Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations and their convergence

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1 Gaussian stationary processes: adaptive wavelet decompositions, discrete approximations and their convergence Gustavo Didier and Vladas Pipiras University of North Carolina at Chapel Hill August 28, 2007 Abstract We establish particular wavelet-based decompositions of Gaussian stationary processes in continuous time. These decompositions have a multiscale structure, independent Gaussian random variables in high-frequency terms, and the random coefficients of low-frequency terms approximating the Gaussian stationary process itself. They can also be viewed as extensions of the earlier wavelet-based decompositions of Zhang and Walter 1994 for stationary processes, and Meyer, Sellan and Taqqu 1999 for fractional Brownian motion. Several examples of Gaussian random processes are considered such as the processes with rational spectral densities. An application to simulation is presented where an associated Fast Wavelet Transform-like algorithm plays a key role. 1 Introduction Consider a real-valued Gaussian stationary process X = {Xt} t having the integral representation Xt = gt udbu = e itx ĝxd Bx, 1.1 where g L 2 is a real-valued function, called a kernel function, ĝ L 2 is its Fourier transform defined by convention as ĝx = e ixu gudu, {Bu} u is a standard Brownian motion and { Bx} x = {B 1 x + ib 2 x} x is a complexvalued Brownian motion satisfying B 1 x = B 1 x, B 2 x = B 2 x, x 0, with two independent Brownian motions {B 1 x} x 0 and {B 2 x} x 0 such that EB = EB = 4π 1. The latter conditions on Bx ensure that the second integral in 1.1 is real-valued and has the same covariance structure as the first integral in 1.1. Many Gaussian stationary processes, especially those of practical interest, can be represented by 1.1. See, for example, ozanov The second author was supported in part by the NSF grant DMS AMS Subject classification. Primary: 60G10, 42C40, 60G15. Keywords and phrases: Gaussian stationary processes, wavelets, simulation, Ornstein-Uhlenbeck process, processes with rational spectral densities, AMA time series, iesz basis.

2 1967 and others. The covariance function t = EXtX0 of X and its Fourier transform are given by t = g g t = 1 ĝ 2 t, x = ĝx 2, 1.2 where g u = g u is the time reversion operation and stands for convolution. The Fourier transform x is also known as the spectral density of X. Note, however, that the two rightmost expressions in 1.2 are not meaningful for general g L 2 because the function may be in neither L 2 nor L 1. Under mild assumptions on g and in a special Gaussian case, Theorem 1 of Zhang and Walter 1994 states that a Gaussian process X in 1.1 has a wavelet-based expansion Xt = a J,n θ J t 2 J n + j=j d j,n Ψ j t 2 j n, 1.3 for any J Z, with convergence in the L 2 Ω-sense for each t. Here, a J = {a J,n } n Z, d j = {d j,n } j J,n Z are independent N 0,1 random variables. The functions θ j and Ψ j are defined through their Fourier transforms as θ j x = ĝx 2 j/2 φ2 j x, Ψj x = ĝx 2 j/2 ψ2 j x, 1.4 where φ and ψ are scaling and wavelet functions, respectively, associated with a suitable orthogonal Multiresolution Analysis MA, in short. For more information on scaling function, wavelet and MA, see for example Mallat 1998, Daubechies 1992, or many others. Moreover, the coefficients a j,n and d j,n in 1.3 can be expressed as a j,n = Xtθ j t 2 j ndt, d j,n = XtΨ j t 2 j ndt, 1.5 with the functions θ j and Ψ j, dual to θ j and Ψ j, defined through θ j x = ĝx 1 2 j/2 φ2 j x, Ψj x = ĝx 1 2 j/2 ψ2 j x. 1.6 Zhang and Walter 1994 call 1.3 a Karhunen-Loève-like KL-like wavelet-based expansion. It is discussed in several textbooks, for example, Walter and Shen 2001, and Vidakovic The sum n a J,nθ J t 2 J n in 1.3 is interpreted as an approximation term at scale 2 J, and the sums n d j,nψ j t 2 j n, j J, are interpreted as detail terms at finer scales 2 j, j J. The KL-like expansion is related to the wavelet-vaguelette expansions of Donoho 1995, the expansions of Benassi and Jaffard 1994, and others, where J = in 1.3 and hence the first approximation term in 1.3 is absent. Though the approximation term n a J,nθ J t 2 J n in 1.3 involves independent N 0,1 random variables a J,n which are convenient to deal with in theory, the term is also unnatural in one important respect. It is customary with wavelet bases that not only an approximation term but also the respective approximation coefficients, the sequence a J,n in this case, approximate the signal at hand. The sequence a J,n does not have this property because it consists of independent random variables and hence cannot approximate a typically dependent stationary process Xt. In this work, we modify the approximation terms as a J,n θ J t 2 J n = 2 X J,n Φ J t 2 J n 1.7

3 so that the new approximation coefficients X J = {X J,n } n Z now have this property, namely, 2 J/2 X J,[2 J t] Xt 1.8 in a suitable sense, as J, where [x] denotes the integer part of x. In the relation 1.7 above, Φ J x = ĝx ĝ J 2 J x 2 J/2 φ2 J x = θ J x ĝ J 2 J x 1.9 with the discrete Fourier transform ĝ J y of a sequence g J = {g J,n }. A discrete Fourier transform of g = {g n } is defined by ĝx = g n e inx, x, and is periodic with the period. The random sequence X J = {X J,n } in 1.7 is defined as X J x = ĝ J xâ J x 1.10 in the frequency domain. Moreover, we expect that X J,n = XtΦ J t 2 J ndt, 1.11 where ĝj 2 Φ J x = J x 2 J/2 φ2 J x ĝx The relation 1.7 can be informally and easily verified by taking the Fourier transform of its two sides. It is well-known e.g. Daubechies 1992 in the deterministic context that 1.8 is a property of the corresponding wavelet basis functions. When G J 2 J x := ĝj2 J x ĝx 1, 1.13 we have Φ J x 2 J/2 φ2 J x 2 J/2 for large J typically, φ0 = φtdt = 1 and hence, by 1.11, we expect that 2 J/2 X J,n = 2J/2 Xxe ix2 J n ΦJ xdx 1 Xxe ix2 J n dx = X2 J n. The conditions for 1.7 and 1.8 will thus involve the function G J given in Though the modification 1.7 appears small, it is fundamental and important in several ways, and surprisingly leads to many research questions. For example, the modification allows for several applications such as simulation considered in Section 8 below. The wavelet-based decomposition 1.3 with 1.7 can also be viewed as a generalization to the wide framework of stationary Gaussian processes of a particular wavelet decomposition of fractional Brownian motion established in Sellan 1995, Meyer, Sellan and Taqqu It thus shows that self-similarity 3

