Density convergence in the Breuer-Major theorem for Gaussian stationary sequences

Yaozhong Hu (1) , David Nualart (1) , Samy Tindel (2, 3) , Fangjun Xu (4)

1 Department of Mathematics [Lawrence, Kansas]
2 BIGS - Biology, genetics and statistics
3 IECL - Institut Élie Cartan de Lorraine
4 School of Finance and Statistics

Consider a Gaussian stationary sequence with unit variance X. Assume that the central limit theorem holds for a weighted sum of elements of the form f(X_k), where f designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of this weighted sum towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of X.

Domaines: Probabilités [math.PR]

Yaozhong Hu, David Nualart, Samy Tindel, Fangjun Xu. Density convergence in the Breuer-Major theorem for Gaussian stationary sequences. Bernoulli, 2015, 21 (4), pp.2336-2350. ⟨10.3150/14-BEJ646⟩. ⟨hal-00959089⟩