Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

The aim of the present paper is to investigate series representations of the Riemann–Liouville process R α , α > 1 / 2, generated by classical orthonormal bases in L 2 [ 0,1 ] . Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of R α via the trigonometric system possesses the optimal convergence rate if and only if 1 / 2 < α ≤ 2. For the Haar system we have an optimal approximation rate if 1 / 2 < α < 3 / 2 while for α > 3 / 2 a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases α > 3 / 2 and α = 3 / 2. However, in this latter case the question whether or not the series representation is optimal remains open. 1 .


Introduction
Let X = (X (t)) t∈T be a centered Gaussian process over a compact metric space (T, d) possessing a.s. continuous paths. Then it admits a representation with (ε k ) k≥1 i.i.d. standard (real) normal random variables and with continuous real-valued functions ψ k on T . Moreover, the right hand sum converges a.s. uniformly on T . Since representation (1.1) is not unique, one may ask for optimal ones, i.e. those for which the error sup t∈T ∞ k=n ε k ψ k (t) (1.2) tends to zero, as n → ∞, of the best possible order. During past years several optimal representations were found, e.g. for the fractional Brownian motion, the Lévy fractional motion or for the Riemann-Liouville process (cf. [7], [2], [5], [8], [1], [18] and [20]). Thereby the representing functions ψ k were either constructed by suitable wavelets or by Bessel functions.
In spite of this progress a, to our opinion, natural question remained unanswered. Are the "classical" representations also optimal ? To make this more precise, suppose that the process X has a.s. continuous paths and admits an integral representation for some interval I ⊆ and with white noise W on I. Given any ONB Φ = (ϕ k ) k≥1 in L 2 (I) we set ψ k (t) := I K(t, x)ϕ k (x) dx .
By the Itô-Nisio-Theorem (cf. [10], Theorem 2.1.1) the sum in (1.1) converges a.s. uniformly on T , thus leading to a series representation of X . For example, if I = [0, 1], then one may choose for Φ natural bases as e.g. the ONB of the trigonometric functions T = {1} ∪ 2 cos(kπ · ) : k ≥ 1 or that of the Haar functions H (see (5.2)). There is no evidence that in some interesting cases these "classical" bases do not lead to optimal expansions as well.
The aim of the present paper is to investigate those questions for the Riemann-Liouville process R α defined by This process is known to have a.s. continuous paths whenever α > 1/2 (cf. [15] for further properties of this process). Thus, for example, representation (1.1) of R α by the basis T leads to R α (t) = ε 0 t α Γ(α + 1) (1.3) and in similar way it may be represented by the Haar system H as (1.4) where the h j,k are the usual Haar functions. The basic question we investigate is whether or not representations (1.3) and (1.4) are optimal. The answer is quite surprising.
For the proof we refer to Theorems 4.1, 5.4, 5.5 and 5.11. As recently shown by M. A. Lifshits (oral communication) representation (1.4) is also not optimal for α = 3/2. Let us recall that the assertions for α > 2 or α ≥ 3/2, respectively, say that these bases are not only non-optimal in their natural order but also after any rearrangement of the bases.
Even in the cases where these representations are not optimal it might be of interest how fast (or slow) the error in (1.2) tends to zero as n → ∞. Here we have lower and upper estimates which differ by log n.
Another process, tightly related to R α , is the Weyl process I α which is stationary and 1-periodic. It may be defined, for example, by (1.5) Here (ε k ) and (ε l ) are two independent sequences of i.i.d. standard (real) normal random variables. We refer to [3] or [14] for more information about this process.
In fact, (1.5) is already a series representation and we shall prove in Theorem 4.8 that it is optimal for all α > 1/2. In comparison with Theorem 1.1 this is quite unexpected. Note that the processes R α and I α differ by a very smooth process (cf. [3]). Moreover, if α > 1/2 is an integer, then their difference is even a process of finite rank.

