Tail Estimation of the Spectral Density under Fixed-Domain Asymptotics

Transcription

1 Tail Estiation of the Spectral Density under Fixed-Doain Asyptotics Wei-Ying Wu, Chae Young Li and Yiin Xiao Wei-Ying Wu, Departent of Statistics & Probability Michigan State University, East Lansing, MI e-ail: Chae Young Li, Departent of Statistics & Probability Michigan State University, East Lansing, MI e-ail: Yiin Xiao, Departent of Statistics & Probability, Michigan State University, East Lansing, MI e-ail: Abstract: Consider a stationary Gaussian rando field on R d with the spectral density fλ that satisfies fλ c Hλ θ as λ for soe nonsingular atrix H. The paraeters c and θ control the tail behavior of the spectral density. c is related to a icroergodic paraeter and θ is related to a fractal index. For data observed on a grid, we propose estiators of c and θ by iniizing an objective function, which can be viewed as a weighted local Whittle likelihood and study their asyptotic properties under fixed-doain asyptotics. AMS 2000 subject classifications: Priary 62M30; secondary 62H11. Keywords and phrases: fixed-doain asyptotics, fractal index, fractal diension, Gaussian rando fields, infill asyptotics, icroergodic, Whittle likelihood. 1. Introduction With recent advances in technology, we are facing enorous aount of data sets. When data sets are observed on a regular grid, spectral analysis is popular Supported by MSU grant 08-IRGP-1532 Supported in part by NSF grant DMS

2 Wu, Li & Xiao/Tail Estiation of the Spectral Density 2 due to fast coputation using the Fast Fourier Transfor. For exaple, paraeters of the spectral density of a stationary lattice process can be estiated using a Whittle likelihood [Whittle 1954], which is ore efficient in ters of coputation copared to the axiu likelihood ethod on a spatial doain. Spatial data on a grid often can be regarded as a realization of a rando field on a lattice. That is, for a rando field, Zs on R d, data are observed at ϕj for J d {1,, j}, where ϕ is a grid length. When ϕ is fixed, asyptotic properties of paraeter estiates on a spectral doain have been studied by any authors [see, e.g., Whittle 1954, Guyon 1982, 1995, Boissy et al and Guo et al. 2009]. For exaple, Guyon 1982 studied asyptotic properties of estiators using a Whittle likelihood or its variants when a paraetric odel is assued for the spectral density of a stationary process on a lattice. Guo et al studied asyptotic properties of estiators of long-range dependence paraeters for anisotropic spatial linear process using a local Whittle likelihood ethod in which a paraetric for near zero frequency is only assued. This is an extension of Robinson 1995 for tie series. For spatial data, however, it is often natural to assue that the data are observed on a bounded doain of interest. More observations on the bounded doain iplies that the distance between observations, ϕ, is decreasing as the nuber of observations increases. This sapling schee requires a different asyptotic fraework, called fixed-doain asyptotics [Stein 1999] or infill asyptotics [Cressie 1993]. The classical asyptotic fraework when the sapling distance is fixed i.e. ϕ is fixed is called increasing-doain asyptotics to differentiate fro fixed-doain asyptotics. It has been shown that the asyptotic results under fixed-doain asyptotics can be different fro the results under increasing-doain asyptotics [see, e.g., Mardia and Marshall 1984, Ying 1991, 1993, Zhang 2004]. For exaple, Zhang 2004 showed not all paraeters in the Matérn covariance function of a stationary Gaussian rando field on R d are consistently estiable when d is saller than or equal to 3, while a reparaeterized quantity of variance and scale paraeters can be estiated consistently by the axiu likelihood ethod. On the other hand, the axiu likelihood estiators MLEs of variance and scale paraeters for a stationary Gaussian process under increasing-doain asyptotics are consistent and asyptotically noral [Mardia and Marshall 1984]. Although not all paraeters can be estiated consistently under fixeddoain asyptotics, a icroergodic paraeter can be estiated consistently [see, e.g., Ying 1991, 1993, Zhang 2004, Zhang and Zieran 2005, Du et al. 2009, Anderes 2010]. The icroergodicity of functions of paraeters

3 Wu, Li & Xiao/Tail Estiation of the Spectral Density 3 deterines the equivalence of probability easures and a icroergodic paraeter is the quantity that affects asyptotic ean squared prediction error under fixed-doain asyptotics. [Stein 1990a, 1990b, 1999]. Although there have been ore asyptotic results available recently under fixed-doain asyptotics, it is still a very few in contrast with vast literature on increasing-doain asyptotics. Also, ost results are for specific odels of covariance functions. For exaple, Ying 1991, 1993 and Chen et al studied asyptotic properties of estiators for a icroergodic paraeter in the exponential covariance function, while Zhang 2004, Loh 2005, Kaufan et al. 2008, Du et al and Anderes 2010 investigated asyptotic properties of estiators for the Matérn covariance function. Moreover, these asyptotic results are established in the spatial doain. Asyptotic work in the spectral doain are even less under fixed-doain asyptotics. Stein 1995 studied asyptotic properties of a spatial periodogra of a filtered version of a stationary Gaussian rando field. Li and Stein 2008 extended results of Stein 1995 and showed asyptotic norality of a soothed spatial cross-periodogra under fixed-doain asyptotics. Regarding the paraeter estiation in the spectral doain under fixed-doain asyptotics, Chan et al proposed a periodogra-based estiator of the fractal diension of a stationary Gaussian rando field when d = 1. In this paper, we propose estiators of paraeters that control the tail behavior of the spectral density for a stationary Gaussian rando field when the data are observed on a grid within a bounded doain and study their asyptotic properties under fixed-doain asyptotics. Let fλ be the spectral density of a stationary Gaussian rando field, Zs on R d and we assue that f λ c Hλ θ as λ, 1.1 where is a usual Euclidean nor, H is a nonsingular atrix and θ > d to ensure integrability of f. That is, we assue a power law for the tail behavior of the spectral density and do not assue any specific paraetric for of the spectral density. Also, this assuption allows a wide range of anisotropic spectral densities by introducing H. The proposed estiators are obtained by iniizing an objective function that can be viewed as a weighted local Whittle likelihood, in which Fourier frequencies near a pre-specified non-zero frequency are considered. This approach is siilar to the local Whittle likelihood ethod introduced by Robinson 1995 for estiating a long-range dependence paraeter in tie series analysis. For a stationary lattice process, Robinson 1995 proposed to estiate a long-range dependence paraeter by iniizing a Whittle

