Spline approximation of random processes and design problems

Journal of Statistical Planning and Inference 84 (2000) 249–262

www.elsevier.com/locate/jspi

Spline approximation of random processes and design problems Oleg Seleznjev1 Faculty of Mathematics and Mechanics, Moscow State University, 119 899, Moscow, Russia Received 29 October 1997; accepted 2 April 1999

Abstract We consider the spline approximation of a continuous (continuously dierentiable) random process with nite second moments based on n observations of the process (and its derivatives). The performance of the approximation is measured by mean errors (e.g., integrated or maximal quadratic mean errors). For Hermite interpolation splines, an optimal rule sets n observation locations (i.e., a design, a mesh). While, for a xed n, an optimal rule is dicult to construct, we nd the sequence of designs with asymptotically optimal properties as n → ∞. We investigate the class of locally stationary random processes whose local behavior is like m-fold integrated c 2000 fractional Brownian motion for a given non-negative m in the quadratic mean sense. Elsevier Science B.V. All rights reserved. MSC: 62M20 Keywords: Approximation; Random process; Spline function; Sacks–Ylvisaker conditions; Asymptotically optimal design; Regular sequence; Fractional Brownian motion

1. Introduction Suppose a random process X (t); t ∈ [0; 1], with nite second moment is observed in a nite number of points. At any unsampled point t, we approximate the value of the process by a spline function. The approximation performance on the entire interval is measured by mean errors. In the following, we deal with two problems: the investigation of the spline interpolation accuracy for X , and the construction of the sequence of sampling designs (i.e., meshes) with optimal, in a certain sense, properties for the Hermite interpolation splines. For the rst problem, we treat Holder’s classes of random functions; for the latter, the class of locally stationary processes is studied. 1 Supported in part by the Swedish government program for East and Central European exchange, by Royal Swedish Academy of Science Grant 1247, by European Community Grant (Esprit Project 21042), and by Russian RFFI Grant 98-01-00524. E-mail address: [email protected] (O. Seleznjev)

c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 9 9 ) 0 0 1 0 8 - 1

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The problems of random process approximation arise, for example, in numerical analysis of random functions, when error bounds are estimated; in time-series analysis for “smoothing” of data; in simulation studies of functional distributions, where the functionals are de ned on realizations of a random process (see Eplett, 1986); in stochastic dierential equations, and in environmental sciences (e.g., the kriging approximation, see Christakos, 1992). For certain types of continuous covariance functions satisfying the Sacks–Ylvisaker conditions (Sacks and Ylvisaker, 1966), optimal designs and asymptotic properties of the best linear unbiased estimators and piecewise linear interpolators are studied by Su and Cambanis (1993), Muller-Gronbach (1996). Some extensions of these results for random elds can be found in Ritter et al. (1995), Muller-Gronbach and Schwabe (1996) and Ritter (1996) contains the most recent results and the very detailed survey of this approach. Closely related problems in the approximation theory of non-random functions are considered in de Boor (1972), Burchard and Hale (1975), Stechkin and Subbotin (1976). For equidistant knots, the piecewise linear interpolation of continuous and continuously dierentiable processes is considered in Piterbarg and Seleznjev (1994), Seleznjev (1996) in quadratic mean (q.m.) and uniform metric. Our aim is here to extend some of these results to certain classes of non-stationary continuous or continuously dierentiable random processes, where non-stationarity means the local stationary condition introduced by Berman (1974). While we study the Hermite spline interpolation of a random process, the proposed approach can be also applied to dierent linear approximation problems (e.g., numerical integration and dierentiation). Various applications of splines in statistics and probability can be found in Karlin et al. (1986), Wahba (1990), and Weba (1992). This paper is organized as follows. In the rst part of Section 2, we consider the approximation of Holder’s classes of random functions, and study the accuracy for the Hermite spline approximation. The proofs are based on the q.m. variant of the Peano kernel theorem. In the second part, we derive several properties of locally stationary processes, which behave locally like m-fold integrated fractional Brownian motion in the q.m. sense. Then, for this class of processes, we obtain the precise form of the density generating the asymptotically optimal regular sequence of sampling designs for Hermite interpolation splines. We also compare the results here to some related works. Section 3 contains the proofs of the statements from Section 2.