4 of fractional Brownian motion, for instance is not a necessary condition for constructing wavelet decompositions in that fashion. See also Pipiras 2004 who explored a similar decomposition for a non-gaussian self-similar process called the osenblatt process. Didier and Pipiras 2006 study analogous wavelet decompositions for stationary time series in discrete time. The decomposition 1.3 with 1.7 can be viewed as being more general than becoming 1.3 when X j = a j are independent, N 0, 1 random variables. For this reason, both decompositions should be viewed under one framework. This is the view taken in the following definition and in a parallel paper Didier and Pipiras Definition 1.1 Decompositions 1.3 and 1.3 with 1.7 will be called adaptive wavelet decompositions. Adaptiveness refers to the fact that the basis functions are chosen based on the dependence structure of the underlying stationary Gaussian process. The rest of the paper is organized as follows. In Section 2, we briefly introduce a wavelet basis to be used in wavelet-based decompositions. In Section 3, we state the assumptions on the discrete deterministic approximations g J and the functions g. In Section 4, we consider several examples of Gaussian stationary processes and their discrete approximations. The KL-like wavelet decomposition 1.3 and its modification 1.7 are proved in Section 5. In particular, we reprove the decomposition 1.3 because inaccurate assumptions were used in Zhang and Walter We show that there is a FWT-like algorithm relating {X j,n } across different scales in Section 6. In Sections 7 and 8, we examine convergence of discrete random approximations X J and illustrate simulation in practice. Finally, in Appendix A, we consider integration of stationary Gaussian processes. 2 Wavelet bases of L 2 We specify here a scaling function φ and a wavelet ψ which will be used below. There are many choices for these functions. We shall work with particular Meyer wavelets Meyer 1992, Mallat 1998 because of their nice theoretical properties. The results of this paper and their proofs rely on specific nice properties of the selected Meyer wavelets. Other wavelet bases could be taken, e.g. the celebrated Daubechies wavelets, and are being currently investigated. Meyer wavelets are also used in Zhang and Walter 1994, Meyer et al and others. Let S be the Schwartz class of C functions f that decay faster than any polynomial at infinity and so do their derivatives, that is, lim t tm dn ft dt n = 0, for any m, n 1. We can choose a scaling function φ S satisfying φx [0, 1], φx = φ x, φx = { 1, x /3, 0, x > 4π/3, φx decreases on [0,. The corresponding CMF u has the discrete Fourier transform { 2 φ2x, x /3, ûx = 0, x > /3. 4

5 The wavelet function ψ associated with φ is such that ψ S and ψx = 1 x x v φ with vx = e ix ûx + π, 2.1 where v is the other CMF. One can verify that, for the Meyer wavelets, ψx = e ix 2 x 2 1/2 φ 2 φx In particular, ψx = 0 for x /3 and x 8π/3. The collection of functions φt k, 2 j/2 ψ2 j t k, k Z, j 0, makes an orthonormal basis of L 2. 3 Basis functions and discrete approximations Let g L 2 be a kernel function appearing in 1.1, and g J = {g J,n } n Z, J Z, be sequences of real numbers such that g J l 2 Z. Following Section 1 see, in particular, 1.13, we shall think of g J as a discrete deterministic approximation of g at scale 2 J. A discrete approximation g J l 2 Z induces a discrete random approximation X J = {X J,n } defined by 1.10, that is, X J,n = g J,k a J,n k 3.1 in the time domain, or symbolically k= X J x = ĝ J xâ J x 3.2 in the frequency domain, where a J = {a J,n } are independent N 0,1 random variables Gaussian white noise. As J, we expect that 2 J/2 X J,[2 J t] approximates Xt defined by 1.1. Conversely, we may think that a random discrete approximation X J of X given by 1.1 can be represented by 3.1 with a sequence g J. Hence, X J also induces a deterministic discrete approximation g J of g. We will make some of the following assumptions on g and g J. Let L p loc consist of functions which are in L p on any compact interval of. Set also G J x = ĝjx ĝ2 J, x. 3.3 x Note that, with the notation 3.3, expressions 1.9 and 1.12 become Φ J x = G J 2 J x 1 2 J/2 φ2 J x, ΦJ x = G J 2 J x2 J/2 φ2 J x 3.4 Assumption 1: Suppose that ĝ 1 L 2 loc. 3.5 Assumption 2: Suppose that, for any J Z, G J, G 1 J L 2 loc

6 Assumption 3: Suppose that, for any J 0 Z, max max sup sup p= 1,1 k=0,1,2 J J 0 x 4π/3 k G J x p x k <. 3.7 Assumption 4: Suppose that, for large x, k ĝx x k const, x k+1 k = 0, 1, Assumption 5: Assume that, for large J, G J 0 1 const 2 J. 3.9 As explained below, Assumptions 1 and 2 ensure that the basis functions used in decompositions are well-defined. Assumptions 3, 4 and 5 will be used to establish the modification 1.7 and to show that X J is an approximation sequence for X in the sense of 1.8. Observe that the functions θ j and Ψ j in 1.4 are well-defined pointwise through the inverse Fourier transform since θ j, Ψ j L 1 for ĝ L 2. By using Assumptions 1 and 2, the functions θ j and Ψ j in 1.6, Φ j in 1.9 and Φ j in 1.12 see also 3.4 are well-defined pointwise through the inverse Fourier transform as well. Moreover, θ j, Ψ j, Φ j, Φ j are in L 2 because their Fourier transforms are in L 2. Appendix A contains some results on defining integrals Xtftdt. See, in particular, the definition of a related function space L 2 g in A.6 of integrands ft. Since θ j, Ψ j L 2 g, the coefficients a j,n and d j,n in 1.5 are well-defined. Using properties of integrals developed in Appendix A, it is easy to see that a j,n and d j,n are independent N 0, 1 random variables. Since Φ j L 2 g, the integral in 1.11 is well-defined as well. Another consequence of the above assumptions are useful bounds on the functions Φ j, Ψ j. We will use these bounds several times below. Lemma 3.1 Under Assumptions 3 and 4 above, we have 2 j/2 Φ j 2 j u, 2 j/2 Φ j 2 j u where a constant C does not depend on j j 0, for fixed j 0. C, u, u 2 Ψ j 2 j u C2 j/2, u, u 2 Proof: By definition of Φ j in 1.9 see also 3.4 and after a change of variables, observe that 2 j/2 Φ j 2 j u = 1 e iux G j x 1 φxdx, u Since supp{ φ} { x 4π/3}, we obtain by Assumption 3 that 2 j/2 Φ j 2 j u C, u,