Approximation by a Fixed Basis
Let be a separable Hilbert space. Then ( , E) denotes the set of those (bounded) operators u from into a Banach space E for which the sum converges a.s. in E for one (then for each) ONB Φ = (ϕ k ) k≥1 in . As above (ε k ) k≥1 denotes an i.i.d. sequence of standard (real) normal random variables. If u ∈ ( , E), we set which is independent of the special choice of the basis.
For u ∈ ( , E) the sequence of l-approximation numbers is then defined as follows: Note that, of course, l(u) = l 1 (u) ≥ l 2 (u) ≥ · · · ≥ 0 and l n (u) → 0 as n → ∞ and that We refer to [1], [9] or [19] for more information about these numbers.
For our purposes we need to specify the definition of the l-approximation numbers as follows. Let Φ = (ϕ k ) k≥1 be a fixed ONB in the Hilbert space . Then we define the l-approximation numbers of u with respect to Φ by Let us state some properties of l Φ n (u) for later use.
Proposition 2.1. Let u ∈ ( , E) and let Φ = (ϕ k ) k≥1 be some ONB in . Then the following are valid.
(2) If u 1 , u 2 ∈ ( , E), then it follows that (3) If v is a (bounded) operator from E into another Banach space F , then it holds Proof. Properties (1), (2) and (4) follow directly from the definition of l Φ n (u). Thus we omit their proofs. In order to verify (3) let us choose a subset N ⊂ with #N < n such that for some given > 0. Then we get which completes the proof of (3) by letting → 0.
Property (5) is an easy consequence of for any elements x k in a Hilbert space . Note, moreover, that all moments of a Gaussian vector are equivalent by Fernique's Theorem (cf. [6]).

Remark 2.1.
It is worthwhile to mention that in general rk(u) < n does not imply l Φ n (u) = 0. This is in contrast to the properties of the usual l-approximation numbers.
In order to define optimality of a given representation in its natural order we have to introduce a quantity tightly related to l Φ n (u). For u ∈ ( , E) and an ONB Φ = (ϕ k ) k≥1 in we set The "o" in the notation indicates that l o,Φ n (u) depends on the order of the elements in Φ while, of course, l Φ n (u) does not depend on it. Clearly l Φ n (u) ≤ l o,Φ n (u) and, moreover, it is not difficult to show (cf. Prop. 2.1 in [9]) that l Φ n (u) ≤ c 1 n −α (log n) β for some α > 0 and β ∈ implies l o,Φ n (u) ≤ c 2 n −α (log n) β where Φ coincides with Φ after a suitable rearrangement of its elements.
We may now introduce the notion of optimality for a given basis (cf. also [9] and [1]). In particular, the approximation error l o,Φ n (u) tends to zero slower than the optimal rate.
For later purposes we state the following result. Since the proof is quite standard we omit it (cf. also Prop. 2.1 in [9]).

Proposition 2.2.
Suppose that there is a constant c 1 > 0 such that for α > 0, β ∈ and all n ∈ Then this implies The following lemma is elementary, but very helpful to prove lower estimates for l Φ n (u).

Then this implies l
Proof. Let N ⊂ be an arbitrary subset of cardinality strictly less than n. Set L := M ∩ N c . Clearly, #L > m − n, hence the assumption leads to Taking on the left hand side of (2.1) the infimum over all subsets N with #N < n proves the assertion.
Let us shortly indicate how the preceding statements are related to the problem of finding optimal series expansions of Gaussian processes. If the process X = (X (t)) t∈T is represented by a sequence Ψ = (ψ k ) k≥1 as in (1.1), then we choose an arbitrary separable Hilbert space and an ONB a unique operator u ∈ ( , C(T )) is defined. Of course, this construction implies l Φ n (u) = l Ψ n (X ) where the latter expression is defined by As shown in [1] we have l n (u) = l n (X ) where Hence representation (1.1) is optimal for X , i.e. there is some c > 0 such that for all n ≥ 1 we have l o,Ψ n (X ) ≤ c l n (X ), if and only if this is so for Φ and u related to Ψ = (ψ k ) k≥1 via (2.2). In the same way Φ is rearrangement non-optimal for u if and only if this is so for the representing functions ψ k of X . Consequently, all our results about series expansions may be formulated either in the language of ONB and operators u ∈ ( , C(T )) or in that of series expansions of centered Gaussian processes X = (X (t)) t∈T with a.s. continuous paths.