1.1. Basic notation Let X = X (t); t ∈ [0; 1], be de ned on the probability space ( ; F; P). Assume that, for every t, the random variable X (t) lies in the normed linear space L2 ( ) = L2 ( ; F; P) of random variables with nite second moments and identi ed equivalent elements with respect to P. We set |||| = (E2 )1=2 for all  ∈ L2 ( ). Henceforth, we consider the approximation by random splines, which is based on the normed linear space Cm [0; 1] of random processes having continuous q.m. derivatives up to order

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m¿0. We de ne the norm for any X ∈ Cm [0; 1] by setting !1=p Z ||X ||p =

1

0

||X (t)||p dt

;

16p¡∞;

and ||X ||∞ = max[0; 1] ||X (t)||. The space C m [0; 1] of non-random functions with continuous derivatives up to order m can be considered as a linear subspace of Cm [0; 1] by usual embedding. Let X (t) be sampled at the distinct design points Tn = (t0 ; t1 ; : : : ; tn ) (also referred to as knots), and the set of all (n + 1)-point designs be denoted by Dn = {Tn : 0 = t0 ¡ t1 ¡ · · · ¡ tn = 1}. We suppress the argument n for the design points tk = tk (n) from Tn ; k = 1; : : : ; n, doing so causes no confusion. The following notation will be also used: P k is the set of polynomials of degree k. For a given function f ∈ C m [0; 1]; m¿0, and a design T = (t0 ; t1 ; : : : ; tn ) let Hk (f; T ); k = 1; 3; : : : ; 2m + 1, denote the corresponding Hermite interpolation splines (i.e., the piecewise Hermite polynomials) Hk( j) (ti ) = f( j) (ti ), where j = 0; : : : ; l (k = 2l + 1) and i = 0; : : : ; n. In particular, H1 (t) is a standard piecewise linear interpolator (also referred to as a broken line). We adopt the convention that the corresponding stochastic counterparts of polynomial and functional sets are denoted through the use of script symbols. We suppose in the following that X ∈ Cm [0; 1] and its rst l6m derivatives (i.e., k = 2l + 162m + 1) can be also sampled {X (j) (t); j = 0; : : : ; l; t ∈ Tn }. Since optimal designs for a xed n are dicult to construct, Sacks and Ylvisaker (1966) developed asymptotic solutions for certain time series models. We use the variant of this approach for linear approximation problems proposed by Su and Cambanis (1993). For Hermite interpolation splines, we de ne asymptotic optimality of the sequence of sampling designs Tn∗ by  lim ||X − Hk (X; Tn∗ )||p inf ||X − Hk (X; T )||p = 1 : n→∞

T ∈Dn

We consider regular sequences (RS) of sampling designs {Tn = Tn (h)} generated by a positive continuous density function h(·) via Z ti h(t) dt = i=n; i = 1; : : : ; n; (1) 0

i.e., the sampling points of Tn are i=n percentiles of the density h(·). We denote this property of {Tn } by: {Tn } is RS(h). If h(·) is uniform over [0; 1] (h(t) ≡ 1), then the regular sampling becomes the equidistant (periodic) sampling including the endpoints.

2. Results First, we state some results about the rate of random process approximation by stochastic splines. Then, the optimal RS is investigated for a given rate of approximation.

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2.1. Holder classes of random processes and Peano kernels Let the mth q.m. (m¿0) derivative of the process X (t) satisfy Holder’s condition for all t; t + s ∈ [0; 1], ||X (m) (t + s) − X (m) (t)||6C|s| ;