7 for a constant C which does not depend on j j 0, for fixed j 0. Using integration by parts in 3.12 twice and Assumption 3, we have 2 j/2 Φ j 2 j u = 1 iux 2 u 2 e x 2 G j x 1 φx dx, u. By Assumption 3 and properties of φ, for any j j 0, 2 j/2 Φ j 2 j u C 2 u 2 x 4π/3 x 2 G jx 1 + x G jx 1 + Gj x 1 dx C u 2, u. The bound 3.10 for Φ j follows from 3.13 and The case of Φ j is proved similarly. To show the bound 3.11, observe from 1.4 that Ψ j 2 j u = 2j/2 e iux ĝ2 j x ψxdx, u Since supp{ ψ} {/3 x 8π/3}, we obtain by Assumption 4 that Ψ j 2 j u C2 j/2 8π/3 /3 dx j x C 2 j/2, u, 3.15 for constants C, C which do not depend on j j 0, for fixed j 0. Using integration by parts in 3.14 and Assumption 4, we have Ψ j 2 j u = 2j/2 iux 2 u 2 e x 2 ĝ2 j x ψx dx, u Hence, by using properties of ψ and Assumption 4, for j j 0, Ψ j 2 j u C2j/2 u 2 2 2j 2 ĝ /3 x 8π/3 x 2 2j x + ĝ 2j x 2j x + ĝ2j x dx C 2 j/2 8π/3 2 2j u j x j j x j/ j dx C, u x u 2 /3 The bound 3.11 follows from 3.15 and Examples We consider here several examples of Gaussian stationary processes together with their possible discrete approximations. Example 4.1 The Ornstein-Uhlenbeck OU process X is perhaps the best-known Gaussian stationary process. It is the only Gaussian stationary process which is Markov. The OU process can be represented by 1.1 with gt = σe λt 1 {t 0}, ĝx = σ λ + ix, 4.1 7

8 for some λ > 0 and σ > 0. At this point, one can approximate either g or X. We do so for the process X because it has a well-known discrete approximation. Observe from 1.1 and 4.1 that, for J, n Z, X2 J n + 1 = e λ2 J X2 J 1 e n + σ 2λ2 J a J,n+1, 2λ where {a J,n } n Z is a Gaussian white noise. Therefore, since we expect 2 J/2 X J,[2 J t] Xt, it appears natural to consider the discrete approximation X J,n = 2 J/2 1 e σ 2λ2 J I e λ2 J B 1 a J,n, 4.2 2λ where B denotes the backshift operator not to be confused with Bm and I = B 0. In other words, X J is an A1 time series see Brockwell and Davis In view of 4.1 and 4.2, the deterministic discrete approximations g J have the discrete Fourier transforms ĝ J x = 2 J/2 1 e σ 2λ2 J 1 e λ2 J e ix λ Furthermore, ĝ and ĝ J satisfy Assumptions 1 to 5. Indeed, Assumptions 1 and 2 hold because, for every J Z, ĝ 1 x = λ + ix, G J x = 2 J/2 1 e 2λ2 J σ 2λ 2 J λ + ix 1 e λ2 J e ix 4.4 and G 1 J are continuous functions on, and thus square-integrable on compact sets. To show Assumption 3, consider the domain D J 0 = {z C : 0 ez 2 J 0 λ, Imz 4π/3}. The functions { z, z C\{i2kπ, k Z}, F z = 1 e z 1, z = 0, and F z 1 are holomorphic and different from zero on the open set D J 0 ɛ = {w C : inf z D J 0 z w < ɛ} D J 0. By setting z = 2 J λ + ix D J 0, we have G J x = C J F z for all J J 0 and x 4π/3, where 0 < c 1 C J c 2 < + for some c 1, c 2. Hence, Assumption 3 must hold. Assumption 4 follows from the relation k ĝx x k = σ ik k!, k = 0, 1, 2,... λ + ix k+1 Finally, Assumption 5 is also satisfied because 1 e G J 0 1 = 2λ2 J 2 J λ 2J/2 1 2λ 1 e λ2 J = λ2 J 1 e λ2 J 1 e λ2 J λ2 J for constants C 1, C 2 > e λ2 J 2 C 1 + e λ2 J e λ2 J λ2 J 1 C 2 2 J 8

9 Example 4.2 Consider a Gaussian stationary process 1.1 with a kernel function g having the Fourier transform ĝx = fx hx. 4.5 Here, with fx = pa k, b k ; x p a k, b k ; x p0, c m ; x, 4.6 k P 1 m P 2 hx = pd k, e k ; x p d k, e k ; x p0, f m ; x 4.7 k Q 1 m Q 2 pa, b; x = ix + ia + b, 4.8 where P 1, P 2, Q 1 and Q 2 are finite sets of indices. It is assumed that polynomials fx and hx have no common roots, and also that k Q 1, e k 0, and m Q 2, f m 0. Note that the polynomials f and h are Hermitian symmetric. Hence, ĝ is also Hermitian symmetric and thus g is real-valued. Kernel functions ĝ as in 4.5 correspond to rational spectral densities ozanov To define a discrete approximation ĝ J of ĝ, consider first pa, b; x, which is a building block of f in 4.6 and h in 4.7. Define a discrete approximation of pa, b; x as and also, in analogy to 3.3, set p J a, b; x = 2 J 1 e 2 J b 2 J ia ix 4.9 P J x = p Ja, b; x pa, b; 2 J x = 1 e 2 J b 2 J ia ix ix + 2 J ia + 2 J b The form 4.9 ensures that p J a, b; xp J a, b; x and p J 0, b; x are Hermitian symmetric functions. Define now a discrete approximation ĝ J of ĝ by 4.5, where p s in 4.6 and 4.7 are replaced by p J s. The function G J is then given by 4.5, where p s in 4.6 and 4.7 are replaced by P J s. We shall now verify that ĝ and ĜJ satisfy Assumptions 1-5. Assumptions 1 and 2 are satisfied because the building blocks p 1, P J and P 1 J for ĝ 1, G J and G 1 J are continuous functions on the real line. To show Assumption 3, it is enough to prove 3.7 for the function P J. Similarly to the case of the Ornstein-Uhlenbeck process, we are interested in the behavior of F and F 1 for z = ix + 2 J a + 2 J b, where x 4π/3 and J J 0. So, define the set { D J 0 = z C : 0 z 2 J 0 b, Iz 4π 3 + z a }, b and note that z = ix + 2 J a + 2 J b D J 0 when x 4π/3 and J J 0. Also, consider the set D J 0 ɛ = {w C : inf z D J 0 z w < ɛ}. The functions F and F 1 are holomorphic on D J 0 ɛ D J 0 for small enough ɛ, and thus Assumption 3 holds. Consider now Assumption 4. The condition 3.8 is satisfied for k = 0 by the definition of ĝ and the implicit assumption ĝ L 2 that is, the polynomial h has a higher degree than the polynomial f. When k = 1, note that ĝx x = f x hx fxh x hx 2 9

