Total variation bounds for Gaussian functionals

Luca Pratelli and Pietro Rigo

Stochastic Processes and their Applications, 2019, vol. 129, issue 7, 2231-2248

Abstract: 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.

Keywords: Fractional brownian motion; Gaussian process; Stable convergence; Total variation distance (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1016/j.spa.2018.07.005

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