4 Wu, Li & Xiao/Tail Estiation of the Spectral Density 4 likelihood over Fourier frequencies near zero since the long-range dependence paraeter controls the behavior of the spectral density near zero. Meanwhile, we are interested in estiating paraeters that govern the spectral density of a rando field when the frequency is very large so that we need to focus on Fourier frequencies that are away fro zero. We establish consistency and asyptotic norality of an estiator of c and an estiator of θ, respectively, when the other paraeter is known. The paraeter c is related to a icroergodic paraeter. For exaple, consider the Matérn spectral density given as fλ = σ 2 α 2ν. 1.2 π d/2 α 2 + λ 2 ν+d/2 The Matérn spectral density has three paraeters, σ 2, α, ν, where σ 2 is the variance paraeter, α is the scale paraeter and ν is the soothness paraeter. Since the Matérn spectral density satisfies fλ σ2 α 2ν π d 2 λ 2ν+d as λ, we have c σ 2 α 2ν /π d/2 and θ 2ν+d, and σ 2 α 2ν is a icroergodic paraeter. Thus, estiating σ 2 α 2ν when ν is known is equivalent to estiate c when θ is known. There are several references that investigate estiation of σ 2 α 2ν in the spatial doain. Zhang 2004 showed that σ 2 and α can be estiated only in the for of σ 2 α 2ν under fixed-doain asyptotics when ν is known and d 3. Du et al investigated asyptotic properties of the MLE and a tapered MLE of σ 2 α 2ν when ν is known, α is fixed and d = 1 for a stationary Gaussian rando field. Anderes 2010 proposed an increent-based estiator of σ 2 α 2ν for a geoetric anisotropic Matérn covariance function and showed that α can be estiated separately when d > 4. The paraeter θ is related to the fractal index or fractal diension. For exaple, for a stationary Gaussian rando field, Zs, s R d, suppose that its covariance function Ct satisfies Ct C0 k t α as t for soe k and 0 < α 2. In this case, α is the fractal index that governs the roughness of saple paths of a rando field and the fractal diension D becoes D = d + 1 α/2. This follows fro Adler 1981, Chapter 8 or Theore 5.1 in Xue and Xiao 2010 where ore general results are proven for anisotropic Gaussian rando fields. When α = 2, it is possible that the saple

5 Wu, Li & Xiao/Tail Estiation of the Spectral Density 5 function ay be differentiable. This can be deterined by the soothness of Ct or in ters of the spectral easure of Zs see, e.g., Adler and Taylor 2007 or Xue and Xiao 2010 for further inforation. By an Abelian type theore, 1.3 holds if the corresponding spectral density satisfies fλ k λ α+d as λ so that θ α + d in our settings. There is a nuber of references that construct estiators based on fractal properties of processes. For exaple, Constantine and Hall 1994 estiated effective fractal diension using variogra for a non-gaussian stationary process on R. Chan and Wood 2004 introduced an increent-based estiator for the fractal diension of a stationary Gaussian rando field on R d with d = 1 or 2. Copared to these works in the spatial doain, the work by Chan et al to estiate the fractal diension is done in the spectral doain. In Section 2, we explain our settings and assuptions, and in Section 3 introduce our estiators and state the ain theores for the asyptotic properties of the proposed estiators. Section 4 discusses soe issues related to our approach and possible extension of the current work. All proofs are given in Appendices. 2. Preliinaries In this paper, we consider a stationary Gaussian rando field, Zs on R d with the spectral density fλ that satisfies 1.1. Define a lattice process Y ϕ J by Y ϕ J ZϕJ, where J Z d, the set of d-diensional integer-valued vectors. The corresponding spectral density of Y ϕ J is f ϕ λ = ϕ λ + 2πQ d f, ϕ Q Z d for λ π, π ] d. fϕ λ has a peak near the origin which is getting higher as ϕ 0. This causes a proble to estiate the spectral density using the periodogra [Stein 1995]. To alleviate the proble, we consider a discrete Laplacian operator to difference the data, which is proposed by Stein The Laplacian operator is defined by ϕ Zs = d {Zs + ϕ e j 2Zs + Zs ϕ e j }, where e j is the unit vector whose jth entry is 1. Depending on the behavior of the spectral density at high frequencies, we need to apply the Laplacian operator

6 Wu, Li & Xiao/Tail Estiation of the Spectral Density 6 iteratively to control the peak near the origin. Define Y τ ϕ J ϕ τ Zs as the lattice process obtained by applying the Laplacian operator τ ties. Then its corresponding spectral density becoes f ϕ τ λ = d 4 sin 2 λ j /2 The liit of f τ ϕ λ as ϕ 0 after scaling by ϕd θ is d ϕ d θ τ f ϕ λ c 4 sin 2 λ j /2 2τ 2τ f ϕ λ. 2.1 Q Z d Hλ + 2πQ θ for λ 0. Define for λ π, π] d, d g c,θ λ = c 4 sin 2 λj 2 2τ Q Z d Hλ + 2πQ θ I λ 0, 2.2 where I A = 1 if A is true and zero, otherwise. The liit function, g c,θ λ is integrable by choosing τ such that 4τ θ > d. When d = 1, siple differencing is preferred as discussed in Stein Then, 4τ will be replaced with 2τ in our results in Section 3. Now suppose that Zs is observed on the lattice ϕj. More specifically, we assue that we observe Yϕ τ J at J T = {1,..., } d after differencing Zs using the Laplacian operator τ ties. We further assue that ϕ = 1 so that the nuber of observations increases within a fixed observation doain. The spectral density of Yϕ τ J can be estiated by a periodogra which is defined using a discrete Fourier transfor of the data. That is, periodogra is defined by I τ λ = 2π d Dλ 2, where Dλ, the discrete Fourier transfor of the data, is given as Dλ = J T Y τ ϕ J exp{ i λ T J}. We consider the periodogra only at Fourier frequencies, 2π 1 J for J T { 1/2,, /2 } d, where x is the largest integer not greater