0 ¡ 61

(2)

for a positive constant C. Denote the class of processes, satisfying (2) over [0; 1] by Cm; (C) = Cm; ([0; 1]; C). For the dierentiable case, m¿1, the estimate of the approximation error can be obtained as a consequence of the following proposition which is a q.m. variant of the well-known Peano kernel theorem (see, e.g., Davis, 1975, p. 69). We formulate this proposition for any closed interval [a; b] ⊆ [0; 1]. De ne linear operators of the following type over Cm [a; b]: ! Z b P P (i) (i) X (s)gi (s) ds + bij X (sij ) ; (3) R(X ) = i6m

j6ki

a

where the integrals are taken in quadratic mean. The functions gi (t) are assumed to be piecewise continuous over [a; b] and all points sij ∈ [a; b]. Further, for any s ∈ [a; b] and a given m, let (t)+ = max(0; t), and ps; m (t) = (t − s)m + ; t ∈ [a; b], be the function of t for a given s. Proposition 1 (q.m. Peano kernel theorem). Let R be of the form (3); and R(Y ) = 0 if Y ∈ P m [a; b]. Then; for all X ∈ Cm+1 [a; b]; Z R(X ) =

b a

X (m+1) (s)Km (s) ds;

(4)

where the Peano kernel Km (s) = R(ps; m )=m!; s ∈ [a; b]: Example 1. For a given point t in [0; 1], denote by Km; k (t; s) the Peano kernel (as a function of s) for the Hermite interpolation. In particular, for the two-point piecewise linear interpolation of X ∈ C1 [0; 1] at the interval [0; 1], we have for a given t,  1 − t; if 06s ¡ t; K0; 1 (t; s) = −t; if t6s ¡ 1 and R(X ) = X (t) − H1 (t) =

R1 0

X (1) (s)K0; 1 (t; s) ds.

More examples and applications of Peano kernels in the conventional (deterministic) approximation theory can be found in Sard (1963) and Davis (1975). Let the mesh size 6c=n; c ¿ 0. Note that for RS, the last property of the mesh size follows directly from the mean-value theorem of integrals. Applying (4), we now obtain the following estimates.

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Proposition 2. Suppose X ∈ Cm; ([0; 1]; C); and Cm; k =maxt∈[0; 1] ||Km−1; k (t; ·)||2 ; where 16m6k62m − 1; then ||X − Hk (X )||∞ 6CCm; k m+ 6C1 n−(m+ ) ;

C1 ¿ 0:

The rate of approximation n−(m+ ) is optimal in a certain sense for the class Cm; (C) (see Seleznjev, 1989; Buslaev and Seleznjev, 1997). For particular classes of random processes Cm; (C), there are estimates for the rate of approximation also in Seleznjev (1991), Weba (1992), and Ritter (1996) (more references can be found in Ritter (1996)). Continuous case (m = 0) is considered in Seleznjev (1996). Note that the constant Cm; k does not depend on X . This is one of the most useful features of Peano kernels for dierent applications in approximation theory and numerical analysis, and hence one can apply the corresponding results from the conventional approximation theory. If, in addition, the sample paths of X are continuously dierentiable up to order m + 1 (for a normal process, the sucient conditions for sample path dierentiability can be found in Dudley (1973)) and k = m, then the following explicit estimate (see, e.g., Davis, 1975, p. 67) can be used: ||X − Hm (X )||∞ 6C2−(m+1) =(m + 1)!||X (m+1) ||∞ m+1 6C1 n−(m+1) ;

C1 ¿ 0:

2.2. Hermite interpolation of a locally stationary process. Optimal sampling We now proceed to the investigation of optimal regular sequences of sampling designs for a piecewise Hermite interpolation. To state the following result, an additional notation will be introduced. Berman (1974) calls a random process Y (t); t ∈ [0; 1], locally stationary if there exists a positive continuous function c(t) such that, for some 0 ¡ 61, lim

s→0

||Y (t + s) − Y (t)|| = c(t)1=2 |s|

uniformly in t ∈ [0; 1]:

(5)

