### Citations

1269 |
Theory of Bessel functions,
- WATSON
- 1966
(Show Context)
Citation Context ... notation is required. Denote by Jν the Bessel function of the first kind with index ν > 0. It is well known that the roots of Jν are real, countable, nowhere dense, and also contain zero (cf. Watson =-=[39]-=-). Throughout the following, let · · ·< ω−1 <ω0 := 0<ω1 < · · · be the ordered (real) roots of the Bessel function J1−H (for convenience, we omit the dependence on the Hurst index H). Define the funct... |

942 |
Stochastic Processes,
- Doob
- 1953
(Show Context)
Citation Context ...nary increments and letKX(s, t) = Cov(Xs,Xt). We have the representation KX(s, t) = ∫∞ −∞F(Is)(λ)F(It)(λ)dµX(λ), where µX is a symmetric Borel measure on R satisfying ∫∞ −∞(1+λ) −2 dµX(λ)<∞ (cf. Doob =-=[8]-=-, Section XI.11). If MX = span{F(It) : t ∈ [0,1]} ⊂ L2(µX), then the RKHS HX associated to the Gaussian process X is given by HX = {F :∃F ∗ ∈MX , such that F (t) = 〈F ∗,F(It)〉L2(µX),∀t ∈ [0,1]} REGRES... |

756 | Regularization of Inverse Problems. - ENGL, HANKE, et al. - 2000 |

683 |
Special Functions and their Applications.
- Lebedev
- 1965
(Show Context)
Citation Context ...shows that {φk :k ∈ Z} is a basis of M and that the sampling formula h= ∑ k akh(2ωk)φk is equivalent to the corresponding result in Theorem 7.2 of [10]. We obtain the expression for a0, using Lebedev =-=[22]-=-, Formula (5.16.1), limλ→0(λ/2) −αJα(λ) = Γ(α+ 1) −1, for all α≥ 0. Furthermore, ak = a−k follows from ωk = −ω−k and the fact that a−1k is just a constant times the derivative of λ 7→ λHJ1−H(λ) evalua... |

651 |
Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance.
- Samorodnitsky, Taqqu
- 1994
(Show Context)
Citation Context ...d in the time domain. For more on that, cf. Pipiras and Taqqu [27]. Set cH := sin(πH)Γ(2H+1). Recall that K(s, t) = E[B H s B H t ], for s, t ∈ [0,1]. Then (cf. Yaglom [40] or Samorodnitsky and Taqqu =-=[32]-=-, equation (7.2.9)), K(s, t) = ∫ F(Is)(λ)F(It)(λ)µ(dλ)(12) with µ(dλ) = cH 2π |λ|1−2H dλ. Given this representation, it is straightforward to describe the corresponding RKHS as follows (cf. Grenander ... |

344 |
An Introduction to Non-Harmonic Fourier Series,
- Young
- 1980
(Show Context)
Citation Context ...al’s identity only holds up to constants in the sense of equivalence of norms. This norm equivalence is usually referred to as frame inequality or near-orthogonality. For more on the topic, cf. Young =-=[41]-=-, Section 1.8. The proof of Lemma 3 is delayed until Appendix B. It relies on a standard result for nonharmonic Fourier series in combination with some bounds on the zeros ωk. Using the previous lemma... |

251 | Nonlinear solution of linear inverse problems by WaveletVaguelette decomposition.
- DONOHO
- 1995
(Show Context)
Citation Context ... discrete regression under dependent errors by∫ · 0 f(u)du+n H−1BH· and sequence model representations were further studied in Johnstone and Silverman [20] and more detailed in Johnstone [19]. Donoho =-=[7]-=- investigates the wavelet-vaguelette decomposition for inverse problems. Since this is very close to the simultaneous orthogonalization presented in Section 3, the connection is discussed in more deta... |

241 |
Introduction to nonparametric estimation.
- Tsybakov
- 2009
(Show Context)
Citation Context ...straints ∫ 1 0 f (ℓ)(s)(s − s2)1/2−H ds = 0, ℓ = 1, . . . , β simplify to the periodic boundary conditions f (q)(0) = f (q)(1), q = 0, . . . , β − 1. In this case, Lemma 5 is well known; cf. Tsybakov =-=[36]-=-, Lemma A.3. In the important case β = 1, the constraint in SobH(β, C̃) is satisfied whenever f = f(1− ·). If we restrict further to these functions, the definition of SobH(1, C̃) does not depend on t... |