A General Approach
We start with a quite general result which is in fact the abstract version of Theorem 5.7 in [9]. Before let us recall the definition of the covering numbers of a compact metric space (T, d). If > 0, then we set With these notation the following is valid. for some γ > 0. Let u ∈ ( , C(T )) and let Φ = (ϕ k ) k≥1 be some fixed ONB in . Suppose there are β > 0 and α > 1/2 such that for all k ≥ 1 and all t, s ∈ T we have Then for each n ≥ 1 it follows that Proof. Let n be a decreasing sequence of positive numbers which will be specified later on. Set N n := N (T, d, n ) and note that (3.1) implies Then there are t 1 , . . . , t N n ∈ T such that T = N n j=1 B j where B j := B(t j , n ) are the open d-balls in T with radius n and center t j . To simplify the notation set J n := 2 n , . . . , 2 n+1 − 1 and write We estimate both terms in (3.6) separately.
Hence the first term in (3.6) can be estimated by In order to estimate the second term in (3.6) we need the following result (cf. Lemma 4.14 in [19]).

Lemma 3.2.
There is a constant c > 0 such that for any N ≥ 1 and any centered Gaussian sequence Z 1 , . . . , Z N one has, We apply (3.8) to Z j = k∈J n ε k (uϕ k )(t j ) and use (3.3). Then similar arguments as in the proof of Theorem 5.7 in [9], p. 686, lead to sup 1≤ j≤N n k∈J n Summing up, (3.6), (3.7) and (3.9) yield Now we choose n := 2 −δn with δ > (α + 1/2)/β. By (3.5) we get log N n ≤ c n and by the choice of δ the first term in (3.10) is of lower order than the second one. This implies as asserted and completes the proof.  [17] and also the proof of Theorem 1 in [16]).
Let us formulate the preceding result in probabilistic language. We shall do so in a quite general way. Let X = (X (t)) t∈T be an a.s. bounded centered Gaussian process on an arbitrary index set T (we do not suppose that there is a metric on T ). Define the Dudley metric d X on T by Since X is a.s. bounded (T, d X ) is known to be compact (cf. [6]). We assume that (T, d X ) satisfies a certain degree of compactness, i.e., we assume that for some γ > 0. Suppose now that where the right hand sum converges a.s. uniformly on T . Note that the ψ k are necessarily continuous with respect to d X . This easily follows from Then the following holds: Then this implies Proof. We will prove the proposition by using Corollary 3.3. Choose and Φ = (ϕ k ) k≥1 as above and construct u as in (2.2). Then (3.1) and (3.3) hold by assumption and it remains to show that (3.2) is satisfied. Yet this follows by the special choice of the metric on T . Indeed, for h ∈ and t, s ∈ T by (3.13) we have Consequently, (3.2) holds with β = 1 and the assertion follows by (2.2) and by Corollary 3.3.

The Trigonometric System
The aim of this section is to investigate whether or not the ONB Equivalently, we may ask whether or not the representation of the Riemann-Liouville process R α = (R α (t)) 0≤t≤1 given by is an optimal one.
We claim that (3.2) holds for all α > 1/2 with In order to verify (3.3) we first mention the following general result: This is a direct consequence of the well-known fact that ∞ 0 s α cos(s) ds as well as ∞ 0 s α sin(s) ds exist for those α.
The following lemma, which is more or less similar to Lemma 4 in [16], shows that (3.3) holds for α ≤ 2.
To complete the proof of Theorem 4.1 we have to show that the basis T is rearrangement nonoptimal for R α whenever α > 2. To verify this we need the following lemma.
Proof. We start with 2 < α < 3. Using (4.5) we get Another integration by parts gives where Consequently, (4.7) and (4.8) lead to which proves our assertion in the case 2 < α ≤ 3 where α = 3 follows by direct calculations.
Suppose now 3 < α < 4. Another integration by parts in the integral defining g k gives g k = c αg k with sup k≥1 g k ∞ < ∞. Then the above arguments lead to (4.6) in this case as well.
We may proceed in that way (in the next step a term of order k −4 appears) for all α > 2. This completes the proof of the lemma. Now we are in position to complete the proof of Theorem 4.1. This is done by using Lemma 2.3. In the notation of this lemma we set M := {1, . . . , 2n} for some given n ≥ 1, hence we have m = 2n. Let L be an arbitrary subset in M with #L > m − n. Using that all moments of Gaussian sums are equivalent it follows that where the last estimate follows by Lemma 4.3. Because of #L > n and L ⊆ M we have Since L ⊆ M was arbitrary with #L > m − n, Lemma 2.3 leads to Yet l n (R α ) ≈ n −α+1/2 log n, thus (4.9) shows that T is rearrangement non-optimal for α > 2.