This property characterizes both stationarity and q.m. smoothness of Y (t) (say, fractional smoothness, and if = 1; c(t) = ||Y 0 (t)||2 ). Denote by Bm; (c(·)) = Bm; ([0; 1]; c(·)) the class of processes X ∈ Cm [0; 1] with the locally stationary mth derivative X m (t), satisfying (5) over [0; 1]. Clearly, Bm; (c(·)) ⊂ Cm; ([0; 1]; C) for some positive C. Let Z ∈ B0; ([0; 1]; cZ (·)); 0 ¡ ¡ 1, be a zero mean process with stationary increments (e.g., fractional Brownian motion with ||Z(t + s) − Z(t)|| = |s| ), and therefore cZ (t) ≡ cZ . Simple non-stationary example of zero mean locally stationary random processes is a time transformation of Z(·): Y (t) = Z(a(t)) for a continuous monotone function a(·), and if a(·) is continuously dierentiable, then cY (t) = cZ (a(t))|a0 (t)|2 , (see Husler, 1994). More examples can be obtained from the following Proposition 3. Let Z ∈ Bm; ([0; 1]; cZ (·)); 06m; 0 ¡ ¡ 1; be a zero mean process and Y (t) = a(t)Z(t) + f(t); where the functions a; f ∈ C m;  ([0; 1]; C); ¡ 61; and a(t) 6= 0; t ∈ [0; 1]. Then Y has also the locally stationary mth derivative; and Y ∈ Bm; ([0; 1]; cY (·)) with cY (t) = cZ (t)a(t)2 .

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For a zero mean random process B(t); t ∈ [0; 1], with the covariance function r(t; s)= (|t|2 + |s|2 − |t − s|2 )=2; 0 ¡ 61, (e.g., fractional Brownian motion) de ne the process Bm (t) by setting Z 1 1 (t − s)m−1 m¿1; B0 (t) = B(t) Bm (t) := + B(s) ds; (m − 1)! 0 which has Bm( j) (0) = 0; j = 0; : : : ; m − 1; Bm(m) (t) = B(t). Let Rk; 2 (Bm )(t) represent the remainder for the two-point Hermite interpolation of Bm (t) by the Hermite spline Hk (Bm )(t) at the interval [0; 1] (cf., Example 1) with the norm bm; k;p =||Rk; 2 (Bm )||p . Let

R := 1=(m+ +1=p), if 16p ¡ ∞, and := 1=(m+ ), if p=∞. Write h∗ (t) := c(t) =2 = 1 c(s) =2 ds; t ∈ [0; 1]. 0 Theorem 1. Let X ∈ Bm; ([0; 1]; c(·)); 0 ¡ ¡ 1; with the mean f ∈ C m;  ([0; 1]; C); ¡ 61; be interpolated by Hermite splines with m6k. (i) If Tn is RS(h); then 1=2 1=p−1= ||p ¿ 0; lim nm+ ||X − Hk (X; Tn )||p = bm; k;p · ||c h

n→∞

(ii) Tn is optimal i Tn = Tn∗ ; where Tn∗ is RS(h∗ ). For the optimal Tn∗ , 1=2 lim nm+ ||X − Hk (X; Tn∗ )||p = bm; k;p · ||c || ¿ 0:

n→∞

So, if the mth derivative of the process has stationary increments, then the asymptotic optimal design is uniform, as one could expect. As an example, we consider in more detail the most frequently used method of this type, namely the piecewise linear interpolation for the integrated (p = 2) and maximal (p = ∞) q.m. errors. Example 2. Let 0 ¡ ¡ 1. For piecewise linear interpolation, one can nd directly (see also Piterbarg and Seleznjev, 1994; Seleznjev, 1996) the following values of the constants bm; 1;p : (i) if p = 2, then 1 1 − ; (2 + 1)( + 1) 6   1 1 1 1; 2 − ; (b1;2 ) = (2 + 1)(2 + 2) 6 (2 + 3)( + 2)

2 (b0; 1;2 ) =

(ii) if p = ∞, then 2 (b0; 1; ∞ ) = s ;

2 (b1; 1; ∞ ) =

1 (2 + 1)(2 + 2)



1 1 − 2 +2 4 2

 ;

where s is the maximal value of the function S(v)=v2 (1−v)+v(1−v)2 −v(1−v); v ∈ 2 −2 − 2−2 , if 0 6 ¡ 1, where 0 is the [0; 1]. It is shown that (b0; 1; ∞ ) = S(1=2) = 2

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255

unique solution of the equation 2 (3 − 2 )21−2 = 1 over the interval [0; 1] ( 0 ≈ 0:1026).

