theorem is proved for wave-functionals defined as the sums of wave amplitudes observed in sample paths of stationary continuously differentiable Gaussian processes. Examples illustrating this theory are given.

scholarly journals Limit theorems for multivariate Brownian semistationary processes and feasible results

2019 ◽  
Vol 51 (03) ◽  
pp. 667-716
Author(s):  
Riccardo Passeggeri ◽  
Almut E. D. Veraart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Laws Of Large Numbers ◽  
Stationary Increments ◽  
Large Numbers ◽  
Weak Laws

AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

scholarly journals Limit theorems for a quadratic variation of Gaussian processes

2011 ◽  
Vol 16 (4) ◽  
pp. 435-452 ◽  
Author(s):  
Raimondas Malukas
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Quadratic Variation

In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.

Central limit theorem for wave-functionals of Gaussian processes

1999 ◽  
Vol 31 (1) ◽  
pp. 158-177 ◽  
Author(s):  
Vladimir Piterbarg ◽  
Igor Rychlik
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Gaussian Processes ◽  
Sample Paths

In this paper a central limit theorem is proved for wave-functionals defined as the sums of wave amplitudes observed in sample paths of stationary continuously differentiable Gaussian processes. Examples illustrating this theory are given.

Asymptotic expansion for the quadratic variations of the solution to the heat equation with additive white noise

2020 ◽  
Vol 21 (02) ◽  
pp. 2150010
Author(s):  
Héctor Araya ◽  
Ciprian A. Tudor
Keyword(s):  
Asymptotic Expansion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Heat Equation ◽  
White Noise ◽  
Malliavin Calculus ◽  
Order Term ◽  
Sharp Estimates ◽  
Additive White Noise ◽  
Quadratic Variations

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are based on sharp estimates of the correlation structure of the solution, which may have their own interest.

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