10 and the condition 3.8 follows since the difference between the degrees of f x and hx, and those of fxh x and hx 2 increased by 1. The case k = 2 can be argued in a similar way. To show 3.9 in Assumption 5, it is enough to prove it for P J 0 = 1 e 2 J b 2 J ia 2 J ia + 2 J b. This can be done by using standard properties of exponentials and using their Taylor expansions. Finally, let us note that the discrete approximations g J based on 4.9 correspond to AMA time series X J Brockwell and Davis Adaptive wavelet decompositions We first reestablish the decomposition 1.3 of Zhang and Walter 1994 by providing a more rigorous proof. Theorem 5.1 Zhang and Walter 1994 Let X be a Gaussian stationary process given by 1.1. Suppose that Assumptions 1 and 2 of Section 3 hold. Then, with the notation of Section 1, the process X admits the following wavelet-based decomposition: for any J Z, Xt = a J,n θ J t 2 J n + d j,n Ψ j t 2 j n 5.1 = j= j=j d j,n Ψ j t 2 j n, 5.2 with the convergence in the L 2 Ω-sense for each t, and independent N 0, 1 random variables a J,n, d j,n that are expressed through 1.5. Proof: Zhang and Walter 1994 Under Assumptions 1 and 2, the basis functions θ J and Ψ j in 5.1 and 5.2 are well-defined pointwise Section 3. The coefficients a j,n, d j,n are well-defined, independent N 0, 1 random variables Section 3. Except for more rigor, the rest of the proof follows that of Zhang and Walter Since the proof is short, we provide it to the reader s convenience. Observe that N 2 K M 2 2 E Xt a J,n θ J t 2 J n d j,n Ψ j t 2 j n n= N 1 j=j n= M 1 N 2 K M 2 = E Xt 2 2 Xta J,n θ J t 2 J n 2 Xtd j,n Ψ j t 2 j n n= N 1 j=j n= M 1 N2 K M a J,n θ J t 2 J n + d J,n Ψ j t 2 j n. 5.3 n= N 1 j=j n= M 1 By using Appendix A and the definition of function θ j Sections 1 and 3, we have EXta J,n = EXt Xsθ J s 2 J nds = 1 e itx ĝx 2 θ J 2 J nxdx = 1 e it 2 J nx ĝx 2 θj xdx = 1 e it 2 J nx 2 J/2 ĝx φ J 2 J xdx = θ J t 2 J n

11 Similarly, we have EXtd j,n = Ψ j t 2 j n. 5.5 Using 5.4, 5.5 and independence of a J,n, d j,n, relation 5.3 becomes 0 N 2 Observe from the definition of θ J that = 1 and similarly n= N 1 θ J t 2 J n 2 θ J t 2 J n = 1 K j=j M 2 ĝx2 J/2 φ2 J + t nxdx = 1 n= M 1 Ψ j t 2 j n e it 2 J nx 2 J/2 ĝx φ J 2 J xdx gu2 J/2 φn 2 J u + tdu, 5.7 Ψ j t 2 j n = 1 gu2 j/2 ψn 2 j u + tdu. 5.8 Since the collection of functions 2 J/2 φn 2 J u + t, 2 j/2 φn 2 j u + t, j J, n Z, makes an orthonormal basis of L 2 for any t, and since 0 = gt 2 dt, we obtain from 5.7 and 5.8 that relation 5.6 converges to 0 as N i, M i i = 1, 2 and K approach infinity. In the next result, we modify the approximation term in the decomposition 5.1 according to 1.7. Theorem 5.2 Let X be a Gaussian stationary process given by 1.1. Suppose that Assumptions 1 and 2 of Section 3 hold. Then, with the notation of Section 1, the process X admits the following wavelet-based decomposition: for any J Z, Xt = X J,n Φ J t 2 J n + j=j d j,n Ψ j t 2 j n. 5.9 The convergence in 5.9 is in the L 2 Ω-sense for each t under Assumption 3, and it is almost sure, uniform over compact intervals of t under Assumptions 3 and 4. The sequence X J = {X J,n } n Z is defined by either 1.11 or 3.1. Proof: We first argue that the definitions 1.11 and 3.1 of X j are equivalent. By using Appendix A, observe that, for X J,n defined by 1.11 and a J,n defined by 1.5, E X J,n N 2 k= N 1 g J,k a J,n k = 1 2 = E ĝ J 2 J x Xt Φ J t 2 J n N 2 k= N 1 g J,k θ J t 2 J n k N 2 g J,k e ix2 J k 2 2 J φ2 J x 2 dx 0, k= N 1 as N i i = 1, 2, since k g J,ke ixk converges to ĝ J x in L 2 π, π and φ has a compact support. dt 2 11

12 To show 5.9, we start with 5.1 and modify its first sum as 1.7, that is, a J,n θ J t 2 J n = X J,n Φ J t 2 J n We first show that, under Assumption 3, the.h.s. converges in the L 2 Ω-sense for fixed t and, under Assumptions 3 and 4, almost surely, uniformly over compacts of t. Observe that, by Lemma 3.1, Φ J t 2 J n C/1 + t 2 J n 2 and, by Lemma 3 in Meyer et al. 1999, X J,n A log2 + n a.s., where a random variable A does not depend on n. The almost sure convergence uniformly on compacts t K follows since log2 + n sup X J,n Φ J t 2 J n A sup 1 + t 2 J n 2 < a.s. t K t K For the convergence in L 2 Ω, observe that, for fixed t, E X J,n Φ J t 2 J n 2 CE X J,n n 2 We shall now prove the equality in Observe that, for each u, Indeed, arguing as above, F m u = m k= m θ J u = k= g J,k Φ J u 2 J k F u = n 2 <. g J,k Φ J u 2 J k k= g J,k Φ J u 2 J k pointwise, and F m x = m k= m g J,k e i2 J kx ĝx ĝ J 2 J x 2 J/2 φ2 J x θ J x 5.13 in L 2, since m k= m g J,ke ikx converges to ĝ J x in L 2 π, π, and ĝx/ĝ J 2 J x is bounded by Assumption 3 on the compact support of φ2 J x. Hence, F m θ J in L 2 and θ J = F a.e. Since both F and θ J are continuous, we obtain Set now, for m 1, a m J,n = { aj,n, n m, 0, n > m, X m J,n = k= g J,k a m J,n k. By using 5.11, we obtain that a m J,n θj t 2 J n = X m J,n ΦJ t 2 J n

13 The L.H.S. of 5.14 converges in L 2 Ω to the L.H.S. of 5.10 and, in fact, also almost surely by the Three Series Theorem. Let us show that the.h.s. of 5.14 converges to the.h.s. of We want to argue next that sup X m J,n A log2 + n a.s m 1 for a random variable A which only depends on J. By using the Lévy-Octaviani inequality e.g. Proposition in Kwapień and Woyczyński 1992, we have P sup X m J,n > a 2P X M J,n, > a 5.16 m=1,...,m for any a > 0 and M 1. By the Three Series Theorem, X M J,n X J,n almost surely, as M. Hence, passing to the limit with M in 5.16, we have P sup X m J,n > a 2P X J,n > a. m 1 The bound 5.15 now follows as in the proof of Lemma 3 in Meyer et al By using Lemma 3.1 and the bound 5.15, the.h.s. of 5.14 converges a.s. to the.h.s. of It is left to show that the second term in 5.9 converges almost surely and uniformly on compacts. By Lemma 3 in Meyer et al. 1999, d j,n A log2 + j log2 + n a.s., where a random variable A does not depend on j, n. By Lemma 3.1, we have Ψ j t 2 j n = Ψ j 2 j 2 j t n C2 j/ j t n 2, for j J. Then, as in the proof of Theorem 2 in Meyer et al. 1999, j=j d j,n Ψ j t 2 j n A 2 j/2 log2 + j A j=j j=j a.s. uniformly over compact intervals of t. 6 FWT-like algorithm log2 + n j t n 2 2 j/2 log2 + j log2 + 2 j t < We show here that discrete approximation sequences X j are related across different scales by a FWT-like algorithm. Proposition 6.1 Let X j and d j be the sequences appearing in 5.9, and let u and v denote the CMFs associated with the orthogonal Meyer MA. Then, under Assumptions 1 4 of Section 3: i econstruction step X j+1 = u j 2 X j + v j 2 d j,