7 Wu, Li & Xiao/Tail Estiation of the Spectral Density 7 than x. A soothed periodogra at Fourier frequencies is defined by 2πJ Î τ = 2πJ + K W h KI τ, K T with weights W h K given by W h K = Λ h 2πK/ L T Λ h 2πL/, 2.3 where Λ h s = 1 s h Λ I { s h} h for a syetric continuous function Λ on R d that satisfies Λs 0 and Λ0 > 0 and I A is the indicator function of the set A. The nor is defined by s = ax{ s 1, s 2,..., s d }. For positive functions a and b, aλ bλ for λ A eans that there exist constants C 1 and C 2 such that 0 < C 1 aλ/bλ C 2 < for all possible λ A. For asyptotic results in this paper, we consider the following assuption on the spectral density f λ. Assuption 1. For a stationary Gaussian rando field Zs on R d, the spectral density fλ satisfies A f λ c Hλ θ as λ, for soe c > 0, θ > d and a nonsingular atrix H, B f λ is twice differentiable and there exists a positive constant C such that for λ > C, fλ 1 + λ θ, fλ λ j 1 + λ θ+1 and fλ λ j λ k 1 + λ θ+2 for j, k = 1,..., d. 3. Main Results Asyptotic properties of a spatial periodogra and a soothed spatial periodogra under fixed-doain asyptotics were investigated by Stein 1995 and Li and Stein They assue that the spectral density f is twice differentiable and satisfies 2.4 for all λ R d, which iplies that the spectral

8 Wu, Li & Xiao/Tail Estiation of the Spectral Density 8 density fλ behaves like 1 + λ θ for all λ. This is uch stronger condition than Assuption 1. This stronger condition allows to find asyptotic bounds for the expectation, variance and covariance of a spatial periodogra at Fourier frequency 2πJ/ by and J for J = 0. Consistency and asyptotic norality of a soothed spatial periodogra at Fourier frequency 2πJ/, however, are only available when li 2πJ/ = µ 0, that is, J should not be close to zero asyptotically. Since we ake use of asyptotic properties of a soothed spatial periodogra at such Fourier frequency, we extend those results in Stein 1995 and Li and Stein 2008 under Assuption 1. We focus on only a soothed spatial periodogra in the following theore, but results for a soothed spatial cross-periodogra can be shown siilarly. Throughout p d the paper, let denote the convergence in probability and denote the convergence in distribution. Theore 3.1. Suppose that the spectral density f of a stationary Gaussian rando field Zs on R d satisfies Assuption 1. Also suppose that 4τ > θ 1 and h = C γ for soe C > 0 where γ satisfies ax{d 2/d, 0} < γ < 1. Further, assue that li 2πJ/ = µ and 0 < µ < π. Let η = d1 γ/2. Then, we have and Î τ 2πJ/ f τ ϕ 2πJ/ η d θ Î τ 2πJ/ g c,θ µ p d N 0, Λ 2 Λ 2 1 d 2π g 2 C c,θµ, 3.2 where Λ r = [ 1,1] d Λ r sds. Reark 3.1. g c,θ is integrable under 4τ > θ d which is satisfied by the condition 4τ > θ 1. 4τ > θ 1 is necessary to show E Îτ 2πJ/ / f ϕ τ 2πJ/ 1 and the condition ax{d 2/d, 0} < γ < 1 is needed to show V ar Îτ 2πJ/ / f ϕ τ 2πJ/ 0 so that 3.1 can be shown. To estiate paraeters, c and θ, we consider the following objective function to be iniized. Lc, θ = { W h K log d θ g c,θ 2πJ + K/ 3.3 K T + 1 d θ I τ 2πJ + K/ g c,θ 2πJ + K/ },

9 Wu, Li & Xiao/Tail Estiation of the Spectral Density 9 where W h K is given in 2.3. In Lc, θ, 2πJ/ is any given Fourier frequency that satisfies J so that 2πJ/ is away fro 0. Lc, θ can be viewed as a weighted local Whittle likelihood. If Λ is a nonzero constant function, W h K 1/ K for K K, where K = {K T : 2πK/ h} and K is the nuber of eleents in the set K. Then, Lc, θ is the for of a local Whittle likelihood for the lattice data {Y τ δ J, J T } in which the true spectral density is replaced with d θ g c,θ. Note that g c,θ λ is the liit of the spectral density of Y τ δ J after scaling by d θ for non-zero λ when ϕ = 1. The suation in Lc, θ is over the Fourier frequencies near 2πJ/ by letting h 0 as. While a local Whittle likelihood ethod to estiate a long-range dependence paraeter for tie series considers Fourier frequencies near zero, we consider Fourier frequencies near a pre-specified nonzero frequency. For exaple, by choosing J such that 2πJ/ = π/2 1 d, where 1 d is the d-diensional vector of ones, Lc, θ considers frequencies only near π/21 d. For the estiation of c, we iniize Lc, θ with a known θ. Thus, the proposed estiator of c when θ is known as θ 0 is given by ĉ = arg in c C Lc, θ 0, where C is the paraeter space of c. ĉ has the explicit expression obtained by Lc, θ 0 / c = 0 : ĉ = K T W h K 1 d θ 0 I τ 2πJ + K/ g 0 2πJ + K/, 3.4 where g 0 g 1,θ0. The following theore establishes the consistency and asyptotic norality of the estiator ĉ. Theore 3.2. Suppose that the spectral density f of a stationary Gaussian rando field Zs on R d satisfies Assuption 1. Also suppose that 4τ > θ 0 1 for a known θ 0 and h = C γ for soe C > 0 where γ satisfies d/d + 2 < γ < 1. Further, assue that J satisfies 2πJ/ = π/2 1 d and the true paraeter c is in the interior of the paraeter space C which is a closed interval. Let η = d1 γ/2. Then, for ĉ given in 3.4, we have ĉ p c, 3.5 and η ĉ c d N 0, c 2 Λ 2 Λ 2 1 d 2π, 3.6 C where Λ r = [ 1,1] d Λ r sds.

