Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes We study the asymptotic behavior of the $$\nu $$ ν -symmetric Riemann sums for functionals of a self-similar centered Gaussian process X with increment exponent $$0<\alpha <1$$ 0 < α < 1 . We prove that, under mild assumptions on the covariance of X, the law of the weak $$\nu $$ ν -symmetric Riemann sums converge in the Skorohod topology when $$\alpha =(2\ell +1)^{-1}$$ α = ( 2 ℓ + 1 ) - 1 , where $$\ell $$ ℓ denotes the largest positive integer satisfying $$\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{-1}$$ ∫ 0 1 x 2 j ν ( d x ) = ( 2 j + 1 ) - 1 for all $$j=0,\dots , \ell -1$$ j = 0 , ⋯ , ℓ - 1 . In the case $$\alpha >(2\ell +1)^{-1}$$ α > ( 2 ℓ + 1 ) - 1 , we prove that the convergence holds in probability. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

Symmetric Stochastic Integrals with Respect to a Class of Self-similar Gaussian Processes

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Abstract

We study the asymptotic behavior of the $$\nu $$ ν -symmetric Riemann sums for functionals of a self-similar centered Gaussian process X with increment exponent $$0<\alpha <1$$ 0 < α < 1 . We prove that, under mild assumptions on the covariance of X, the law of the weak $$\nu $$ ν -symmetric Riemann sums converge in the Skorohod topology when $$\alpha =(2\ell +1)^{-1}$$ α = ( 2 ℓ + 1 ) - 1 , where $$\ell $$ ℓ denotes the largest positive integer satisfying $$\int _{0}^{1}x^{2j}\nu (\mathrm{d}x)=(2j+1)^{-1}$$ ∫ 0 1 x 2 j ν ( d x ) = ( 2 j + 1 ) - 1 for all $$j=0,\dots , \ell -1$$ j = 0 , ⋯ , ℓ - 1 . In the case $$\alpha >(2\ell +1)^{-1}$$ α > ( 2 ℓ + 1 ) - 1 , we prove that the convergence holds in probability.

Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: May 30, 2018

References

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