240 | Wavelet threshold estimators for data with correlated noise.
- Johnstone, Silverman
- 1995
(Show Context)
Citation Context ...rownian motion (fBM) with Hurst parameter H ∈ (0,1) as a Gaussian process (BHt )t≥0 with covariance function (s, t) 7→Cov(BHs ,BHt ) = 12(|t|2H + |s|2H−|t−s|2H). In Wang [38], Johnstone and Silverman =-=[20]-=- and Johnstone [19] it has been argued that Yt = ∫ t 0 f(u)du+ nH−1BHt , t ∈ [0,1],BH a fBM(2) is a natural candidate for a continuous version of (1) for H ≥ 1/2. By projecting (Yt)t∈[0,1] onto a suit... |

226 |
Abstract Inference.
- Grenander
- 1981
(Show Context)
Citation Context ..., equation (7.2.9)), K(s, t) = ∫ F(Is)(λ)F(It)(λ)µ(dλ)(12) with µ(dλ) = cH 2π |λ|1−2H dλ. Given this representation, it is straightforward to describe the corresponding RKHS as follows (cf. Grenander =-=[17]-=-, page 97): let M denote the closed linear span of {F(It) : t ∈ [0,1]} in the weighted L2-space L2(µ), then H= {F :∃F ∗ ∈M, such that F (t) = 〈F ∗,F(It)〉L2(µ),∀t ∈ [0,1]}, where 〈g,h〉L2(µ) := ∫ g(λ)h(... |

213 |
Correlation theory of stationary and related random functions. Vol. I. Basic results. Springer Series in Statistics.
- YAGLOM
- 1987
(Show Context)
Citation Context ...s could equally well be described in the time domain. For more on that, cf. Pipiras and Taqqu [27]. Set cH := sin(πH)Γ(2H+1). Recall that K(s, t) = E[B H s B H t ], for s, t ∈ [0,1]. Then (cf. Yaglom =-=[40]-=- or Samorodnitsky and Taqqu [32], equation (7.2.9)), K(s, t) = ∫ F(Is)(λ)F(It)(λ)µ(dλ)(12) with µ(dλ) = cH 2π |λ|1−2H dλ. Given this representation, it is straightforward to describe the corresponding... |

210 |
Asymptotic equivalence of nonparametric regression and white noise.
- Brown, Low
- 1996
(Show Context)
Citation Context ...ates of convergence for regression under long-range dependent noise were first considered by Hall and Hart [18] using kernel estimators. Inspired by the asymptotic equivalence result of Brown and Low =-=[2]-=-, Wang [38] makes the link between discrete regression under dependent noise and experiment E2,n(Θ), in which the path of the integral of f is observed plus a scaled fBM. Passing from the discrete to ... |

123 | M.:Asymptotic equivalence of density estimation and Gaussian white noise.
- Nussbaum
- 1996
(Show Context)
Citation Context ...s shows that finding an ONB of M (cf. Theorem 2) is highly nontrivial and it remains unclear in which sense M could admit a multiresolution decomposition. Besides that, Brown and Low [2] and Nussbaum =-=[25]-=- established nonparametric asymptotic equivalence as own research field. Since then, there has been considerable progress in this area. Asymptotic equivalence for regression models was further general... |

96 | Integrations questions related to fractional Brownian motion. Probab. Theory Rel.
- Pipiras, Taqqu
- 2000
(Show Context)
Citation Context ... rely on the spectral representation as it avoids some technical issues. In principle, however, all results could equally well be described in the time domain. For more on that, cf. Pipiras and Taqqu =-=[27]-=-. Set cH := sin(πH)Γ(2H+1). Recall that K(s, t) = E[B H s B H t ], for s, t ∈ [0,1]. Then (cf. Yaglom [40] or Samorodnitsky and Taqqu [32], equation (7.2.9)), K(s, t) = ∫ F(Is)(λ)F(It)(λ)µ(dλ)(12) wit... |

55 |
Wavelets, generalized white noise and fractional integration: the syn- thesis of fractional Brownian motion.
- Meyer, Sellan, et al.
- 1999
(Show Context)
Citation Context ... the strong assumption that the noise process is completely 24 J. SCHMIDT-HIEBER decorrelated by a wavelet decomposition. Multiscale representations that nearly whiten fBM are known (cf. Meyer et al. =-=[24]-=-, Section 7), but it is unclear whether fBM admits an exact wavelet decomposition. One possibility to extend the result to regression under fractional noise is to give up on orthogonality and to deal ... |

49 | Nonparametric statistical inverse problems.
- Cavalier
- 2008
(Show Context)
Citation Context ...Z, εk i.i.d.∼ N (0,1).(3) Here, θk(f) denote the Fourier coefficients and σk > 0 are weights. Models of type (3) have been extensively studied in statistical inverse problems literature (cf. Cavalier =-=[5]-=-). In this work, we investigate these approximations and its limitations under Le Cam distance (cf. Appendix E in the supplementary material [33] for a summary of the topic). The Le Cam distance allow... |