10 Wu, Li & Xiao/Tail Estiation of the Spectral Density 10 Reark 3.2. We can prove Theore 3.2 for J such that li 2πJ/ = µ and 0 < µ < π instead of the specific choice of 2πJ/ = π/21 d, which we have chosen for siplicity of the proof. Reark 3.3. Without uch difficulty, Theore 3.2 can be also proved when we replace θ 0 with a consistent estiator ˆθ as long as the estiator ˆθ satisfies ˆθ θ 0 = o p log 1 p which iplies ˆθ θ 0 1. When we choose Λ as a constant function and C = 1/2π 2, we have η ĉ c d N 0, 2 d c 2 π d. For the Matérn spectral density given in 1.2 with d = 1, Du et al showed that for any fixed α 1 with known ν, the MLE of σ 2 satisfies n 1/2 ˆσ 2 α 2ν 1 σ 2 0α 2ν 0 d N 0, 2σ 2 0α 2ν 0 2, 3.7 where n is the saple size, and σ0 2 and α 0 are true paraeters. Note that is the saple size of Yϕ τ which is the τ ties differenced lattice process of Zs so that = n 2τ for the siple differencing and = n 4τ for the Laplace differencing. Since π 1/2 c = σ 2 α 2ν for d = 1, we have the sae asyptotic variance as in 3.7. However, our approach has a slower convergence rate since η < 1/3 when d = 1 as we used partial inforation. This is also the case for a local Whittle likelihood ethod in Robinson To estiate θ, we assue that c is known as c 0. The proposed estiator of θ is then given by ˆθ = arg in θ Θ Lc 0, θ, 3.8 where Θ is the paraeter space of θ. The consistency and the convergence rate of the proposed estiator ˆθ are given in the following Theore. Theore 3.3. Suppose that the spectral density f of a stationary Gaussian rando field Zs on R d satisfies Assuption 1. Also suppose that 4τ > θ 1 and h = C γ for soe C > 0 where γ satisfies d/d + 2 < γ < 1. Further, assue that J satisfies 2πJ/ = π/2 1 d and the true paraeter θ is in the interior of the paraeter space Θ which is a closed interval. Then, for ˆθ given in 3.8, we have ˆθ p θ. 3.9 In addition, ˆθ θ = o p log

11 Wu, Li & Xiao/Tail Estiation of the Spectral Density 11 Reark 3.4. The consistency of ˆθ is not enough to prove the asyptotic distribution of ˆθ since we have θ in the exponent of in the expression of Lc, θ. For the proof of the asyptotic distribution, we need the rate of convergence given in Fro Theore 3.3, we can now show the following Theore for the asyptotic distribution of ˆθ. Theore 3.4. Under the conditions of Theore 3.3, we have log η d ˆθ θ N 0, Λ d 2 2π Λ 2, 1 C where η = d1 γ/2. Reark 3.5. Note that we have a different convergence rate for ˆθ copared to the convergence rate for ĉ given in Theore 3.2. The additional ter log is fro the fact that θ is in the exponent of in the expression of Lc, θ. 4. Discussion We proposed estiators of c and θ that govern the tail behavior of the spectral density of a stationary Gaussian rando field on R d. The proposed estiators are obtained by iniizing the objective function given in 3.3. As entioned in Section 3, this objective function is siilar to the one used in the local Whittle likelihood ethod when a kernel function Λ in W h K is constant. When we replace d θ g c,θ with f τ ϕ λ and reove W hk in the expression given in 3.3, it can be thought of a Whittle approxiation to the likelihood of Y τ ϕj. This approxiation, however, has not been verified under fixed-doain asyptotics. One ight think that we can apply a siilar technique to prove the validity of a Whittle approxiation to the likelihood under increasing-doain asyptotics since Y τ ϕj is a lattice process. However, the spectral density f τ ϕ λ of Y τ ϕj converges to zero as ϕ 0, which requires a different approach and further investigation is needed. The weights in 3.3 is controlled by h, a bandwidth, which can be interpreted as a proportion of Fourier frequencies to be considered in the objective function. In our theores, we assue h = C γ for soe constant C. In proofs, we ake use of the properties of a soothed spatial periodogra Îτ. Thus, we could find the optial bandwidth that iniizes the ean squared error of Îτ. However, finding the ean squared error of Îτ needs explicit first order asyptotic expressions of the bias and variance of Îτ λ, which are not yet available. It will

12 Wu, Li & Xiao/Tail Estiation of the Spectral Density 12 be ore useful when we can estiate c and θ together or estiate θ when c is unknown. Due to the for of g c,θ, proving their asyptotic properties under fixed-doain asyptotics is challenging but worthwhile investigating further. The Assuption 1 A iplies that the spectral density is regularly varying at infinity with exponent θ. Together with a soothness condition Assuption 1 B, it satisfies assuptions A1 and A2 in Stein 2002, which includes slowly varying tail behavior. Assuptions A1 and A2 in Stein 2002 guarantee that there is a screening effect, that is, one can get a nearly optial predictor at a location s based on the observations nearest to s see Stein 2002 for further details. Then, our result that one can estiate tail behavior using only local inforation can be seen as a kind of analogue to a screening effect. Appendix A: The properties of g c,θ λ Soe properties of g c,θ λ are discussed in this Appendix. These properties will be used in the proofs given in Appendix B. Recall that 2τ d g c,θ λ = c 4 sin 2 λ j /2 Hλ + 2πQ θ Q Z d for λ 0. For a function g c,θ λ, let g denote the gradient of g with respect to λ and let ġ and g denote the first and second derivatives of g c,θ λ with respect to θ, respectively. That is, g = g/ λ 1,, g/ λ d, ġ = g c,θ λ/ θ and g = 2 g c,θ λ/ θ 2. LetA ρ = [ π, π] d \ ρ, ρ d for a fixed ρ that satisfies 0 < ρ < 1. Since we assue that the paraeter space Θ is a closed interval in Section 3, let Θ = [θ L, θ U ] and θ L > d. Although Lea A.1 can be shown for any fixed ρ with 0 < ρ < 1, we further assue that ρ is sall enough so that all Fourier frequencies near π/21 d considered in Lc, θ are contained in A ρ. Lea A.1. The following properties hold for g c,θ λ. a g c,θ λ is continuous on Θ A ρ. b There exist K L and K U such that for all θ, λ Θ A ρ, 0 < K L g c,θ λ K U <. A.1 c There exist K L and K U such that for all λ A ρ and all θ 1, θ 2 Θ, 0 < K L g c,θ1 λ/g c,θ2 λ K U <. A.2

13 Wu, Li & Xiao/Tail Estiation of the Spectral Density 13 d g, ġ, g, ġ/g and ġ/g are uniforly bounded on Θ A ρ. Proof. We prove the Lea when H is an identity atrix for siplicity. The results with a general nonsingular atrix H can be followed without uch difficulty. To show a, it is enough to show continuity of Q Z λ + d 2πQ θ on Θ A ρ { d 2τ since 4 sin2 λ j /2} is continuous on Aρ. It can be easily shown that Q Z d, Q >n λ + 2πQ θ converges to zero uniforly on Θ A ρ as n, which iplies the unifor convergence of Q Z d, Q n λ + 2πQ θ to g 1,θ λ. Thus, continuity of g c,θ λ follows fro the continuity of λ + 2πQ θ. To prove the reaining parts, we find the upper and lower bounds of Q Z d λ + 2πQ θ. For all θ, λ Θ A ρ, we have Q Z d λ + 2πQ θ π θu > 0 and Q Z d λ + 2πQ θ λ + 2πQ θ L + ϵ θ U Q Z d \{0} 2π d ϵ d θ L /d θ L + ϵ θ U, where the last inequality follows fro λ + 2πQ θ L λ + 2πy θ L dy Q Z d \0 y 1 2π d z θ L dz z ϵ = 2π d x d 1 x θ L dx = 2π d ϵ d θ L /θ L d, A.3 since θ L > d. Thus, we have 0 < k L λ + 2πQ θ k U <, A.4 Q Z d x ϵ where k L = π θ U and k U = 2π d ϵ d θ L /θ L d + ϵ θ U. Then, b follows fro A.4, 2τ d 4d sin 2 ϵ/2 2τ 4 sin 2 hλ j /2 4d 2τ,

