# Optimal Gamma Approximation on Wiener Space

@article{Azmoodeh2019OptimalGA, title={Optimal Gamma Approximation on Wiener Space}, author={E. Azmoodeh and P. Eichelsbacher and L. Knichel}, journal={arXiv: Probability}, year={2019} }

In \cite{n-p-noncentral}, Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we provide an optimal rate of convergence in the $d_2$-distance in terms of the maximum of the third and fourth cumulants analogous to the result for normal approximation in \cite{n-p-optimal}. In order to achieve our goal, we introduce a novel operator theory approach to Stein's method… Expand

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#### References

SHOWING 1-10 OF 59 REFERENCES

On the Rate of Convergence to a Gamma Distribution on Wiener Space

- Mathematics
- 2018

In [NP09a], Nourdin and Peccati established a neat characterization of Gamma approximation on a fixed Wiener chaos in terms of convergence of only the third and fourth cumulants. In this paper, we… Expand

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

- Mathematics
- 2014

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… Expand

Convergence in law in the second Wiener/Wigner chaos

- Mathematics
- 2012

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 only… Expand

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

- Mathematics
- 2012

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by… Expand

Stein’s method on Wiener chaos

- Mathematics
- 2007

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 Gaussian… Expand

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

- Mathematics
- 2019

Abstract We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of l 2 ( N ∗ ) . We use this bound to estimate the Wasserstein-2 distance between… Expand

Noncentral convergence of multiple integrals

- Mathematics
- 2009

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 Wiener… Expand

Generalization of the Nualart-Peccati criterion

- Mathematics
- 2016

The celebrated Nualart–Peccati criterion [Ann. Probab. 33 (2005) 177–193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {Xn}n≥1 of multiple… Expand

Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach

- Mathematics
- 2014

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

Asymptotic independence of multiple Wiener-It\^o integrals and the resulting limit laws

- Mathematics
- 2011

We characterize the asymptotic independence between blocks consisting of multiple Wiener-It\^{o} integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem… Expand