Malliavin-Stein method for variance-gamma approximation on Wiener space

@article{Eichelsbacher2014MalliavinSteinMF,
  title={Malliavin-Stein method for variance-gamma approximation on Wiener space},
  author={P. Eichelsbacher and Christoph Thale},
  journal={Electronic Journal of Probability},
  year={2014},
  volume={20},
  pages={1-28}
}
We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of… Expand

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References

SHOWING 1-10 OF 29 REFERENCES
Stein’s method on Wiener chaos
We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general GaussianExpand
Gamma limits and U-statistics on the Poisson space
Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presentedExpand
Normal Approximations with Malliavin Calculus: From Stein's Method to Universality
Preface Introduction 1. Malliavin operators in the one-dimensional case 2. Malliavin operators and isonormal Gaussian processes 3. Stein's method for one-dimensional normal approximations 4.Expand
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos
We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. OurExpand
Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions
We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure ofExpand
Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach
We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables,Expand
Noncentral convergence of multiple integrals
Fix ν>0, denote by G(v/2) a Gamma random variable with parameter v/2, and let n≥2 be a fixed even integer. Consider a sequence (F_k) of square integrable random variables, belonging to the nth WienerExpand
Stein's Method and the Laplace Distribution
Using Stein's method techniques, we develop a framework which allows one to bound the error terms arising from approximation by the Laplace distribution and apply it to the study of random sums ofExpand
Variance-Gamma approximation via Stein's method
Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class ofExpand
Convergence in law in the second Wiener/Wigner chaos
Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset $L_0$ of $L$ satisfying that, for any $F_\infty$ in $L_0$, the convergence of onlyExpand
...
1
2
3
...