Remark. (i) Let us compare the above results for piecewise Hermite interpolation with those for best linear unbiased estimates (BLUEs). Up until recently, the only investigated and available case was a certain class of continuous random processes whose local behavior is like Brownian motion, at least in the q.m. sense. The conditions, which de ne this class of processes, have been initially proposed by Sacks and Ylvisaker (1966) for close regression problems. This case has been considered in two papers, Su and Cambanis (1993) and Muller-Gronbach (1996). These conditions and the investigation method allow them to nd the asymptotically optimal sampling design density for BLUEs in the linear approximation problem. As examples, when these conditions do not hold, one can consider fractional Brownian motion with 6= 12 or stationary zero-mean process with the stable-type covariance function exp{−c|t − s|2 }, where c ¿ 0; ∈ (0; 1]; 6= 12 . For the case = 12 , let R(t; s) be a covariance function of a zero-mean random process, and for the partial derivative R(0; 1) (t; s) , the values of R(0; 1) (t; t+) and R(0; 1) (t; t−) are nite. The main result of Su and Cambanis (1993) (as well as Muller-Gronbach, 1996) is formulated in terms of the function A(t) = R(0; 1) (t; t−) − R(0; 1) (t; t+). This function can be found in dierent optimal sampling problems (see, e.g., Cambanis, 1985). Clearly, in our notation, we 1=2 = 6−1=2 , which does agree with have A(t) = c(t); p = 2; m = 0; = 12 , with b0; 1;2 Su and Cambanis (1993). Notice that the BLUE and the piecewise linear interpolator have the same optimal performance for this case (see Su and Cambanis, 1993). The result is not so unexpected, since, for normal stationary processes and equidistant meshes, equivalence in terms of the maximal q.m. error for regression and ordinary broken lines has been also shown in Belyaev and Simonyan (1979). Therefore, the results of piecewise Hermite interpolation may serve at least as upper bounds for those of BLUEs. Let us remark that the construction of BLUEs requires the precise knowledge of the covariance function, and hence are not robust (see Su and Cambanis, 1993). Thus, the simple nonparametric piecewise polynomial estimates, and especially broken lines, may be useful. (ii) Note that the type of optimal density, h∗ (t) = const · c =2 (t), does not depend on the used method (for non-random close results, see de Boor (1972); Burchard and Hale (1975)). It provides further support to the conjecture of Eubank et al. (1982) (for linear approximation problems, see also Muller-Gronbach and Ritter, 1997) that densities of such type are optimal for general classes of random processes. (iii) In practice, the function c(t) and the values of the corresponding derivatives of an initial approximated random process must be estimated by using conventional splines de ned on the ner (equidistant) mesh of knots. Then, these approximated values and Hermite splines will be ultimately used (see Micchelli and Wahba, 1981; Stechkin and Subbotin, 1976, p. 165). The problem of estimating the function A(·) is addressed in Istas (1996) and Muller-Gronbach and Ritter (1997).

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We now proceed to the cases not included in Theorem 1 but important for some applications. The present theory dealing with optimal RSs of sampling designs for BLUEs requires that the process has exactly the nite number of continuous derivatives (see, e.g., Cambanis, 1985). Therefore, in nitely dierentiable processes (e.g., the processes with band-limited spectra) cannot be investigated by these methods, since BLUEs t the smoothness of arbitrary order, and hence no optimal samplings are available. This property is essential for BLUEs, but this is not the case for piecewise polynomials. For interpolation by splines of order k and smoothness m (in particular, for Hermite splines) in the conventional approximation theory, the smoothness of order m = k + 1 is an upper bound for the optimal rate of approximation (cf., the ordinary Taylor polynomials). We investigate these cases for stochastic Hermite splines (for the deterministic case, cf., Kornejchuk (1984, p. 306)) in the following theorem where the notation of Theorem 1 is used. Let B(t; s) is the beta function for t; s ¿ 0. Theorem 2. Let X ∈ Bm;1 ([0; 1]; c(·)); with the mean f ∈ C m+1;  ([0; 1]; C); 0 ¡ 61; be interpolated by the Hermite splines = 1=(k + 1 + 1=p) (i) If m¿k and Tn is RS(h); then 1=2 1=p−1= ||p ¿ 0: lim nk+1 ||X − Hk (X; Tn )||p = bk;1 k;p · ||c h