Corollary 4.4.
If α > 2, then are constants c 1 , c 2 > 0 only depending on α such that Proof. The left hand estimate in (4.10) was proved in (4.9). In order to verify the right hand one we use property (3) of Proposition 2.1. If α > 2 this implies This completes the proof.

Remark 4.2.
We conjecture that for α > 2 the right hand side of (4.10) is the correct asymptotic of l T n (R α ).
Proof. We start with the case 1/2 < α ≤ 1. By using the same method as in the proof of Lemma 5.6 in [9] we have where as before ϕ k ∈T are ordered in the natural way. Condition (3.2) holds by the same arguments as in the proof of Theorem 4.1. Thus Corollary 3.3 applies and proves thatT is optimal.
To treat the case α > 1 we need the following lemma.
Lemma 4.6. If α > 1, then it follows that Proof. Suppose first 1 < α < 2 and write with sup k g k ∞ < ∞. Clearly, from this we derive (recall α > 1) that On the other hand, completing the proof of the lemma in that case.
If α = 2 the assertion follows by direct calculations and for α > 2 as in the proof of Theorem 4.1 we integrate by parts as long as we get in (4.12) an exponent of s which is in (−1, 0). where we used (4.11) in the last step. The set L ⊂ M was arbitrary with #L > n, hence, by Lemma 2.3 we obtain lT n (R α ) ≥ c n −1/2 .
In view of l n (R α ) ≈ n −α+1/2 log n this implies thatT is rearrangement non-optimal whenever α > 1 completing the proof.
Finally we investigate series representations of the Weyl process. Recall the definition of the Weyl operator of fractional integration. For α > 1/2, the Weyl operator I α is given on exponential functions for all t ∈ [0, 1] by where for α / ∈ , the denominator has to be understood as  Proof. Let e n (u) denote the n-th (dyadic) entropy number of an operator u from into a Banach space E (cf. [4] for more information about these numbers). As proved in [11], Proposition 2.1, whenever an operator u ∈ ( , E) satisfies e n (u) ≤ c 1 n −a (log n) β for some a > 1/2 and β ∈ , then this implies Moreover, as shown in [3], for any α > 1/2 it follows that e n (R α − I α ) ≤ c 1 e −c 2 n 1/3 .
In particular, we have e n (R α − I α ) ≤ c γ n −γ for any γ > 0. Thus, by the above implication, for any γ > 0 holds l n (R α − I α ) ≤ c γ n −γ as well. Of course, since l 2n−1 (I α ) ≤ l n (I α − R α ) + l n (R α ) by l n (R α ) ≈ n −α+1/2 log n we get l n (I α ) ≤ c n −α+1/2 log n. The reverse estimate is proved by exactly the same methods. This completes the proof.

Remark 4.3.
The upper estimate l n (I α ) ≤ c n −α+1/2 log n may also be derived from Theorem 4.8 below.
Proof. Direct calculations give as well as

Consequently, condition (3.3) in Proposition 3.4 holds for I α and T .
We claim now that for 1/2 < α < 3/2 it follows that for all h ∈ L 0 2 [0, 1] and t, s ∈ [0, 1]. This is probably well-known, yet since it is easy to prove we shortly verify (4.13). It is a direct consequence of for any k ≥ 1 and > 0 small enough. Clearly, (4.13) shows that (3.2) holds with β = α − 1/2 as long as 1/2 < α < 3/2. The identity I α 1 • I α 2 = I α 1 +α 2 implies the following: Suppose that for some α > 1/2 and β ∈ (0, 1]. Then for any α ≥ α estimate (4.14) is also valid (with the same β). This, for example, follows from the fact that (4.14) is equivalent to Remark 4.4. It may be a little bit surprising that for all α > 1/2 the basis T is optimal for I α whilẽ T is not for R α in the case α > 1. Recall that l n (I α − R α ) tends to zero exponentially and, if α ∈ , then I α and R α differ only by a finite rank operator, i.e., we even have l n (I α − R α ) = 0 for large n. The deeper reason for this phenomenon is that l T n (I α − R α ) tends to zero slower than l T n (I α ).

