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Prediction of Fractional Processes with Long-range Dependence

We introduce a class of Gaussian processes with stationary increments which exhibit long-range dependence. The class includes fractional Brownian motion with Hurst parameter H>1/2 as a typical example. We establish infinite and finite past prediction formulas for the processes in which the predictor coefficients are given explicitly in terms of the MA and AR coefficients.


Introduction
Let (X(t) : t ∈ R) be a centered Gaussian process with stationary increments, defined on a probability space (Ω, F , P ), that admits the moving-average representation We will also assume some extra conditions such as lim t→0+ c(t) = ∞, (1.5) , t → ∞, (1.6) where ℓ(t) is a slowly varying function at infinity and H is a constant such that (1.7) 1/2 < H < 1.
Other important examples of (X(t)) are the processes with long-range dependence which, unlike fBm, have two different indices H 0 and H describing the local properties (path properties) and long-time behavior of (X(t)), respectively (see Example 2.4 below). Let t 0 , t 1 and T be real constants such that For I = (−∞, t 1 ] or [−t 0 , t 1 ], we write P I X(T ) for the predictor of the future value X(T ) based on the observable (X(s) : s ∈ I) (see Section 3 below). One of the fundamental prediction problems for (X(t)) is to express P I X(T ) using the segment (X(s) : s ∈ I) and some deterministic quantities. Another is to express the variance of the prediction error P ⊥ I X(T ) := X(T ) − P I X(T ). Results of this type become important tools in the analysis of non-Markovian processes and systems modulated by them (see, e.g., Norros et al. [23], Anh et al. [3], Inoue et al. [19] and Inoue and Nakano [18]). One of our main purposes here is to derive such results for (X(t)).
We establish the following infinite and finite past prediction formulas for (X(t)) (see Theorems 3.8 and 4.12 below): The significance of (1.11) and (1.12) is that the predictor coefficients b(t, s) and h(t, s) are given explicitly in terms of the MA(∞) coefficient c(t) and AR(∞) coefficient a(t) of (X(t)). We will find that a(t) has a nice integral representation similar to (1.3) (see (3.3) below). It turns out that the existence of such a nice AR(∞) coefficient, in addition to the nice MA(∞) coefficient, is a key to the solution to the prediction problems above.
We apply the results above to the proof of Baxter's inequality for (X(t)), which concerns the L 1 -estimate of the difference between the predictor coefficients b(t, s) and h(t, s). The original inequality of Baxter [4] is an assertion for stationary time series (Y n : n ∈ N) with short memory. It takes the form where K is a positive constant, and φ j and φ n,j are the infinite and finite past predictor coefficients in respectively, with P (−∞,−1] Y 0 and P [−n,−1] Y 0 being defined similarly. See Berk [5], Cheng and Pourahmadi [7], and Inoue and Kasahara [17] for related work; for a textbook account, see Pourahmadi [26,Section 7.6.2]. Using the explicit representations of b(t, s) and h(t, s), we can prove an analogue of (1.13) for (X(t)) which are continuous-time stationary-increment processes with long-range dependence. For fBm with 1/2 < H < 1, the predictor coefficients b(t, s) and h(t, s) are given in Gripenberg and Norros [9] (see (3.13) and (5.3) below). See [23] and [25] for different proofs. Fractional Brownian motion has a variety of nice properties, and the methods of proof of [9,23,25] naturally rely on such special properties of fBm, hence are not applicable to (X(t)). The method of this paper is based on the alternating projections to the past and future (see Section 4.1 below). As for fBm with 0 < H < 1/2, its infinite and finite past prediction formulas also exist, and are due to Yaglom [29] and Nuzman and Poor [25], respectively (see also Anh and Inoue [2]); see Inoue and Anh [15] for an extension to these results, which have different forms from (1.11) and (1.12) since no stochastic integrals appear there.
We provide the basic properties and examples of (X(t)) in Section 2. We consider the infinite and finite past prediction problems for (X(t)) in Sections 3 and 4, respectively. In Section 5, we prove an analogue of Baxter's inequality for (X(t)), using the results in Sections 3 and 4.