14 Wu, Li & Xiao/Tail Estiation of the Spectral Density 14 and by setting K L c 4d sin 2 ϵ/2 2τ k L and K U c 4d 2τ k U. c follows fro observing that Q Zd λ + 2πQ θ has bounds that are unifor on Θ A ρ as given in A.4. lower and upper For d, we have g { d } 2τ 1 λ i = c 4τ 4 sin 2 λ j /2 sinλ i λ + 2πQ θ Q Z d { d } 2τ 1 θ 4 sin 2 λ j /2 λ i + 2πQ i λ + 2πQ θ 2 Q Z d K Q Z d λ + 2πQ θ K k U for soe constant K > 0 and k U given in A.4, which iplies unifor boundedness of g on Θ A ρ. For the unifor bound of ġ and g, we first copute ġ and g: ġ = { d } 2τ c 4 sin 2 λ j /2 λ + 2πQ θ log λ + 2πQ, Q Z d g = { d } 2τ c 4 sin 2 λ j /2 λ + 2πQ θ log λ + 2πQ 2. Q Z d Since we can find x 0 and K such that for a given β > 0, log x Kx β for all x > x 0, we can show that there exist n 0, K 1 and K 2 that satisfy ġ K 1 + K 2 Q Z d, Q n 0 λ + 2πQ θ+β for soe fixed β > 0. When we choose β = θ L θ/2, we can show that Q Z d, Q n 0 λ + 2πQ θ+β < using a siilar arguent to show A.3, which leads to unifor boundedness of ġ. Siilarly, we can show unifor boundedness of g. The unifor boundedness of ġ/g follows fro unifor boundedness of ġ and

15 Wu, Li & Xiao/Tail Estiation of the Spectral Density 15 b. To show unifor boundedness of ġ/g, consider Q Z ġ/g = λ + d 2πQ θ 2 λ i + 2πQ i 1 θ log λ + 2πQ λ i Q Z λ + 2πQ θ d Q Z λ + 2πQ θ log λ + 2πQ d + Q Z d λ + 2πQ θ 2 θ λ + 2πQ θ 2 λ i + 2πQ i Q Z d Since denoinators in the expression of ġ/g / λ i have unifor lower bounds as shown in A.4, it is enough to find unifor bounds of nuerators to show unifor boundedness of ġ/g / λ i. By observing that λ i +2πQ i λ + 2πQ and λ + 2πQ 1 K for soe K > 0 on A ρ, we can show that each nuerator in the expression of ġ/g / λ i is uniforly bounded on Θ A ρ using a siilar arguent to show unifor boundedness of ġ. Appendix B: Proofs of Theores in Section 3 Proof of Theore 3.1. If fλ satisfies 2.4 for all λ, 3.1 and 3.2 hold by results in Stein 1995 and Li and Stein To prove 3.1 and 3.2 when 2.4 holds only for large λ, we need to show that the effect of fλ on λ C is negligible. Consider a spectral density kλ which satisfies kλ c Hλ θ as λ and kλ is twice differentiable and satisfies 2.4 for all λ. Also assue that kλ fλ for λ > C. Let I f,τ λ be the periodogra at λ fro the observations under fλ and Note that a f,τ,ϕ J, K = 2π d d R d { d } 2τ 4 sin 2 ϕλ j /2 fλ sin 2 ϕλ j /2 sin ϕλ j /2 + πj j / sin ϕλ j /2 + πk j / dλ. E I f,τ 2πJ/ = a f,τ,ϕ J, J, V ar I f,τ 2πJ/ = a f,τ,ϕ J, J2 + a f,τ,ϕ J, J2.

16 Wu, Li & Xiao/Tail Estiation of the Spectral Density and 3.2 follow fro Theores 3, 6 and 12 in Li and Stein 2008 when these Theores hold for f under Assuption 1. The key part of proofs of these Theores under Assuption 1 is to show E I f,τ 2πJ/ f ϕ τ 2πJ/ = 1 + O β 1 B.1 V ar I f,τ 2πJ/ f τ ϕ 2πJ/2 = 1 + O β2, B.2 for soe β 1, β 2 > 0. Once B.1 and B.2 are shown, the other parts of proofs are siilar to the proofs in Li and Stein Since results in Stein 1995 and Li and Stein 2008 hold for kλ, we have B.1 and B.2 for kλ. Then, B.1 and B.2 for fλ follow fro a f,τ,ϕ J, ±J ak,τ,ϕ J, ±J = O d 4τ, B.3 for J that satisfies J and 2J/ Z d. B.3 holds since a f,τ,ϕ J, ±J ak,τ,ϕ J, ±J { d } 2τ = 2π d 4 sin 2 ϕλ j /2 fλ kλ λ C d sin 2 ϕλ j 2 sin ϕλ j /2 + πj j / sin ϕλ j /2 ± πj j / dλ 2π d d v d 4τ λ C { d } 2τ 4 sin 2 ϕλ j /2 fλ kλ sin 2 ϕλ j /2 sin ϕλ j /2 + πj j / sin ϕλ j /2 ± πj j / dλ for soe positive constant v since kλ fλ for λ > C and ϕλ j /2 ± πj j / stays away fro zero and π when is large. Proof of Theore 3.2. To show weak consistency of ĉ, we consider upper and lower bounds of ĉ. Let K U = argax K T,W h K 0 g 0 2πJ + K/ and K L = argin K T,,W h K 0 g 0 2πJ + K/. Recall that g 0 = g 1,θ0. Then,