n→∞

The sequence of designs Tn is optimal i Tn =Tn∗ ; where Tn∗ is RS(h∗ ). For the optimal Tn∗ ; 1=2 lim nk+1 ||X − Hk (X; Tn∗ )||p = bk;1 k;p · ||c || ¿ 0;

n→∞

1=p where bk;1 =(k + 1)!; 16p ¡ ∞; and bk;1 k;p = B(p(k + 1)=2 + 1; p(k + 1)=2 + 1) k;∞ = −(k+1) =(k + 1)!. 2 (ii) If m ¡ k; then limn→∞ nm+1 ||X − Hk (X; Tn∗ )||p = 0.

Example 3. Consider the approximation by Hermite splines with a given number of interpolating knots. In order to compare the asymptotical approximation accuracy for the optimal and uniform (i.e., hu (t) ≡ 1) sampling designs, we de ne the following coecient p : p = lim ||X − Hk (X; Tn (h))||p =||X − Hk (X; Tn∗ )||p = ||c1=2 ||p =||c1=2 || : n→∞

The latter equality follows from Theorems 1 and 2. Let Xi (t) = ci (t)1=2 Bm (t); t ∈[0; 1]; i = 1; 2, where Bm (t) is integrated fractional Brownian motion. (i) Let c1 (t) = 1=(t + a)2d ; a ¿ 0; d ¿ 1=2, and p = 2. By de nition, we obtain that, for 0 ¡ ¡ 1=d, 2 (a) = ((a1−2d − (a + 1)1−2d )=(2d − 1))1=2 (((a + 1)1−d − a1−d )=(1 − d ))−1= ∼ (1 − d )1= (2d − 1)−1=2 a−(d−1=2)

as a → 0:

(6)

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Fig. 1. The behavior of the uniform=optimal coecients 2 (a) (solid line) for X1 (t) = (t + a)−3 B3 (t); = 12 , and ∞ (a) (dashed line) for X2 (t) = et=(2a) B1 (t); = 1.

(ii) Let c2 (t) = et=a ; a ¿ 0; p = ∞. Then, ∞ (a) = ( =(2a))1= (1 − e− =(2a) )−1= ∼ ( =2)1= a−1=

as a → 0:

(7)

Fig. 1 shows the behavior of the uniform/optimal coecients for (i) and (ii), when a ∈ [0:05; 0:3]. Let d = 3, m = 3, and = 12 (i.e., = 14 ) for (i), and m = 1 and = 1 (i.e., = 12 ) for (ii). 3. Proofs Proof of Proposition 1. The proof is evident and repeats that for non-random functions (see, e.g., Davis, 1975, p. 69). In fact, applying Taylor’s theorem of q.m. calculus (see, e.g., Loeve, 1979) yields ! Z b 1 (m+1) m R X (s)(t − s)+ ds (8) R(X ) = m! a since the linear functional R(·) vanishes on P m [a; b]. Finally, we have assumed form (3) for R(·), and therefore the integral in (8) and R(·) are interchangeable. We shall use several properties of Peano kernels which follow directly from the de nition. ∗ Proposition 4. Let Km; k (t; s); t; s ∈ [0; 1]; be a Peano kernel for the two-point Hermite interpolation; m¿1.

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R1

K ∗ k (t; s) ds = 0; R0 1 m−1; ∗ (ii) If m ¡ k62m − 1; then 0 sKm−1; k (t; s) ds = 0; R1 ∗ (iii) If k = m; then 0 sKm−1; k (t; s) ds = t (k+1)=2 (1 − t)(k+1)=2 =(k + 1)!. (i) If m6k62m − 1; then

