Total variation bounds for Gaussian functionals

Let X={Xt:0≤t≤1} be a centered Gaussian process with continuous paths, and In=an2∫01tn−1(X12−Xt2)dt where the an are suitable constants. Fix β∈(0,1), cn>0 and c>0 and denote by Nc the centered Gaussian kernel with (random) variance cX12. Under a Holder condition on the covariance function of X, there is a constant k(β) such that ‖P(cnIn∈⋅)−E[Nc(⋅)]‖≤k(β)(ann1+α)β+|cn−c|cfor alln≥1,where ‖⋅‖ is total variation distance and α the Holder exponent of the covariance function. Moreover, if ann1+α→0 and cn→c, then cnIn converges ‖⋅‖-stably to Nc, in the sense that ‖PF(cnIn∈⋅)−EF[Nc(⋅)]‖→0for every measurable F with P(F)>0. In particular, such results apply to X= fractional Brownian motion. In that case, they strictly improve the existing results in Nourdin et al. (2016) and provide an essentially optimal rate of convergence.