17 Wu, Li & Xiao/Tail Estiation of the Spectral Density 17 we have K T W h KI2πJ τ + K/ d θ 0 g0 2πJ + K U ĉ / K T W h KI2πJ τ + K/ d θ 0 g0 2πJ + K L, / which can be rewritten as c Îτ 2πJ/ d θ0 g c,θ0 2πJ + K U / ĉ c Îτ 2πJ/ d θ 0 gc,θ0 2πJ + K L / B.4 with probability one. Note that both g c,θ0 2πJ + K U / and g c,θ0 2πJ + K L / converge to g c,θ0 π/21 d by continuity of g c,θ λ and d θ0 Î2πJ/ τ converges to g c,θ0 π/21 d in probability by Theore 3.1. Thus, it follows that ĉ converges to c in probability. For the asyptotic distribution of ĉ, note that we have Îτ η 2πJ/ g d θ0 c,θ0 π/21 d d N 0, Λ 2 Λ 2 1 d 2π gc,θ 2 C 0 π/21 d B.5 fro Theore 3.1 and η g c,θ0 2πJ + K E / g c,θ0 π/21 d 0, B.6 for E = U or L, since 4τ > θ 0 1, h = C γ and follows fro B.5 and B.6. d d+2 < γ < 1. Then, 3.6 To prove Theore 3.3, we consider following leas. Lea B.1. Consider a function h x = logx+d x 1, where d is a positive function of a positive integer. Also assue that d 1 as. Then, for a given r with 0 < r < 1, there exist δ r > 0 and M r > 0 such that for all M r, h x > δ r, for any x Z r {z : z 1 > r, z > 0}.

18 Wu, Li & Xiao/Tail Estiation of the Spectral Density 18 Proof. It can be easily shown that for any positive integer, h x is a convex function on 0, and iniized at x = 1/d with h 1/d 0. Let h x = logx + x 1. Since d 1, for any r 0, 1, there exists M r > 0 such that for all M r, we have 1/d 1 r and in{h 1 r, h 1+r} > 1/2 in{h 1 r, h 1 + r} > 0. Hence for all x Z r, we have h r x in{h 1 r, h 1 + r} > 1/2 in{h 1 r, h 1 + r} δ r. Lea B.2. For a positive integer and θ Θ, we have where A = log Lc 0, θ Lc 0, θ 0 A + B + C, θ θ 0 gc0,θ B = log 0 2πJ + S / 2πJ + S M / C = g 2πJ +S c0,θ 0 Î τ 2πJ + B.7 g 2πJ +S c0,θ d θ 0 gc0 2πJ +K M,θ 0 g 2πJ +S θ θ c0,θ 0 0 1, g 2πJ +S c0,θ Î τ 2πJ/ d θ0 2πJ + K M / In B.7-B.9, K M, K, S M and S are defined as, B.8 g c0,θ2πj + S M / g c0,θ2πj + S / 1 g c 0,θ 0 2πJ + K M / 2πJ + K / K M = arg ax {K T,W h K 0} 2πJ + K/, K = arg in {K T,W h K 0} 2πJ + K/, S M = arg ax {K T,W h K 0} log gc0,θ 0 2πJ + K/ g c0,θ2πj + K/ g c0,θ S = arg in 0 2πJ + K/ {K T,W h K 0} g c0,θ2πj + K/. Furtherore,, B.9. sup B = o1, B.10 θ Θ C = o p 1, B.11 where B.11 holds under the conditions of Theore 3.3.

19 Wu, Li & Xiao/Tail Estiation of the Spectral Density 19 Proof. Fro the expression of Lc, θ given in 3.3, we have Lc 0, θ Lc 0, θ 0 = { W h K log d θ g c0,θ 2πJ + K/ K T log d θ 0 2πJ + K/ } + 1 I W h K 2πJ τ + K/ d θ g c0 K T,θ2πJ + K/ 1 I W h K 2πJ τ + K/ d θ0 g c0,θ K T 0 2πJ + K/ = W h K log θ θ g 0 c 0,θ 0 2πJ + K/ g c0,θ 2πJ + K/ K T + I W h K 2πJ τ + K/ d θ0 g c0,θ K T 0 2πJ + K/ θ θ g 0 c 0,θ 0 2πJ + K/ g c0,θ2πj + K/ I W h K 2πJ τ + K/ d θ 0 gc0,θ K T 0 2πJ + K/ log θ θ g 0 c 0,θ 0 2πJ + S M / g c0,θ 2πJ + S M / + I W h K 2πJ τ + K/ d θ 0 gc0,θ K T 0 2πJ + K M / = log g θ θ0 c 0,θ 0 2πJ + S / g c0,θ2πj + S / I2πJ τ + K/ d θ0 2πJ + K / W h K K T g θ θ0 c 0,θ 0 2πJ + S M / g c0,θ2πj + S M / + Î2πJ/ τ d θ0 2πJ + K M / g c0,θ2πj + S / g c 0,θ 0 2πJ + K M / 2πJ + K / θ θ g 0 c 0,θ 0 2πJ + S / =: H.

20 Wu, Li & Xiao/Tail Estiation of the Spectral Density 20 H is further decoposed as H = log θ θ g 0 c 0,θ 0 2πJ + S / g c0,θ2πj + S / θ θ 0 g c 0,θ 0 2πJ + S / Î τ + 2πJ/ d θ 0 gc0,θ 0 2πJ + K M / gc0,θ + log 0 2πJ + S / g c0,θ2πj + S M / g c0,θ 0 2πJ + S M / g c0,θ2πj + S / + Î τ 2πJ/ d θ 0 gc0,θ 0 2πJ + K M / g c0,θ2πj + S / 1 1 g c 0,θ 0 2πJ + K M / 2πJ + K / which is A + B + C given in B.7-B.9. Note that 2πJ + K M /, 2πJ + K /, 2πJ + S M / and 2πJ + S / converge to π/21 d as. Note also that the convergence of 2πJ +S M / and 2πJ +S / holds for θ uniforly on Θ, because h 0. The continuity of g c0,θ in Lea A.1 iplies that gc0,θ log 0 2πJ + S / g c0,θ2πj + S M / 0 B.12 g c0,θ 0 2πJ + S M / g c0,θ2πj + S / holds for θ uniforly on Θ, therefore, sup Θ B = o1. Also, we have, d θ 0 Î τ 2πJ//g c0,θ 0 2πJ + K M / p 1, since d θ0 Î τ 2πJ//g c0,θ 0 π/21 d converges to one in probability by Theore 3.1 and 2πJ + K M / converges to π/21 d. Thus, together with 1 g c 0,θ 0 2πJ + K M / 2πJ + K / 0, B.13 C converges to zero in probability. Proof of Theore 3.3. Let Ω, F, P be the probability space where a stationary Gaussian rando field Zs is defined. To ephasize dependence on, we use ˆθ instead of ˆθ in this proof. Note that we have P Lc 0, ˆθ Lc 0, θ 0 0 = 1 B.14 for each positive integer by the definition of ˆθ. We are going to prove consistency of ˆθ by deriving a contradiction to B.14 when ˆθ does not converge