Proof. Let pm (t) = t m =m!. By the de nition of a Peano kernel, we have Z 1 Km−1; k (t; s) ds ≡ 0; t ∈ [0; 1] Rk; 2 (pm )(t) = 0

since Rk; 2 vanishes on P m and (i) follows. In a similar way, we obtain (ii). The last property follows from the well-known property of the Hermite interpolation (see, e.g., Davis, 1975, p. 67): there exists a point t1 ∈ [0; 1] such that Rk; 2 (f)(t) = f(k+1) (t1 )t (k+1)=2 (t − 1)(k+1)=2 =(k + 1)! by setting f(t) = t k+1 =(k + 1)!. Proof of Proposition 2. Applying Propositions 1 and 4(ii) to the Hermite spline approximation, we have, for t ∈ [tj−1 ; tj ], !2 Z tj 2 m m (X (s) − X (tj−1 ))Km−1; k (t; s) ds ||X (t) − Hk (X )(t)|| = E tj−1

6 C 2 h2 j

Z

tj

tj−1

Km−1; k (t; s)2 ds;

and the assertion follows by changing the variables. Proof of Theorem 1. First, we investigate the asymptotic behaviour of the q.m. error ek; n (t):=||X (t) − Hk (t)||2 for any t ∈ [tj−1 ; tj ], i = 1; : : : ; n, when the number of knots n tends to in nity. We shall use an explicit formula for the continuous case (m = 0) and the q.m. Peano kernel representation for the dierentiable case (m¿1). Further, we nd the asymptotic form of ||(X − Hk (X; Tn )||p for any density h(·) generating a regular sequence. Finally, assertion (ii) will follow directly from (i) as a property of means. We are coming now to the detailed presentation of the proof. Throughout the proof, the next property of c(·) will be used: c(t + s) = c(t)(1 + rt (s));

rt (s) → 0

as s → 0 uniformly in t ∈ [0; 1]

(9)

which follows directly from the uniform continuity and positiveness of c(·). Consider at rst the continuous case and piecewise linear interpolation by a Hermite spline H1 (·) (i.e., a broken line). We denote the incremental variance of a random process Z(·) by dZ (t; s) = ||Z(t) − Z(s)||2 . Suppose t ∈ [tj−1 ; tj ]. We represent the q.m. error e1; n (t) as follows: e1; n (t) = ||X (t) − (1 − t)X (tj−1 ) − t X (tj )||2 = (1 − t)d(t; tj−1 ) + td(t; tj ) − (1 − t)t d(tj−1 ; tj );

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where t = (t − tj−1 )=hj ; hj :=tj − tj−1 (06t61). From (9) and the local stationarity condition (5), we now obtain   e1; n (t)2 = c(tj )h2 j + ((1 − t )t

2

+ t(1 − t)2 − (1 − t)t)(1 + rn; j (t))

 = c(tj )h2 j S(t )(1 + rn; j (t));

(10)

where n = max{|rn; j (t)|; t ∈ [tj−1 ; tj ]; j = 0; : : : ; n} = o(1) as n → ∞ and S(t) = ||B(t) − t B(1)||2 = ||R1;2 (B)(t)||2 ;

||S||p = b0; 1;p :

Note that b0; 1;p ¿ 0, otherwise fractional Brownian motion is degenerated at [0; 1] (see P ug, 1982; Seleznjev, 1996). Thus, we get

 e1; n (t) = c(tj )h2 j S(t )(1 + o(1))

as n → ∞ uniformly in t ∈ [0; 1]:

(11)

In a similar way, by using the explicit form of Hermite polynomials, we obtain that for k = 2m + 1; ek; n (t) = c(tj )h2m+2 ||Rk; n (X )(t)||2 (1 + o(1)) as n → ∞ uniformly in t ∈ [0; 1]. In the dierentiable case, for X ∈ Cm [0; 1] and m6k62m − 1, Proposition 1 gives the following form of the remainder, for t ∈ [tj−1 ; tj ], Z 1 Z tj X (m) (s)Km−1; k (t; s) ds = X (m) (s)Km−1; k (t; s) ds Rk; n (X )(t) = tj−1

0

Z =

tj

tj−1

(X (m) (s) − X (m) (tj−1 ))Km−1; k (t; s) ds:

(12)

The last equality is a consequence of Proposition 4(i). Substituting the new variable s = (s − tj )=hj in (12) yields Z 1 m ∗ (Y (s)  − Y (0))Km−1;  d s; (13) Rk; n (X )(t) = hj k (t; s) 0