Some useful notations and some preliminary results
Recall (cf. (1.4)) that for any parameter α > 1/2, the Riemann-Liouville process can be written as where R α is the Riemann-Liouville operator and the h j,k 's are the usual Haar functions, i.e.
and where the series converges almost surely uniformly in t (i.e. in the sense of the norm · ∞ ). For any t ∈ [0, 1] and J ∈ we set, and In order to conveniently express the coefficients (R α h j,k )(t), for any reals ν and x we set Then it follows from (4.1), (5.2) and (5.6) that one has for every integers j ∈ 0 := ∪ {0} and 0 ≤ k ≤ 2 j − 1 and real t ∈ [0, 1], Let us now give some useful lemmas. The following lemma can be proved similarly to Lemma 1 in [2], this is why we omit its proof.
Let us now give a lemma that allows to control the increments of the Riemann-Liouville process. This result is probably known however we will give its proof for the sake of completeness.

Lemma 5.2.
(i) For any α ∈ (1/2, 3/2), there is a random variable C 2 > 0 of finite moment of any order such that almost surely for every t 1 , t 2 ∈ [0, 1], (ii) For any α > 3/2, there is a random variable C 3 > 0 of finite moment of any order such that one has almost surely for every t 1 , t 2 ∈ [0, 1], Proof. (of Lemma 5.2) Part (ii) is a straightforward consequence of the fact that the trajectories of R α are continuously differentiable functions when α > 3/2. Let us now prove part (i). First observe (see for instance relation (7.6) in [12]) that inequality (5.8) is satisfied when the Riemann-Liouville process is replaced by the fractional Brownian motion (fBm) (B H (t)) 0≤t≤1 with Hurst index H := α − 1/2. Recall that for some c α > 0 (again H and α are related via where the process (Q α (t)) t∈[0,1] is called the low-frequency part of fBm and up to a positive constant it is defined as Finally, it is well-known that the trajectories of the process (Q α (t)) t∈ [0,1] are C ∞ -functions. Therefore, it follows from (5.9) that (R α (t)) 0≤t≤1 satisfies (5.8) as well.
Remark 5.1. It is very likely that (5.8) also holds for α = 3/2. But in this case our approach does not apply. Observe that α = 3/2 corresponds to H = 1 and (5.9) is no longer valid.
For any reals γ > 0 and t ∈ [0, 1] and for any integers j ∈ 0 and 0 ≤ k ≤ 2 j − 1 we set Observe that (5.10) and (5.7) imply that Furthermore, we denote by k j (t) the unique integer satisfying the following property: with the convention that k j (1) = 2 j − 1.
The following lemma allows us to estimate (R α h j,k )(t) suitably.
(ii) There is a constant c 4 > 0, only depending on γ, such that the inequality holds when 0 ≤ k ≤ k j (t).

Optimality when 1/2 < α < 1
The goal of this section is to prove the following theorem.
Theorem 5.4. Suppose 1/2 < α < 1. Then there is a random variable C 7 > 0 of finite moments of any order such that one has almost surely, for every J ∈ , In particular, this implies that in this case representation (5.1) possesses the optimal approximation rate.