Basic properties and examples
In this section, we assume (1.2)-(1.4) and Then, as in [15, Therefore, for a one-dimensional standard Brownian motion (W (t) : t ∈ R) with W (0) = 0, we may define the centered stationary-increment Gaussian process (X(t) : t ∈ R) by (1.1).
For s > 0 and t ∈ R, we put ∆ s X(t) := X(t + s) − X(t). Then, by definition, (∆ s X(t) : t ∈ R) is a stationary process.
Lemma 2.2. Let H 0 ∈ (1/2, 1) and ℓ 0 (·) a slowly varying function at infinity. We assume In particular, we have From Lemma 2.2, we see that the index H 0 describes the path properties of (X(t)) (see Adler [1,Section 8.4] Similarly, (2.2) implies ; and so all the conditions above are satisfied. The resulting process (X(t)) is fBm (B H (t)): where (x) + := max(0, x) for x ∈ R. The representation (2.5) of fBm is due to the pioneering work of Mandelbrot and Van Ness [21].
Example 2.4. Let f (·) be a nonnegative, locally integrable function on (0, ∞). For H 0 , H ∈ (1/2, 1) and slowly varying functions ℓ 0 (·) and ℓ(·) at infinity, we assume Thus all the conditions above are satisfied. As we have seen above, the indices H 0 and H describe the path properties and long-time behavior of (X(t)), respectively.

Infinite past prediction problems
In this section, we assume (1.1)-(1.5), (2.1) and Notice that, for the processes (X(t)) in Examples 2.3 and 2.4, all these conditions are satisfied. We also assume (1.10).
We write M (X) for the real Hilbert space spanned by (X(t) : t ∈ R) in L 2 (Ω, F , P ), and · for its norm. Let I be a closed interval of R such as [−t 0 , t 1 ], (−∞, t 1 ], and [−t 0 , ∞). Let M I (X) be the closed subspace of M (X) spanned by (X(t) : t ∈ I). We write P I for the orthogonal projection operator from M (X) to M I (X), and P ⊥ I for its orthogonal complement: . Note that, since (X(t)) is a Gaussian process, we have We define the AR(∞) coefficient a(t) of (X(t)) by We define the positive kernel b(t, s) by Then, by [15,Lemma 3.4], the following equalities hold: This is the class of functions f for which we can define the stochastic integral I f (s)dX(s). We define a subclass H 0 I of H I (X) by Each member of f ∈ H 0 I a simple function on I. We The general case follows easily from this.
Proof. By Proposition 3.2 and the monotone convergence theorem, we have Thus the proposition follows.
For a real-valued function f on I, where {f + n } and {f − n } are arbitrary sequences of non-negative simple functions on From the definition above, we see that I f (s)dX(s) ∈ M I (X) for f ∈ H I (X). The next proposition follows immediately from Proposition 3.3.
Proposition 3.5. The equality (3.6) also holds for f ∈ H I (X).