48 |
Integrals of Bessel Functions,
- LuKE
- 1962
(Show Context)
Citation Context ...2− 2H) = 24H−3HΓ(H +1/2)Γ(3− 2H) (1−H)Γ2(1−H)Γ(3/2−H) . This proves (31). Next, let us show that (φk)k is L 2(µ)-normalized, that is ‖φk‖L2(µ) = 1. This is immediately clear for k = 0 since (cf. Luke =-=[23]-=-, Section 13.2)∫ ∞ 0 |J1−H(λ)|2λ−1 dλ= 1/(2− 2H). To compute the normalization constant for k 6= 0, the last equality in the proof of [10], Theorem 7.2 gives ‖S1(2ωk, ·)‖2L2(µ) = σ−2(ωk), where for k ... |

44 |
Nonparametric regression with long-range dependence. Stochastic Process.
- HALL, P, et al.
- 1990
(Show Context)
Citation Context ...̂ − f‖2L2[0,1]]. E [ ∑ |k|≤Mn |θk − θ̂k|2 ] + ∑ |k|>Mn |θk|2. Choosing Mn = O(n −(1−H)/(β+1−H)), the rate becomes n−2β(1−H)/(β+1−H) in accordance with Wang [38] and, for β = 2, H ≥ 1/2, Hall and Hart =-=[18]-=-. Surprisingly, faster rates can be obtained if H is small. The ill-posedness is overcompensated by the gain in the noise level. REGRESSION UNDER FRACTIONAL NOISE 17 4.3. Necessary conditions. In this... |

44 | Wavelet shrinkage for correlated data and inverse problems : adaptivity results. Statistica Sinica 9
- Johnstone
- 1999
(Show Context)
Citation Context ...) with Hurst parameter H ∈ (0,1) as a Gaussian process (BHt )t≥0 with covariance function (s, t) 7→Cov(BHs ,BHt ) = 12(|t|2H + |s|2H−|t−s|2H). In Wang [38], Johnstone and Silverman [20] and Johnstone =-=[19]-=- it has been argued that Yt = ∫ t 0 f(u)du+ nH−1BHt , t ∈ [0,1],BH a fBM(2) is a natural candidate for a continuous version of (1) for H ≥ 1/2. By projecting (Yt)t∈[0,1] onto a suitable basis, one is ... |

43 | Function estimation via wavelet shrinkage for long–memory data
- Wang
- 1996
(Show Context)
Citation Context ...e definition of a fractional Brownian motion (fBM) with Hurst parameter H ∈ (0,1) as a Gaussian process (BHt )t≥0 with covariance function (s, t) 7→Cov(BHs ,BHt ) = 12(|t|2H + |s|2H−|t−s|2H). In Wang =-=[38]-=-, Johnstone and Silverman [20] and Johnstone [19] it has been argued that Yt = ∫ t 0 f(u)du+ nH−1BHt , t ∈ [0,1],BH a fBM(2) is a natural candidate for a continuous version of (1) for H ≥ 1/2. By proj... |

39 | Asymptotic equivalence theory for nonparametric regression with random design. Dedicated to the memory of Lucien Le Cam.
- Brown, Cai, et al.
- 2002
(Show Context)
Citation Context ...≥0 with covariance function (s, t) 7→Cov(BHs ,BHt ) = 12(|t|2H + |s|2H−|t−s|2H). In Wang [38], Johnstone and Silverman [20] and Johnstone [19] it has been argued that Yt = ∫ t 0 f(u)du+ nH−1BHt , t ∈ =-=[0,1]-=-,BH a fBM(2) is a natural candidate for a continuous version of (1) for H ≥ 1/2. By projecting (Yt)t∈[0,1] onto a suitable basis, one is further interested in an equivalent sequence space representati... |

37 |
Reproducing Kernel Hilbert Spaces of Gaussian priors,
- Vaart, Zanten
- 2008
(Show Context)
Citation Context ... process (Xt)t∈T , there is a Girsanov formula with the associated RKHS H playing the role of the Cameron–Martin space. Lemma 1 (Example 2.2, Theorem 2.1 and Lemma 3.2 in van der Vaart and van Zanten =-=[37]-=-). Let (Xt)t∈T be a Gaussian process with continuous sample paths on a compact metric space T and H the associated RKHS. REGRESSION UNDER FRACTIONAL NOISE 5 Denote by Pf the probability measure of t 7... |