14 where the filters u j and v j are defined through their discrete Fourier transforms ii Decomposition step û j x = ĝj+1x ĝ j 2x ûx, v jx = ĝ j+1 x vx; 6.2 X j = 2 u d j X j+1, d j = 2 v d j X j+1, 6.3 where x stand for the time reversal of a sequence x, and the filters u d j and vd j are defined through their discrete Fourier transforms by û d ĝ j 2x j x = ûx, v d 1 j x = vx. 6.4 ĝ j+1 x ĝ j+1 x The convergence in 6.1 and 6.3 is in the L 2 Ω-sense, and also absolute almost surely. Proof: Observe first that the filters u j, v j, u d j, vd j are well-defined since û j, v j, û d j, vd j L2 π, π. The latter follows by writing 1ûx, û j x = G j+1 x G j 2x vj x = G j+1 xĝ2 j+1 x vx, û d j x = G j+1 x 1 G j 2xûx, v d j x = G j+1 x 1 ĝ2 j+1 x 1 vx 6.5 see 3.3, and using Assumptions 1 and 3. i To show 6.1, we need to prove X j+1,n = k= X j,k u j,n 2k + k= d j,k v j,n 2k. 6.6 We first prove the convergence in 6.6 in the L 2 Ω-sense. Observe that, by using 1.11, 1.5 and Appendix A, K 2 E X j+1,n X j,k u j,n 2k + d j,k v j,n 2k = E Xt Φ j+1 t 2 j 1 n = 1 k= K K k= K ĝx 2 e i2 j 1 nx Φj+1 x Φ j x = 1 Φ j t 2 j ku j,n 2k K k= K K k= K e i2 j kx u j,n 2k Ψ j x Ψ j t 2 j kv j,n 2k dt K k= K 2 e i2 j kx v j,n 2k 2 dx e i2 j 1 nxĝ j+1 2 j 1 x2 j+1/2 φ2 j 1 x ĝ j 2 j x2 j/2 φ2 j x K k= K Hence, it is sufficient to prove that ĝ j 2 j x2 j/2 φ2 j x e i2 j kx u j,n 2k 2 j/2 ψ2 j x k= K k= K e i2 j kx u j,n 2k + 2 j/2 ψ2 j x 14 e i2 j kx v j,n 2k 2 dx. k= e i2 j kx v j,n 2k

15 = e i2 j 1 nxĝ j+1 2 j 1 x2 j+1/2 φ2 j 1 x 6.7 with the convergence in L 2. We only consider the case n = 2p the case n = 2p + 1 may be treated in an analogous fashion. Then, relation 6.7 becomes ĝ j 2 j x 2 j/2 φ2 j x m= e i2 j mx u j,2m + 2 j/2 ψ2 j x m= e i2 j mx v j,2m The L.H.S. of 6.8 is = ĝ j+1 2 j 1 x 2 j+1/2 φ2 j 1 x. 6.8 ĝ j 2 j x 2 j/2 φ2 j xûj2 j 1 x + û j 2 j 1 x + π +2 j/2 ψ2 j x v j2 j 1 x + v j 2 j 1 x + π 2 2 = j/2 φ2 j x ĝ j+1 2 j 1 x û2 j 1 x + ĝ j+1 2 j 1 x + π û2 j 1 x + π j/2 ψ2 j x ĝ j+1 2 j 1 x v2 j 1 x + ĝ j+1 2 j 1 x + π v2 j 1 x + π = ĝ j+1 2 j 1 x 2 j/2 φ2 j x + û2 j 1 x + π xû2 j j/2 ψ2 j x v2 j 1 x + v2 j 1 x + π j/2 φ2 j xû2 j 1 x + π + ψ2 j x v2 j 1 x + π ĝ j+1 2 j 1 x + π ĝ j+1 2 j 1 x. This is also.h.s. of 6.8 since 2 j/2 φ2 j xû2 j 1 x + û2 j 1 x + π j/2 ψ2 j x v2 j 1 x + v2 j 1 x + π 2 = 2 j+1/2 φ2 j 1 x this is the Fourier transform of the last relation in the proof of Theorem 7.7 in Mallat 1998 and, with y = 2 j 1 x, φ2 j xû2 j 1 x + π + ψ2 j x v2 j 1 x + π = φ2yûy + π + ψ2y vy + π = 2 1/2 φy ûyûy + π + vy vy + π = 0, where we used the facts φ2y = 2 1/2 φyûy 7.30 in Mallat 1998, ψ2y = 2 1/2 φy vy 7.57 in Mallat 1998 and ûyûy + π + vy vy + π = 0 Theorem 7.8 in Mallat We now show that the convergence in 6.6 is also absolute almost surely. By using Assumptions 3, 4, and integration by parts twice, we may conclude that u j,k, v j,k C1 + k 2 1, k Z. By Lemma 3 in Meyer et al. 1999, X j,k, d j,k A log2 + k a.s., where a random variable A does not depend on k. The absolute convergence a.s. now follows. ii The proof of 6.3 follows by similar arguments. We need to prove that X j,n = k= X j+1,k u d j,k 2n

16 and d j,n = k= X j+1,k v d j,k 2n To show 6.9 with convergence in the L 2 Ω-sense, it suffices to prove that But the.h.s. of 6.11 is e i2 j nxĝ j 2 j x2 j/2 φ2 j x = ĝ j+1 2 j+1 x2 j+1/2 φ2 j+1 x ĝ j+1 2 j+1 x2 j+1/2 φ2 j+1 x k= m= e ik2 j+1x u d j,k 2n e i2n+m2 j+1x u d j,m = ĝ j+1 2 j+1 x2 j+1/2 φ2 j+1 xe in2jxû d j 2 j+1 x = 2 j+1/2 φ2 j+1 xe in2jxĝ j 2 j xû2 j+1 x, 6.12 which is also the L.H.S. of 6.11 by using φ2y = 2 1/2 φyûy. The proof of the equality 6.10 in the L 2 Ω-sense is similar. The absolute almost surely convergence of 6.9 and 6.10 may be deduced by arguments analogous to those for the absolute almost surely convergence of Convergence of random discrete approximations We will also assume the following: Assumption 6: Suppose that there are β N {0} and α 0, 1] such that, for any compact K, Xt Xs X 1 st s... X β s t sβ β! A t s β+α for all t, s K, a.s., 7.1 where a random variable A depends only on K. As usual, f k denotes the kth derivative of f. Note that 7.1 implies, for some random variable B, Xt Xs B t s γ for all t, s K, with γ = 1 β + α. 7.2 Condition 7.1 in Assumption 6 is satisfied by many Gaussian stationary processes. It follows, in particular, from the two conditions: and X β is α-hölder a.s. 7.3 Xt C1 + t β+α a.s. 7.4 By Theorem and a discussion on pp in Cramér and Leadbetter 1967, 7.3 follows from 0 x 2β+2α log1 + x ĝx 2 dx <