Optimality when 1 < α < 3/2
The goal of this subsection is to show that the following theorem holds.
Theorem 5.5. Suppose 1 < α < 3/2. Then there is a constant c 8 > 0 such that for every J ∈ one has In particular, this implies that also in this case representation (5.1) possesses the optimal approximation rate.
First we need to prove some preliminary results.
Proposition 5.6. If 1/2 < α < 3/2, there exists a constant c 9 > 0 such that one has for any J ∈ , Proof. (of Proposition 5.6) It follows from (5.4), (5.11) and parts (i) and (ii) of Lemma 5.3, that Lemma 5.7. For any α ∈ (1, 3/2), there exists a random variable C 10 > 0 of finite moment of any order such that one has almost surely for any real t ∈ [0, 1] and any integer J ∈ , We refer to (5.12) for the definition of the integer k J (t).
In order to be able to prove Lemma 5.7 we need the following lemma.
Lemma 5.8. For any real α > 1, there exists a constant c 11 > 0 such that for all t ∈ [0, 1], J ∈ , j ∈ 0 and k ∈ 0 satisfying 0 ≤ j ≤ J and 0 ≤ k ≤ k j (t), Proof. (Proof of Lemma 5.8) It is clear that (5.20) holds when t = k J (t)2 −J , so we will assume that t = k J (t)2 −J . By applying the Mean Value Theorem, it follows that there exists a ∈ ( k J (t)2 −J , t) such that Observe that one has k j (a) = k j (t) since a ∈ k J (t) . Thus, putting together, (5.21), (5.10) and (5.13) in which we replace t by a and γ by α − 1, we obtain the lemma.
We are now in position to prove Lemma 5.7.
Proof. (of Lemma 5.7) By using Lemma 5.1 and the fact that On the other hand, it follows from Lemma 5.8 that Proof. (of Theorem 5.5) Putting together Lemma 5.9, Lemma 3.2, the fact that σ 2 J ≥ sup 0≤K<2 J ,K∈ 0 σ 2 J (2 −J K) and Proposition 5.6 one obtains the theorem.

The case α = 3/2
Recall (cf. [9]) that l n (R 3/2 ) ≈ n −1 log n; this clearly implies that The goal of this subsection is to show that a slightly stronger result holds, namely the following theorem.
Theorem 5.10. There exists a constant c 14 > 0 such that, for any n ∈ and any set N ⊆ {( j, k) ∈ 2 : 0 ≤ k ≤ 2 j − 1} satisfying #N < n one has where J n ≥ 2 is the unique integer such that 2 J n −2 ≤ n < 2 J n −1 .
In fact by using the same technics as before one can prove that 2 −J J + 1 is the right order of σ J i.e. one has for some constant c 15 > 0 and all J ∈ 0 , σ J ≤ c 15  Clearly, Putting together (5.11) in which we replace α by 3/2, (5.14) in which we replace γ by 3/2, the fact that k J n (l/2 J n ) = l for any integer l satisfying 0 ≤ l ≤ 2 J n − 1 and (5.24), it follows that with the convention that = · · · = 0 whenever we have 2 J n − 2 ≤ k ≤ 2 J n − 1. Finally combining (5.25) with (5.26) we obtain the theorem

Non-optimality of the Haar basis for α > 3/2
The goal of this subsection is to prove the following theorem.
Theorem 5.11. If α > 3/2, then we have: (i) For any t 0 ∈ (0, 1] there exists a constant c 16 > 0 such that, for each n ∈ and each set N ⊆ {( j, k) ∈ 2 : 0 ≤ k ≤ 2 j − 1}, satisfying #N < n one has There exists a constant c 17 > 0 such that for each J ∈ one has A straightforward consequence of Theorem 5.11 is that the Haar basis H is rearrangement nonoptimal for R α when α > 3/2. More precisely: Corollary 5.12. If α > 3/2, then there are two constant 0 < c ≤ c , only depending on α, such that for each n ≥ 2 one has c n −1 ≤ l H n (R α ) ≤ c n −1 log n . Let us now assume that 3/2 < α < 2; then (5.31), the fact that is an increasing function on [0, k J n (t 0 )] and (5.12) imply that   In order to be able to prove part (ii) of Theorem 5.11 we need some preliminary results. Let us assume that 3/2 < α < 2; then the fact that is an increasing function on [0, k J n (t 0 ) + 1) implies where c 20 > 0 is a constant only depending on α. Next let us assume that α ≥ 2; then the fact that is an nonincreasing function on [−1, k J n (t 0 )] entails that k j (t) one gets that On the other hand, it follows from Lemma 5.8 that Let us assume that 3/2 < α < 2; then one has where c 23 = c 11 1 − 2 3/2−α −1 ∞ l=1 l α−3 < ∞. Next let us assume that α ≥ 2. By using the same technics as in the proof of Proposition 5.13 one can show that there is a constant c 24 > 0, only depending on α, such that for each j ∈ , k j (t)