3.3.
Infinite past prediction formulas. We denote by D(R) the space of all φ ∈ C ∞ (R) with compact support, endowed with the usual topology. For a random distribution Y (cf. [11,Section 2] and [3, Section 2]), we write DY for its derivative. For t ∈ R, we write M (−∞,t] (Y ) for the closed linear hull of . Notice that M I (X) here coincides with that defined above.
As in [15,Proposition 2.4], we have the next proposition.
Proposition 3.6. The derivative DX of (X(t)) is a purely nondeterministic stationary random distribution, and Here is the infinite past prediction formula for , the following assertions hold: it follows from (3.5) and the Fubini-Tonelli theorem that, for s < t, Thus we obtain (a). By Proposition 3. This and Proposition 3.5 yield By (3.7), (3.8) and Proposition 3.5, the integral on the right-hand side is Thus (b) follows.
By putting f (s) = I (t1,T ] (s) in Theorem 3.7 (b), we immediately obtain the next infinite past prediction formula for (X(t)).
Using the Hilbert space isomorphism θ : M (X) → M (X) characterized by θ(X(t)) = X(−t) for t ∈ R, we obtain the next theorem from Theorem 3.7 (see the proof of [3, Theorem 3.6]). Theorem 3.9. For t ∈ [0, ∞) and f ∈ H [t,∞) (X), the following assertions hold: . Then The proof of Proposition 3.10 is the same as that of [3, Proposition 3.5], whence we omit it. We need Theorem 3.9 and Proposition 3.10 in the next section. Example 3.11. As in Example 2.3, we consider fBm (B H (t)) with 1/2 < H < 1. Then the MA(∞) coefficient c(t) is given by By the change of variable u = sv, where we have used the equality and so, from Theorem 3.8, we see that, for 0 ≤ t < T , This prediction formula was obtained in [9, Theorem 3.1] by a different method.
It should be noted that {P n } ∞ n=1 is merely an alternating sequence of projection operators, first to M (−∞,t1] (X), then to M [−t0,∞) (X), and so on. This sequence plays a key role in the proof of the finite past prediction formula for (X(t)).
For t, s ∈ (0, ∞) and n ∈ N, we define b n (t, s) = b n (t, s; t 2 ) iteratively by We may assume that f ≥ 0. By Theorem 3.7, (a) holds for n = 1. By the Fubini-Tonelli theorem, we have, for s > −t 0 , Hence, by Theorem 3.9, we have (b) for n = 2. Repeating this procedure, we obtain the proposition.
Let f ∈ H [t1,∞) (X). By Proposition 4.1, we may define the random variables G n (f ) by     We are about to investigate the limit of (4.2) as n → ∞ (see Lemma 4.9 below).  From the proof of the next proposition, we see that these integrals converge absolutely. Recall (W * (t)) from (3.10).

Proof.
We may assume f ≥ 0. By the Fubini-Tonelli theorem, we have, for t > 0, Thus the assertion holds for n = 0. Now we assume that n ≥ 1. Since we have we obtain the assertion, again using the Fubini-Tonelli theorem.
We need the next L 2 -boundedness theorem.
The proof of Theorem 4.8 is similar to that of [15,Theorem 5.1], whence we omit it.
By putting z = iy in (3.2), we get By Karamata's Tauberian theorem (cf. [6, Theorem 1.7.6]) applied to this, (2.3) implies This and the monotone density theorem give The next lemma is a key to our arguments.
Thus the lemma follows.
We can now state the conclusions of the arguments above.
We can prove Theorem 4.10 using Proposition 4.2 and Lemma 4.9. Since the proof is similar to that of [3, Theorem 4.6], we omit the details.

4.2.
Finite past prediction formulas. We define h(s, u) = h(s, u; t 2 ) by Here is the finite past prediction formula for ∞ t1 f (s)dX(s). Theorem 4.11. Let f ∈ H [t1,∞) (X). Then the following assertions hold: Proof. We may assume that f ≥ 0. By Theorem 4.10 (b), Proposition 4.2 and Lemma 4.9, we have, in M (X), where, for 0 < s < t 2 and u > 0, we define h n (s, u) = h n (s, u; t 2 ) by  Here are the solutions to the finite past prediction problems for (X(t)).

Representation in terms of β.
We define a positive function β(t) by We next derive the representation of the finite past prediction coefficient h(s, u) = h(s, u; t 2 ) in terms of β(t) (and c(t) and a(t)). We need this result in Section 5.2. See [16,17,14] for the usefulness of such expressions in terms of β(t) in the discrete-time setting.
Thus the proposition follows.
For example, we see from (5.2) that, formally, which is (b) of the proposition above. Since we can prove the two assertions rigorously as in the proof of [17, Proposition 3.2], we omit the details. Theorem 5.1 follows immediately from the next more precise result.   as t 0 → ∞. Therefore, by Lemma 5.5 below, we obtain the lemma. Proof. Though the lemma is a general result, we give a proof based on the results for fBm. Thus we take fBm (B H (t)) as (X(t)). Then we have (3.11) and (3.12). Also, by [9], we have (3.13) and Thus the lemma follows.