17 There is also an equivalent condition in terms of the autocovariance function of a stationary Gaussian process. Condition 7.4 is always satisfied for stationary Gaussian processes that are bounded on compact intervals, such as for those satisfying 7.3. In fact, a stronger condition holds: Xt C log2 + t a.s., 7.6 where C is a random variable. To see this, note that the discrete-time sequence X k = sup t [k,k+1 Xt is stationary. Moreover, by Theorem 2 in Lifshits 1995, p. 142, for some m and σ > 0, P X 0 m + τ 21 Φτ/σ, τ > 0, where Φ is the distribution function of standard normal law. In other words, the right tail of the distribution function of X n decays at least as fast as that of the distribution function of standard normal law. The bound 7.6 can then be obtained as 3.15 in Lemma 3 of Meyer et al We shall need some assumptions stronger than parts of Assumptions 3 and 5. Assumption 3*: Suppose that, for any J 0 Z, max sup k=0,1,...,β+[α]+2 Assumption 5*: Assume that, for large J, sup J J 0 x 4π/3 k G J x x k <. 7.7 G J 0 1 const 2 β+1j. 7.8 As in Lemma 3.1, under Assumption 3*, we have 2 j/2 Φ j 2 j u where a constant C does not depend on j j 0, for fixed j 0. In addition, we will suppose the following: Assumption 7: If β 1 in Assumption 6, suppose that G n J C, u, u β+[α]+2 0 = n G J 0 = 0, n = 1,..., β xn The next result establishes convergence of random discrete approximations. Proposition 7.1 Under Assumptions 2,3,5 of Section 3 and Assumption 6 above, we have sup 2 J/2 X J,[2 J t] Xt A 1 2 Jγ a.s., 7.11 t K where K is a compact interval and A 1 is a random variable that does not depend on J. If, in addition, Assumptions 3*,5* and 7 above hold, then sup 2 J/2 X J,[2 J t] X[2 J t]2 J A 2 2 Jβ+α a.s., 7.12 t K where a random variable A 2 does not depend on J. 17

18 Proof: Suppose without loss of generality that K = [0, 1]. In view of Assumption 6, it is enough to show 7.12 or that 2 J/2 X J,k Xk2 J A 2 Jβ+α a.s. sup k=0,...,2 J Note by Assumption 7 and the properties of the scaling function φ that u n n n Φ J udu = i Φn J 0 = i n x n G J x2 J/2 φ2 x J x=0 = 0, for n = 1,..., β. By using Appendix A, Assumptions 5* and 6 with 7.9, we have 2 J/2 X J,k Xk2 J 2 J/2 Xt Xk2 J... X β k2 J t k2 J β Φ J t k2 J dt β! +Xk2 J G J 0 1 A2 J/2 t k2 J β+α Φ J t k2 J dt + B2 β+1j = A 2 Jβ+α = A2 J/2 u β+α Φ J u du + B2 β+1j setting u = 2 J v v β+α 2 J/2 Φ J 2 J v dv A 2 Jβ+α v β+α 1 + v β+[α]+2 dv = A 2 Jβ+α. According to Proposition 7.1, the discrete approximations 2 J/2 X J,[2 J t] converge to the process Xt. Note also that, when β 1, the convergence is faster on the dyadics than on the whole interval. An interesting question is whether the faster convergence rate β + α can be obtained on an interval for some other approximation based on X J,[2 J t]. For a function f, defined on either or Z, consider the operator p h fa = p k=0 p 1 p k fa + kh, p N, k where a, h are in either or Z, respectively. When f = f k is a function on Z, we write p f k = p 1 fk and p f = p 1 f0. In view of the condition 7.1, to obtain the faster rate β + α on a whole interval, it is natural to try an approximation which includes the terms mimicking the β derivatives in 7.1. Thus, for β 1, consider the approximations X β,j t = 2 J/2 X J,[2 J t] + 2 J/2 β p X J,[2 J t] t [2 J t]2 J p 2 Jp, 7.13 p! with the idea that 2 J/2 p X J,[2 J t] X p t2 Jp for large J. For example, when β = 1, p=1 X 1,J t = 2 J/2 X J,[2 J t] + 2 J/2 X J,[2 J t]+1 X J,[2 J t] 2 J t [2 J t]2 J. When β = 0, we get X 0,J t = 2 J/2 X J,[2 J t]. 18

19 Although intuitive, the approximation X β,j in 7.13 may not converge to Xt at the faster rate β +α on compact intervals see emark 7.1 below. It turns out, though, that a modification of 7.13 does attain that rate. In order to build such approximation, we will make use of Lemma 7.1 below, stated without proof. Observe from 7.12 that replacing 2 J/2 p X J,[2 J t] by p X[2 J t]/2 J in the approximation 7.13 makes an error of the desired faster rate α + β. 2 J Lemma 7.1 shows that, after suitable correction, p X[2 J t]/2 J approximates X p [2 J t]/2 J 2 J and then X p t at the desired rate α + β. This correction needs to be taken into account when considering a modification to X β,j. For any x, define the function s x on Z by s x k = x + k, k Z. Note that p s j 0 = p k=0 p k 1 p k k j, j N recall from above that p s j 0 = p 1 sj 0 0. Lemma 7.1 Let β N, α 0, 1 and G be an open interval. If f : G is a Lipschitz function of order β + α in the sense of 7.1, then, for N p β, a G, we have as h 0. p h fa β p h p f p f p+j a a = h j p s p+j 0 + Oh β+α p, 7.14 p + j! j=1 Define the approximation function X β,j t by X β,j t = X 0,J + β p=1 X p,j t p! t [2 J t]2 J p, 7.15 where and X 0,J := 2 J/2 X J,[2 J t], Xβ,J := 2J/2 β X J,[2 J t] 2 Jβ X p,j := 2J/2 p β p X J,[2 J t] 2 J + j=1 X p+j,j p + j! 2 Jj p s p+j 0, p = 1, 2,..., β 1. Proposition 7.2 Under stronger assumptions of Proposition 7.1, we have sup X β,j t Xt A2 Jβ+α a.s., 7.16 t K where K is a compact interval and A is random variable that does not depend on J. Proof: If the relation X p,j X p [2 J t]2 J = O2 Jβ+α p 7.17 holds for p = 0, 1, 2,..., β, then, by Assumption 6, X β,j t Xt X β,j t X[2 J t]2 J β p=1 X p [2 J t]2 J p! t [2 J t]2 J p 19