33 |
Hilbert spaces of entire functions,
- Branges
- 1968
(Show Context)
Citation Context ...nonharmonic Fourier series. This approach is based on the one-to-one correspondence between mass distributions of vibrating strings and certain measures which was developed in Krĕın [21], de Branges =-=[6]-=-, Dym and McKean [9] and Dzhaparidze et al. [11]. Let us sketch the construction. Recall that X is a continuous, centered Gaussian process with stationary increments and letKX(s, t) = Cov(Xs,Xt). We h... |

25 | H.H.: Asymptotic equivalence of spectral density estimation and Gaussian white noise.
- Golubev, Nussbaum, et al.
- 2010
(Show Context)
Citation Context ...nd to deal with nearly orthogonal wavelet decompositions instead. This, however, causes various new issues that are very delicate and technical. One might view the methods developed in Golubev et al. =-=[14]-=- and Reiß [29] as first steps toward such a theory, as both deal with similar problems, however in very specific settings. APPENDIX A: PROOFS FOR SECTION 2 Proof of Lemma 2. Write νf = dPf/dP0 and νg ... |

22 | Asymptotic equivalence for nonparametric regression
- Grama, Nussbaum
- 2002
(Show Context)
Citation Context ...there has been considerable progress in this area. Asymptotic equivalence for regression models was further generalized to random design in Brown et al. [1], non-Gaussian errors in Grama and Nussbaum =-=[16]-=- and higher-dimensional settings in Carter [3] and Reiß [28]. Rohde [30] considers periodic Sobolev classes, improving on condition (5) in this case. Carter [4] establishes asymptotic equivalence for ... |

20 | Asymptotic equivalence for nonparametric regression with multivariate and random design
- Reiß
(Show Context)
Citation Context ...c equivalence for regression models was further generalized to random design in Brown et al. [1], non-Gaussian errors in Grama and Nussbaum [16] and higher-dimensional settings in Carter [3] and Reiß =-=[28]-=-. Rohde [30] considers periodic Sobolev classes, improving on condition (5) in this case. Carter [4] establishes asymptotic equivalence for regression under dependent errors. The result, however, is d... |

19 | Asymptotic equivalence for inference on the volatility from noisy observations
- Reiß
(Show Context)
Citation Context ...ng in the equivalence E4,n(Θ) = E5,n(Θ) in the proof of Theorem 1. The use of the interpolation function (30) for asymptotic equivalence appears implicitly already in the proof of Theorem 2.2 in Reiß =-=[29]-=-. Approximation of discrete regression under dependent errors by∫ · 0 f(u)du+n H−1BH· and sequence model representations were further studied in Johnstone and Silverman [20] and more detailed in Johns... |

18 |
On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise
- Rohde
(Show Context)
Citation Context ...e for regression models was further generalized to random design in Brown et al. [1], non-Gaussian errors in Grama and Nussbaum [16] and higher-dimensional settings in Carter [3] and Reiß [28]. Rohde =-=[30]-=- considers periodic Sobolev classes, improving on condition (5) in this case. Carter [4] establishes asymptotic equivalence for regression under dependent errors. The result, however, is derived under... |

14 | Krein’s spectral theory and the PaleyWiener expansion for fractional Brownian motion.
- Dzhaparidze, Zanten
- 2005
(Show Context)
Citation Context ... orthogonality property of (gk)k. This part relies essentially REGRESSION UNDER FRACTIONAL NOISE 11 on the explicit orthogonal decomposition of the underlying RKHS H due to Dzhaparidze and van Zanten =-=[10]-=- (cf. also Appendix B). Theorem 2 (Dzhaparidze and van Zanten [10], Theorem 7.2). Recall that · · ·< ω−1 < ω0 := 0< ω1 < · · · are the ordered zeros of the Bessel function J1−H . For k ∈ Z, define φk(... |

11 | Stochastic volatility and fractional Brownian motion
- Gloter, Hoffmann
- 2004
(Show Context)
Citation Context ...) and hence, E0[(Uf −Ug)νf ] = ( 1− 〈g, f〉H‖f‖2 H ) E0[(Uf)νf ] = ‖f‖2H − 〈g, f〉H. Plugging this into the formula for dKL(Pf , Pg), the result follows. For a similar result, cf. Gloter and Hoffmann =-=[13]-=-, Lemma 8. A.1. Completion of Theorem 1. The remaining parts for the proof of Theorem 1 follows from Propositions A.1 and A.2 below. If X= (X1, . . . ,Xn) is a stationary process with spectral density... |