21 Wu, Li & Xiao/Tail Estiation of the Spectral Density 21 to θ 0 in probability. Suppose that ˆθ does not converge to θ 0 in probability. Then, there exist ϵ > 0, δ > 0 and M 1 such that for M 1, P ˆθ θ 0 > ϵ > δ. We define D = {ω Ω : ˆθ ω θ 0 > ϵ}. By Lea B.2, we have Lc 0, ˆθ ω Lc 0, θ 0 A + B + C, where A, B and C are given in B.7-B.9 with θ = ˆθ ω, ω D. We are going to show that there exist { k }, a subsequence of {} and a subset of D k on which A k + B k + C k is bounded away fro zero for large enough k. Note that A = h ˆθ θ 0 2πJ + S /, g c0,ˆθ 2πJ + S / where h is defined in Lea B.1 with d = Î τ 2πJ/ d θ 0 gc0,θ 0 2πJ + K M /, B.15 where K M is defined in Lea B.2. By Theore 3.1 and the convergence of g c0,θ 0 2πJ + K M / to g c0,θ 0 π/21 d, p we have d 1. Then, there exists {k }, a subsequence of {} such that d k converges to one alost surely. By B.13 in the proof of Lea B.2, alost sure convergence of d k iplies that C k defined in B.9 converges to zero alost surely. To use Lea B.1, we need unifor convergence of d k. By Egorov s Theore Folland, 1999, there exists G δ Ω such that d k and C k converge uniforly on G δ and P G δ > 1 δ/2. Let H k = D k G δ. Note that P H k > δ/2 > 0 for k M 1. On the other hand, there exists a M 2, which does not depend on ω, such that for k M 2, ˆθ k θ 0 2πJ + S k / k k g c0,ˆθ k 2πJ + S k / k 1 > 1 B.16 2 for all ω D k, because of the unifor boundedness of /g c0,θ. Then, by Lea B.1 with r = 1/2, there exist δ r > 0 and M r ax{m 1, M 2 } such that

22 Wu, Li & Xiao/Tail Estiation of the Spectral Density 22 for k M r, A k = log ˆθ k θ 0 k 2πJ + S k / k g c0,ˆθ k 2πJ + S k / k B.17 + > δ r d θ0 Î τ k 2πJ/ k k g c0,θ 0 2πJ + K M / k k θ 0 g c0,θ 0 2πJ + S k / k ˆθ k g c0,ˆθ k 2πJ + S k / k 1 uniforly on H k. By the unifor convergence of B on Θ shown in Lea B.2, there exists a M 3 such that for k M 3, B k < δ r 4 with θ = ˆθ k ω uniforly on H k. The unifor convergence of C k allows us to find M 4 such that for k M 4, B.18 on G δ C k < δ r 4 B.19 uniforly on H k. Therefore, for k ax{m r, M 3, M 4 }, we have A k + B k + C k A k B k C k > δ r /2 on H k which leads Lc 0, ˆθ k Lc 0, θ 0 > δ r 2 B.20 on H k. Since P H k > δ/2 > 0, it contradicts to B.14 which copletes the proof. Here, we do not need P k H k > 0 since B.14 should holds for any > 0. To show the convergence rate of ˆθ given in 3.10, it is enough to show that ˆθ θ p 0 1 which is equivalent to show that 2πJ + S / g c0,ˆθ 2πJ + S / ˆθ θ 0 g c0,θ 0 2πJ + S / g c0,ˆθ 2πJ + S / p 1, B.21 p 1. B.22 B.21 follows fro the consistency of ˆθ and the continuity of g c0,θ shown in Lea A.1. To show B.22, we consider a siilar arguent to show consistency

23 Wu, Li & Xiao/Tail Estiation of the Spectral Density 23 of ˆθ. For siplicity, we reset notations such as r, δ, δ r, M and D, etc. used in the proof of consistency. Suppose that B.22 does not hold. Then, there exists r > 0, δ > 0 and M 1 such that P ˆθ θ 0 2πJ + S / g c0,ˆθ 2πJ + S / 1 > r > δ for all M 1. On the other hand, there exists { k }, a subsequence of {}, such that d k 1, B k 0 and C k 0 alost surely, where d is given in B.15, B and C are given in B.8 and B.9 with θ = ˆθ ω. Then, by Egorov s Thoere, there exists Ω δ Ω such that P Ω δ > 1 δ/2 and d k, B k and C k are uniforly convergent on Ω δ. As in Lea B.1, for a k, a nonzero solution of h k b k = 0, where b = ˆθ θ 0 g c0,θ 0 2πJ + S / g c0,ˆθ 2πJ + S /, there exists M 2 such that a k 1 r uniforly on Ω δ for all k M 2. Now, define { } D = ω : ˆθ θ 0 2πJ + S / g c0,ˆθ 2πJ + S / 1 > r. Note that P D k Ω δ δ/2 > 0 for all k ax{m 1, M 2 }. Siilarly to the proof of Lea B.1, for each k ax{m 1, M 2 }, there exists δ r > 0 such that A k > δ r for all ω D k Ω δ. This iplies that P A k > δ r δ/2 for each k ax{m 1, M 2 }. Note that δ r does not depend on k which can be seen in Lea B.1. Meanwhile, there exists M 3 such that for k M 3, B k δ r /4, C k δ r /4 for all ω Ω δ. Hence we have P Lc 0, ˆθ Lc 0, θ 0 > δ r /2 δ/2 for k ax{m 1, M 2, M 3 }, which contradicts to B.14. Thus, B.22 is proved. Note that an alternative proof of the consistency of ˆθ is available [Wu 2011]. To proof Theore 3.4, we consider the following Lea.