20 + Xt X[2J t]2 J β p=1 X p [2 J t]2 J p! t [2 J t]2 J p = O2 Jβ+α, which proves elation 7.17 holds for p = 0 by Proposition 7.1. To show 7.17 for β 1, we argue by backward induction. For p = β, by Proposition 7.1 and Lemma 7.1, we have X β,j X β [2 J t]2 J 2 J/2 β X J,[2 J t] 2 Jβ β X[2 J t]2 J 2 Jβ β X[2 J t]2 J + 2 Jβ X β [2 J t]2 J = O2 Jβ+α β. Assume by induction that 7.17 holds for p + 1,..., β 1, β with p 1. Then, by Proposition 7.1 and Lemma 7.1, we obtain that X p,j X p [2 J t]2 J 2 J/2 p X J,[2 J t] 2 Jp p X[2 J t]2 J 2 Jp β p + j=1 + p X[2 J t]2 J β p 2 Jp X p [2 J t]2 J X p+j [2 J t]2 J β p 2 Jj p s p+j 0 p + j! j=1 j=1 X p+j,j X p+j [2 J t]2 J p + j! p + j! 2 Jj p s p+j 0 2 Jj p s p+j 0 = O2 Jβ+α p. emark 7.1 When β = 2, the approximation X β,j becomes XJ,[2 X 2,J = 2 J/2 X J,[2 J t] +2 J/2 J t]+1 X J,[2 J t] 2 J + XJ,[2J t]+2 2X J,[2J t]+1 + X J,[2J t] 2 2 J t [2 J t]2 J +2 J/2 X J,[2 J t]+2 2X J,[2 J t]+1 + X J,[2 J t] 2 J 2 t [2 J t]2 J Compare 7.18 with the approximations X 2,J given in Observe that, if X 2,J also converges to X at the rate 2 + α, then O2 J2+α = X 2,J X 2,J = 2 J/2 X J,[2 J t]+2 2X J,[2 J t]+1 + X J,[2 J t] 2 2 J t [2 J t]2 J or, by using 7.12 in Proposition 7.1, O2 J2+α = X[2J t] + 22 J 2X[2 J t] + 12 J + X[2 J t]2 J 2 J t [2 J t]2 J or, by using Taylor expansions, O2 J2+α = 2X t 1 X t 2 2 J t [2 J t]2 J, with t 1 = t 1 J and t 2 = t 2 J that are close to t. The last relation may not be satisfied under our assumptions, showing that one cannot expect X 2,J to converge to X at the rate 2 + α. 20

21 Although the approximations X β,j converge to X at the faster rate β+α, these approximations do not necessarily have continuous paths. Indeed, it can be easily verified that X β,j is continuous when β = 1, 2 but not so when β = 3. For a fixed β 2, it may be desirable to have not only a continuous but also a C β 1 approximation X β,j. Moreover, in analogy to 7.15, in order to have the faster convergence, we would expect the p-th derivative of the approximation X β,j at [2 J t]2 J to approximate the p-th derivative of the process X at t. We generally found such C β 1 approximations difficult to construct. One difficulty is the following. As in 7.15, we may seek an approximation X β,j which is a polynomial of order β on an interval [2 J t]2 J, [2 J t]2 J + 1. Since X β,j is globally C β 1, we would require its derivatives X p β,j, p = 0, 1,..., β 1, to be equal to prescribed values at the endpoints [2 J t]2 J and [2 J t]2 J +1. equiring this yields 2β equations that a polynomial X β,j must satisfy. Since a polynomial of order β has only β + 1 coefficients, this is not possible in general. Despite this difficulty, we have found the following general scheme to yield C β 1 approximations, at least for the first several values of β 2. To construct a C 1 approximation X 2,J, we could require first that its derivative 2 J/2 X 1 2,Jt = 2 J/2 X1,J t based on the sequence X J,[2 J t] 2 J = X J,[2 J t] 2 J + 1 XJ,[2 J t]+1 2 J 2 J X J,[2J t] 2 J t [2 J t]2 J = X J,[2 J t] 2 J + 2 X J,[2J t] 2 J 2 t [2 J t]2 J Observe that, by construction using continuous approximation X 1,J, X 1 2,J is continuous. Moreover, X 1 2,J approximates X 1 t, and X 2 2,J on the interval [2 J t]2 J, [2 J t]2 J + 1 approximates X 2 t. Integrating 7.19 and requiring it to be continuous yields the following approximation 2 J/2 X 2,J t = X J,[2 J t]+1 + X J,[2 J t] 2 + X J,[2 J t] 2 J t [2 J t]2 J + 2 X J,[2 J t] 2 J 2 t [2 J t]2 J Note that X 2,J differs from X 2,J by the constant term. Similarly, to construct a C 2 approximation X 3,J, we could require that X 1 3,Jt = X 2,J t based on the sequence X J,[2 J t] 2 J. Integrating the resulting expression and requiring it to be continuous yields X 3,J t = 1 6 X J,[2 J t] X J,[2 J t] X J,[2 J t] X J,[2 J t] 2 J t [2 J t]2 J 2 X J,[2 J t] 2 J 2 t [2 J t]2 J X J,[2 J t] 6 2 J 3 t [2 J t]2 J the subindex 2 in 2 X J,[2 J t] is not a typo. The approximation 7.21 is C 2 and its derivatives of orders p = 0, 1, 2, 3 approximate those of the process X. 21

22 We expect that the above scheme yields C β 1 approximations X β,j for any β 2. However, as explained in emark 7.2 below, we cannot expect these approximations to convergence at the faster rate β + α. This is perhaps not surprising, because the discontinuous approximations in 7.15 are already nontrivial. emark 7.2 One cannot expect the approximation X 2,J in 7.21 to converge to X at the faster rate 2 + α. Indeed, if this rate were achieved, we would have see 7.18 O2 J2+α = X 2,J t X 2,J t = X J,[2 J t]+1 X J,[2 J t] X J,[2 J t]+2 2X J,[2 J t]+1 + X J,[2 J t] 2 2 J t [2 J t]2 J or, by using 7.11, O2 J2+α = X[2 J t] + 12 J X[2 J t]2 J X[2J t] + 22 J 2X[2 J t] + 12 J + X[2 J t]2 J 2 J t [2 J t]2 J or, by Taylor expansions, O2 J2+α = X [2 J t]2 J 2 J X t 1 2 2J X t 2 2 J t [2 J t]2 J, with t 1 = t 1 J and t 2 = t 2 J close to t, or, by expanding X [2 J t]2 J further, O2 J2+α = X t2 J X t 1 2 2J X t 2 2 J t [2 J t]2 J. This relation may not be satisfied under our assumptions on X. 8 Simulation: the case of the OU process We will illustrate here how the results of Sections 6 and 7 can be used to simulate a stationary process X. We consider only the case of the OU process in Example 4.1. ecall from that example that the discrete approximations taken for the OU process are ĝ J x = 2 J/2 1 e σ 2λ2 J 1 e λ2 J e ix λ and the corresponding discrete random approximations X J are suitable A1 time series in 4.2. With the choice 8.1 of approximations, observe that the filters u j and v j used in reconstruction 6.1 become 2 1/2 û j x = 1 + e λ2 j+1 e ix ûx, 8.2 2λ2 j e v j x = 2 j+1/2 1 e σ 2λ2 j+1 1 e λ2 j+1 e ix 1 vx λ Suppose one wants to simulate the OU process on the interval [0, 1]. The idea is to begin by generating a discrete approximation X 0 at scale 2 0. This step is easy as X 0 is an A1 time 22