9 | A continuous Gaussian approximation to a nonparametric regression in two dimensions
- Carter
(Show Context)
Citation Context ...ea. Asymptotic equivalence for regression models was further generalized to random design in Brown et al. [1], non-Gaussian errors in Grama and Nussbaum [16] and higher-dimensional settings in Carter =-=[3]-=- and Reiß [28]. Rohde [30] considers periodic Sobolev classes, improving on condition (5) in this case. Carter [4] establishes asymptotic equivalence for regression under dependent errors. The result,... |

8 |
Representations of fractional Brownian motion using vibrating strings. Stochastic Process.
- Dzhaparidze, Zanten, et al.
- 2005
(Show Context)
Citation Context ...sed on the one-to-one correspondence between mass distributions of vibrating strings and certain measures which was developed in Krĕın [21], de Branges [6], Dym and McKean [9] and Dzhaparidze et al. =-=[11]-=-. Let us sketch the construction. Recall that X is a continuous, centered Gaussian process with stationary increments and letKX(s, t) = Cov(Xs,Xt). We have the representation KX(s, t) = ∫∞ −∞F(Is)(λ)F... |

6 | Asymptotically sufficient statistics in nonparametric regression experiments with correlated noise
- Carter
(Show Context)
Citation Context ...-Gaussian errors in Grama and Nussbaum [16] and higher-dimensional settings in Carter [3] and Reiß [28]. Rohde [30] considers periodic Sobolev classes, improving on condition (5) in this case. Carter =-=[4]-=- establishes asymptotic equivalence for regression under dependent errors. The result, however, is derived under the strong assumption that the noise process is completely 24 J. SCHMIDT-HIEBER decorre... |

6 |
On some cases of effective determination of the density of an inhomogeneous cord from its spectral function. Doklady Akad
- Krĕın
- 1953
(Show Context)
Citation Context ...leading again to nonharmonic Fourier series. This approach is based on the one-to-one correspondence between mass distributions of vibrating strings and certain measures which was developed in Krĕın =-=[21]-=-, de Branges [6], Dym and McKean [9] and Dzhaparidze et al. [11]. Let us sketch the construction. Recall that X is a continuous, centered Gaussian process with stationary increments and letKX(s, t) = ... |

6 |
Representation Formulae for the Fractional Brownian Motion. Séminaire de
- Picard
(Show Context)
Citation Context ...hogonalization of the spaces is the crucial tool to verify the second approximation condition of Theorem 1 on Sobolev balls. A slightly simpler characterization of the RKHS H can be given (cf. Picard =-=[26]-=-, Theorem 6.12), but it remains unclear whether it can lead to a comparable simultaneous diagonalization. For more, see the discussion in Section 5. 4. Asymptotic equivalence: Main results. 4.1. Asymp... |

3 |
Self-similar probability distributions. Theory Probab
- unknown authors
- 1976
(Show Context)
Citation Context ...that dKL(Pv, Pw)≤ c(n1−2H ∨ 1)(v−w)t(v−w). Proof. Denote the spectral density of fractional Gaussian noise with Hurst index H by fH . fGN is stationary and from the explicit formula of fH (cf. Sinăı =-=[34]-=-), we find that fH(λ)∼ cHλ1−2H for λ ↓ 0, and that fH is bounded away from zero elsewhere. Using Lemma A.1, λn(Cov(Yn))≥ ( 1− 1 π ) inf λ∈[1/n,π] fH(λ)& n 2H−1 ∧ 1. From the general formula for the Ku... |

1 |
On optimal regularization methods for fractional differentiation
- Tautenhahn, Gorenflo
- 1999
(Show Context)
Citation Context ... element g ∈ S(Θ) such that (15) holds. Source conditions are a central topic in the theory of deterministic inverse problems (cf. Engl et al. [12] for a general treatment and Tautenhahn and Gorenflo =-=[35]-=- for source conditions for inverse problems involving fractional derivatives). A similar construction is employed in fractional calculus, by defining the domain of a fractional derivative as the image... |

1 |
Representations of Gaussian processes with stationary increments
- Zareba
- 2007
(Show Context)
Citation Context ...e sum in L2(µX). Furthermore, for any k ∈ Z, νk = −ν−k, S(νk, νk) = S(ν−k, ν−k) and |νk|= 2|k|π(1 + o(1)), for |k| →∞. Proof. This follows largely from Dzhaparidze et al. [11], Theorem 3.5 and Zareba =-=[42]-=-, Lemma 2.8.7. It remains to show that νk = −ν−k and S(νk, νk) = S(ν−k, ν−k). Notice that by the reproducing property S(νk, νk) = ‖S(νk, λ)‖2L2(µX ) ≥ 0. To see that νk =−ν−k, observe that from [11], ... |