24 Wu, Li & Xiao/Tail Estiation of the Spectral Density 24 Lea B.3. Under the conditions of Theore 3.3, let η = d1 γ/2, we have a b η I W h K 2πJ τ + K/ d θ 0 gc0,θ K T 0 2πJ + K/ 1 D N 0, Λ d 2 2π Λ 2, 1 C I τ 2πJ +K W h K 1 K T d θ0 g 2πJ +K c0,θ 0 ġ 2πJ +K c0,θ 0 = O g 2πJ p η +K c0,θ 0 B.23 B.24 Proof. To prove B.23, we find the asyptotic distribution of its lower and upper bounds. It can be easily shown that L η I W h K 2πJ τ + K/ d θ 0 gc0,θ K T 0 2πJ + K/ 1 U, where L = η Î2πJ/ τ d θ 0 gc0,θ 0 2πJ + K M / 1, B.25 U = η Î2πJ/ τ d θ0 g c0,θ 0 2πJ + K / 1 B.26 with K M and K as defined in Lea B.2. We rewrite L as L = η Î2πJ/ τ d θ 0 gc0,θ 0 π/21 d 1 π/21 d 2πJ + K M / g c0,θ + 0 π/21 d g c0,θ 0 2πJ + K M / 1. By Lea A.1 and γ > d/d + 2, we have π/21 d g c0,θ 0 2πJ + K M / η π/21 d 2πJ + K M / 1 1, 0.

25 Wu, Li & Xiao/Tail Estiation of the Spectral Density 25 Thus, by Theore 3.1, Siilarly, we can show L U d N d N 0, Λ 2 Λ 2 1 0, Λ 2 Λ 2 1 d 2π. C d 2π. C The convergence of lower and upper bounds to the sae distribution iplies B.23. To show B.24, we rewrite the LHS of B.24 as 1 K T W h K I τ 2πJ + K/ d θ 0 gc0,θ 0 2πJ + K/ ġc0,θ 0 2πJ + K/ g c0,θ 0 2πJ + K/ = W h Kġc0,θ02πJ + K/ g c0,θ K T 0 2πJ + K/ ġc 0,θ 0 π/21 d π/21 d I W h K 2πJ τ + K/ d θ0 g c0,θ K T 0 2πJ + K/ ġc 0,θ 0 2πJ + K/ 2πJ + K/ ġc 0,θ 0 π/21 d. π/21 d By Lea A.1 and γ > d/d + 2, we can show that η W h Kġc0,θ02πJ + K/ g c0,θ K T 0 2πJ + K/ ġc 0,θ 0 π/21 d π/21 d 0. Also, it can be easily shown that L I W h K 2πJ τ + K/ ġ c0,θ 0 2πJ + K/ d θ0 g c0,θ K T 0 2πJ + K/ 2πJ + K/ U, where with L = U = Î τ 2πJ/ d θ 0 gc0,θ 0 π/21 d Î τ 2πJ/ d θ 0 gc0,θ 0 π/21 d π/21 d ġ c0,θ 0 2πJ + P / gc 2, 0,θ 0 2πJ + P / π/21 d ġ c0,θ 0 2πJ + P M / gc 2, 0,θ 0 2πJ + P M / ġ c0,θ P M = arg ax 0 2πJ + K/ {K T,W h K 0} gc 2 0,θ 0 2πJ + K/, ġ c0,θ P = arg in 0 2πJ + K/ {K T,W h K 0} gc 2 0,θ 0 2πJ + K/.

26 Wu, Li & Xiao/Tail Estiation of the Spectral Density 26 By Lea A.1, γ > d/d + 2 and Theore 3.1, we can show that 2 d η L ġc 0,θ 0 π/21 d d ġc0,θ N 0, 0 π/21 d Λ 2 2π g c0,θ 0 π/21 d g c0,θ 0 π/21 d Λ 2, 1 C 2 d η U ġc 0,θ 0 π/21 d d ġc0,θ N 0, 0 π/21 d Λ 2 2π. π/21 d π/21 d C This copletes the proof of B.24. Λ 2 1 Proof of Theore 3.4. Let L = L/ θ and L = 2 L/ θ 2. To show the asyptotic distribution of ˆθ, we consider the Taylor expansion of Lc 0, ˆθ around θ 0, Lc 0, ˆθ = Lc 0, θ 0 + Lc 0, θˆθ θ 0, where θ lies on the line segent between ˆθ and θ 0. Since Lc 0, ˆθ = 0, we have log η ˆθ θ 0 = log η Lc0, θ 1 Lc0, θ 0. Thus, it is enough to show log 1 η Lc0, θ 0 log 2 Lc0, θ d N 0, Λ 2 Λ 2 1 d 2π, B.27 C p 1. B.28 Since Lc 0, θ 0 = log + W h Kġc0,θ02πJ + K/ g c0,θ K T 0 2πJ + K/ W h KI2πJ τ + K/ K T log d θ 0 2πJ + K/ + d θ0ġ c0,θ 0 2πJ + K/ d θ0 g c0,θ 0 2πJ + K/ 2 I = log W h K 2πJ τ + K/ d θ 0 gc0,θ K T 0 2πJ + K/ 1 + I τ W h K 1 2πJ + K/ ġc0,θ 0 2πJ + K/ d θ0 g c0,θ K T 0 2πJ + K/ 2πJ + K/, we see that B.27 follows fro Lea B.3.

27 Wu, Li & Xiao/Tail Estiation of the Spectral Density 27 Next we prove B.28. After soe siplification, we have Lc 0, θ = log 2 I W h K 2πJ τ + K/ d θg K T c0, θ2πj + K/ 2 log W h K Iτ 2πJ + K/ġ c0, θ2πj + K/ d θg 2 2πJ + K/ K T c 0, θ + 2 I2πJ τ + K/ġ 2 c W h K 0 2πJ + K/, θ d θg 3 2πJ + K/ K T c 0, θ + I W h K 1 2πJ τ + K/ gc0, θ2πj + K/ d θg K T c0, θ2πj + K/ g c0, θ2πj + K/ ġ 2 c W h K 0, θ 2πJ + K/ g 2 2πJ + K/ K T c 0, θ =: E 1 + E 2, where E 1 is the first ter with log 2 and E 2 is the last four ters in the expression of Lc 0, θ. First, we want to show that It can be easily shown that log 2 E 1 p 1. B.29 LB log 2 E 1 U, where with L = U M = Î2πJ/ τ θ θ 0 π/21 d d θ 0 gc0,θ 0 π/21 d g c0, θ2πj + P M /, Î2πJ/ τ θ θ 0 g c0,θ 0 π/21 d d θ 0 gc0,θ 0 π/21 d g c0, θ2πj + P / P M = arg ax {K T,W h K 0}g c0, θ2πj + K/, P = arg in {K T,W h K 0}g c0, θ2πj + K/. By Theore 3.1, 3.10 in Theore 3.3 and Lea A.1, we can show that both L and U converge to one in probability, which in turn iplies B.29. In a siilar way, we can show that log 1 E 2 = O p 1. Thus, together with B.29, we can show B.28, which copletes the proof.

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