23 1.5 Approximations to OU process 0.5 Log sup difference between approximations X J diff t j Figure 1: Approximations X J and the logarithms of their sup differences. series. Then, substituting X 0 into 6.1, one may get the approximation X 1, and continuing recursively from X 1 now, the approximation X J for arbitrary fixed J 1. Note that applying 6.1 recursively each time essentially involves just simulating independent N 0, 1 random variables and computing filters u j and v j. Proposition 7.1 ensures that the properly normalized X J approximate the OU process uniformly over [0, 1] and exponentially fast in J. We illustrate this in Figure 1 for the OU process with λ = 1, σ = 1. The plot on the left depicts the consecutive approximations X j from X 0 at scale 2 0 to X J at the finest scale 2 J with J = 11. In the right plot, we present the sup-differences between consecutive approximations X j 1 and X j, j = 2,..., 11, on the log scale. The decay in that plot confirms that normalized approximations X J converge to the OU process exponentially fast in J. Several comments should be made on how approximations X j are obtained in Figure 1. Though theoretically unjustified, we use not Meyer but the celebrated Daubechies CMFs with N = 8 zero moments. The advantage of these CMFs is that they have finite length equal to 2N. In particular, the filters u j in 8.3 are then also finite of length 2N + 2 for any j. The filters v j, however, are not finite and are truncated in practice, disregarding those elements that are smaller than a prescribed level δ = Let us also note that applications of 6.1 involve more elements of X j than those plotted in Figure 1. This is achieved by taking the initial approximation X 0 of suitable length. Some indication on how this is done, can be seen from the analogous simulation of fractional Brownian motion in Pipiras Finally, let us indicate another interesting feature of the above simulation. Focus on the filters v j defined by 8.3. They have infinite length and are truncated in practice. It may seem from the definition 8.3 that v j have to be taken of very long length as j increases because the elements of the filter 1 e λ2 j+1 e ix 1 = e λ2 j+1k e ixk decay extremely slowly for larger j. In fact, the opposite turns out to be true. As j increases, the filters v j can essentially be taken of finite length 2N 2, and things get even better for larger j in a way! To explain why this happens, recall e.g. Mallat 1998, Theorem 7.4 that N zero moments k=0 23

24 translates into the factorization vx = 1 e ix N v 0,N x, 8.4 where, in the case of Daubechies CMF v, the filters v 0,N have also finite length. An explanation follows by observing that 1 e ix 1 e λ2 j+1 e = a j ix k e ixk 1, or a j 0 1, a k 0 0, k 1, as j. More precisely, 1 e ix 1 e λ2 j+1 e 1 = e ix 1 e λ2 j+1 = 1 e λ2 j+1 e λ2 j+1 k 1 e ixk, ix 1 e λ2 j+1 e ix so that the elements a j k, k 1, are bounded by 1 e λ2 j+1 λ2 j+1 0, as j. k=0 k=1 A On the integration of stationary Gaussian processes Let {Xt} t be a Gaussian stationary process given by 1.1. We define here the integral Xtftdt, A.1 for suitable functions f and state its properties as used throughout the paper. No proofs will be given as most of them are standard. Our strategy will be to define A.1 both pathwise and as an L 2 Ω limit and to show that the two definitions coincide in relevant cases. In the pathwise case, the integral A.1 will be denoted by I ω f i.e. defined ω-wise, and, in the L 2 Ω case, it will be denoted by I 2 f. For simplicity, we assume that the sample paths of X are continuous. Path continuity is not a stringent assumption since, by Belayev s alternative Belayev 1960, either the sample paths of a Gaussian stationary process are continuous or very badly-behaved in the sense of possessing discontinuities of the second type. Assume first that ft = n i=1 f i1 [ai,b i t is a step function. For such function, the stochastic integral A.1 may be defined pathwise as the ordinary iemann integral n n bi I ω f i 1 [ai,b i ] = Xtdt. A.2 a i i=1 Lemma A.1 The integral A.2 has the following properties: for step functions f, f 1 and f 2, and with the notation If = I ω f: P1 If is a Gaussian random variable with mean zero. P2 The following moment formulae hold: EIf 2 = 1 ĝx 2 fx 2 dx; A.3 [ ] E If 1 If 2 [ ] E IfXt = 1 = 1 i=1 P3 For real c and d, Icf 1 + df 2 = cif 1 + dif ĝx 2 f1 x f 2 xdx; e itx ĝx 2 fxdx. A.4 A.5

25 An extension of the integral A.1 to more general functions f can be achieved by an argument of approximation in L 2 Ω. Consider the space of deterministic functions { L 2 g := f L 2 : fx } 2 ĝx 2 dx < A.6 with the inner product Denote also equipped with the ordinary L 2 Ω inner product f 1, f 2 L 2 g := 1 f 1 x f 2 x ĝx 2 dx. A.7 { } IX s = I ω f : f is a step function, A.8 EI ω f 1 I ω f 2. A.9 The space IX s and the restriction of L2 g to step functions are isometric since, for elementary functions f 1 and f 2, EI ω f 1 I ω f 2 = 1 f 1 x f 2 x ĝx 2 dx = f 1, f 2 L 2 g. A.10 Thus, a natural way to define the integral I 2 for a given f L 2 g is to take a sequence of step functions l n that approximate f in the L 2 g norm, and set I 2 f as the corresponding L 2 Ω limit of I ω l n. The following result can be proved as Lemma 5.1 in Pipiras and Taqqu Lemma A.2 For every function f L 2 g, there is a sequence {l n } of step functions such that f l n L 2 g 0. Given f L 2 g, we may use Lemma A.2 to define A.1 as I 2 f = liml 2 ΩI ω l n, A.11 where {l n } is a sequence of step functions such that f l n L 2 g 0. This definition does not depend on the approximating sequence of f. The integral I 2 f has the following properties. Theorem A.1 The map I 2 : f I 2 f defined by A.11 is an isometry between the spaces L 2 g and I X = {I 2 f : f L 2 g}. Moreover, I 2 f = I ω f a.s. for step functions f, and the integral I 2 f satisfies the properties P1, P2 and P3 of Lemma A.1 with If = I 2 f and f, f 1, f 2 L 2 g. It is possible to define A.1 also pathwise for more general integrand functions. As discussed in Section 7, for a Gaussian stationary process {Xt} t, we have, almost surely, Xt C log2 + t, t, A.12 where C is a random variable. Consider the space L := {f L 1 : log2 + t ft dt < }. For f L, in view of A.12 we may define I ω f = Xtftdt A.13 pathwise as an improper iemann integral. One can show that I 2 f = I ω f a.s. for f L L 2 g. 25

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27 Vidakovic, B. 1999, Statistical Modeling by Wavelets, John Wiley & Sons Inc., New York. Walter, G. G. & Shen, X. 2001, Wavelets and Other Orthogonal Systems, Studies in Advanced Mathematics, second edn, Chapman & Hall/CC, Boca aton, FL. Zhang, J. & Walter, G. 1994, A wavelet-based KL-like expansion for wide-sense stationary random processes, IEEE Transactions on Signal Processing 427, Gustavo Didier and Vladas Pipiras Dept. of Statistics and Operations esearch UNC at Chapel Hill CB#3260, Smith Bldg. Chapel Hill, NC 27599, USA didier@unc.edu, pipiras@ .unc.edu 27