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From Reviews: 0MR4398348 62M10 62G05 62G20 62M15
Li, Linyuan (1-NH-MS) ; Zhang, Biao [Zhang, Biao2 ] (1-NH-MS)
Nonlinear wavelet-based estimation to spectral density for stationary
non-Gaussian linear processes. (English summary)Appl. Comput. Harmon. Anal. 60 (2022), 176–204.

P∞
Let {Xt }nt=1 be a realization of a stationary linear process Xt = j=0 ϕj Zt−j , where
the innovation {Zt } is an i.i.d. sub-Gaussian process with zero mean and unit variance.
In this paper, convergence rates of the mean integrated squared errors (MISEs) for two
wavelet-based spectral estimators are derived.
Let f (ω), ω ∈ [−π, π], be the spectral density of {Xt }. The first estimator for f (ω) isjj0 −1 2j −1
j1 2X−1noXγb(0) X X b
bjk 1 βbjk > δjk Ψjk (ω),βjk Ψjk (ω) +f (ω) = √ +β2π j=0 k=0
j=j0 k=0
PnPn
where γb(h) = (1/n) t=|h|+1 (Xt − X)(Xt−|h| − X) with X = (1/n) t=1 Xt being thesample autocovariance of {Xt }, {Ψjk (·)} is an orthonormal 2π-periodic wavelet basis,
bjk are empirical wavelet coefficients, and δjk is a thresholding constant. Notice thatthe β(j0 , j1 ) are set to diverge to infinity more slowly than the sample size n. The estimatorf (ω) is infeasible because the thresholding constant δjk depends on unknown quantities.
Then, a fully operational estimator fb(ω) with δjk replaced by its empirical counterpartis also proposed as the second estimator.
Convergence properties of similar wavelet estimators have been investigated in the literature [e.g., M. H. Neumann, J. Time Ser. Anal. 17 (1996), no. 6, 601–633; MR1424908;
P. Fryzlewicz, G. P. Nason and R. von Sachs, J. Time Ser. Anal. 29 (2008), no. 5, 868–880; MR2450900]. What distinguishes this article from others is the application of theHanson-Wright inequality [e.g., M. Rudelson and R. Vershynin, Electron. Commun.
Probab. 18 (2013), no. 82 (Theorem 1.1); MR3125258] to evaluate the convergence
bjk . This inequality is a general concentration one
rates of empirical wavelet coefficients β
bjk arefor quadratic forms of sub-Gaussian random variables, and the fact that the βquadratic functions of {Xt } establishes their convergence rates.
The main result of this article is(2s/(2s+1) )Z πln n2sup Ef̌ (ω) − f (ω) dω = O
ns (M )
f ∈Bp,q−πsfor f̌ ∈ {f , fb}, where Bp,q(M ) is a Besov ball of radius M > 0 with s > 0, p, q ∈[1, ∞). The rate (ln n/n)2s/(2s+1) in MISE is nearly optimal within a logarithmic term,compared with the MISE-optimal rate n−2s/(2s+1) of standard nonparametric regression
estimators.
Masayuki Hirukawa

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