∗ where Y (s)  = X (tj−1 + hj s),  and Km−1;l (t; s)  denotes the corresponding Peano kernel for the two-point Hermite interpolation at [0; 1], (m)

∗  = 1=(m − 1)!Rk; 2 ((· − s)  m−1 Km−1; k (t; s) + ):

From (13), we obtain that the q.m. error can be represented as follows: ek; n (t) = ||Rk; n (X )(t)||2 Z 1Z 1 ∗ ∗ = h2m E[(Y (v) − Y (0))(Y (w) − Y (0))]Km−1; j k (t; v)Km−1; k (t; w) dv dw 0

= 1=2hm j

Z

0 1

0

Z 0

1

∗ (dY (v; 0) + dY (w; 0) − dY (v; w))Km−1; k (t; v)

∗ ×Km−1; k (t; w) dv dw:

By using (9) again, we nd ek; n (t) = 1=2 ·

c(tj )h2m+2 j

Z ·

0

1

Z 0

1

∗ (|v|2 + |w|2 − |v − w|2 )Km−1; k (t; v)


∗ × Km−1; k (t; w) dv dw (1 + o(1))

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O. Seleznjev / Journal of Statistical Planning and Inference 84 (2000) 249–262



Z

= c(tj )h2m+2 E j

!2  ∗  (1 + o(1)) B(v)Km−1; k (t; v) dv

1

0

= c(tj )h2m+2 S(t)(1 j

+ o(1))

as n → ∞

(14)

uniformly in t ∈ [0; 1], where S(t) = ||Rk; 2 (Bm )(t)||2 . Relation (14) corresponds to that for the continuous case, and with similar arguments, we have bm; k;p ¿ 0. Further, the integral mean-value theorem and the de nition of the RS density h(·) imply that hj = 1=(h(wj )n)

for some wj ∈ [tj−1 ; tj ];

j = 1; : : : ; n:

(15)

Now (11), (14), and (15) together give the following representation of ||X −Hk (X; Tn )||p both for continuous and dierentiable cases: Z tj n P p ek; n (t)p=2 dt ||(X − Hk (X; Tn )||p = j=1 n P

=

j=1

tj−1

Z

c(tj )p=2 hmp+ p+1 j

0

n P

p −(m+ )p = (bm; k;p ) n

j=1 n P

p −(m+ )p = (bm; k;p ) n

1

j=1

! p=2

S(t)

d t (1 + o(1)) !

c(tj )p=2 =h(wj )(m+ )p hj

(1 + o(1))

! c(wj )

p=2

=h(wj )

1−p=

hj

(1 + o(1))

as n → ∞. Since the function c(t)p=2 =h(t)1−p= is Riemann integrable, we get lim n

n→∞

m+

||X −

Hk (X; Tn )||p = bm; k;p

Z 0

1

!1=p p=2

c(t)

= ||c1=2 h1=p−1= ||p

1−p=

=h(t)

dt

as n → ∞

(16)

and assertion (i) follows. For p = ∞, assertion (i) follows similarly. We proceed to prove (ii). By using the standard argument based on the monotone property of integral means (see Hardy et al., 1964, p. 143), we obtain ||c

1=2 1=p−1=

h

Z ||p ¿

0

1

!1=

=2

−1

c(t) h(t)

h(t) dt

= ||c1=2 ||

unless c(t) =2 h(t)−1 ≡ const. as was to be shown. Proof of Theorem 2. The proof of (i) repeats those of Theorem 1 for dierentiable case. The explicit form of bk;1 k;p is a consequence of Proposition 4(ii). In fact, if

O. Seleznjev / Journal of Statistical Planning and Inference 84 (2000) 249–262

16p ¡ ∞, then  Z 1 bk;1 =  k;p

Z

0

Z =

0

1

0

1

!p=2

261

1=p dt 

∗ sKm−1;l (t; s) ds

!1=p t

p(k+1)=2

(1 − t)

p(k+1)=2

dt

=(k + 1)!

= B(p(k + 1)=2 + 1; p(k + 1)=2 + 1)1=p =(k + 1)!

(17)

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