Total variation bounds for Gaussian functionals

Let $X=\X_t:0le tle 1$ be a centered Gaussian process with continuous paths, and $I_n=raca_n2,int_0^1 t^n-1 (X_1^2-X_t^2),dt$ where the $a_n$ are suitable constants. Fix $etain (0,1)$, $c_n>0$ and $c>0$ and denote by $N_c$ the centered Gaussian kernel with (random) variance $cX_1^2$. Under an Holder condition on the covariance function of $X$, there is a constant $k(eta)$ such that egingather* ormPigl(sqrtc_n,I_nincdotigr)-Eigl[N_c(cdot)igr],le k(eta),Bigl(raca_nn^1+alphaBigr)^eta+racabsc_n-ccquad extfor all nge 1, endgather* where $ ormcdot$ is total variation distance and $alpha$ the Holder exponent of the covariance function. Moreover, if $raca_nn^1+alpha ightarrow 0$ and $c_n ightarrow c$, then $sqrtc_n,I_n$ converges $ ormcdot$-stably to $N_c$, in the sense that egingather* ormP_Figl(sqrtc_n,I_nincdotigr)-E_Figl[N_c(cdot)igr], ightarrow 0 endgather* for 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 citeNNP16 and provide an essentially optimal rate of convergence.