Operator Fractional Brownian Motion and Martingale Differences

It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays important role in both applications and theory. In this paper, we study the relationship between them. We will construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.


Introduction
Fractional Brownian motion (FBM) is a continuous Gaussian process with stationary increments. It is one of the well-known self-similar processes. Some studies of financial time series and telecommunication networks have shown that this kind of process with longrange dependency-memory might be a better model in some cases than the traditional standard Brownian motion. Due to its applications in the real world and its interesting theoretical properties, fractional Brownian motion has become an object of intense study. One of those studies concerns obtaining its weak limit theorems; see, for example, Enriquez [7], Niemine [15], Sottinent [17], Li and Dai [12] and the reference therein.
Based on the study of FBMs, many authors have proposed a generalization of it, and have obtained many new processes. An extension of FBMs is the operator fractional Brownian motion(OFBM). OFBMs are multivariate analogues of one-dimensional FBMs. They arise in the context of multivariate time series and long range dependence (see, for example, Chung [1], Davidson and de Jong [4], Dolado and Marmol [6], Robinson [16], and Marinucci and Robinson [13]). Another context is that of queuing systems, where reflected OFBMs model the size of multiple queues in particular classes of queuing models. They are also studied in problems related to, for example, large deviations (see Delgado [5], and Konstantopoulos and Lin [9]). Similar to those for FBMs, weak limit theorems for OFBMs [ [2,3] and the references therein.
It is well known that a martingale difference sequence is extremely useful because it imposes much milder restrictions on the memory of the sequence than under independence, yet most limit theorems that hold for an independent sequence will also hold for a martingale difference sequence. In recent years, some researchers have used this type of sequences to construct approximation sequences of some known processes. For example, Nieminen [15] studied the limit theorems for FBMs based on martingale difference sequences. This is a natural motivation for this paper. The direct motivation is the recent works by Dai [2,3], in which, based on a sequence of I.I.D. random variables, the author presented some weak limit theorems for some special kinds of OFBMs.
In this short paper, we establish a weak limit theorem for a special case of OFBMs, which comes from Maejima and Mason [14]. The rest of this paper is organized as follows. In Section 2, we recall OFBMs and martingale-difference sequences, and present the main result of this paper. Section 3 is devoted to prove the main result of this paper.

Operator fractional Brownian motion and Martingale-differences
In this section, we first introduce a special type of OFBMs. Let End(R d ) be the set of linear operators on R d (endomorphisms) and Aut(R d ) be the set of invertible linear operators (automorphisms) in End(R d ). For convenience, we will not distinguish an operator D ∈ End(R d ) from its associated matrix relative to the standard basis of R d . As usual, for c > 0, Throughout this paper, we will use x to denote the usual Euclidean norm of x ∈ R d . Without confusion, for A ∈ End(R d ), we also let A = max x =1 Ax denote the operator norm of A. It is easy to see that for A, B ∈ End(R d ), and Let σ(A) be the collection of all eigenvalues of A. We denote Let x ′ denote the transpose of a vector x ∈ R d . We now extend the fractional Brownian motion of Riemann-Liouville type studied by Lévy [11, p. 357] to the multivariate case.
As is standard for the multivariate context, we assume that RL-OFBM is proper. A random variable in R d is proper if the support of its distribution is not contained in a proper hyperplane of R d . In this short note, we want to obtain an approximation of RL-OFBMs. Inspired by Nieminen [15], we want to construct an approximation sequence of RL-OFBM X by martingale differences. Let for some C ≥ 1.
Lemma 2.1 Under the condition (2.6) and the condition the processes converge in distribution to a Brownian motion B, as n → ∞.
3 Such a type of sequences is very useful, since it is very easy to obtain it in the real world. See, Nieminen [15], for example.
Below, we extend Lemma 2.1 to the d-dimensional case. Define i , F n i } is still a sequence of square integrable martingale differences on the probability space Ω, F , P . Inspired by Lemma 2.1, we have the following lemma.
Lemma 2.2 Under conditions (2.6) and (2.7), the sequence of processes η n (t) converges in law to a d-dimensional Brownian motion W , as n → ∞.
Noting that W i (u), i = 1, · · · , d, are mutually independent, and so are ξ Our main objective in this paper is to explain and prove the following theorem.
In the rest of this paper, most of the estimates contain unspecified constants. An unspecified positive and finite constant will be denoted byK, which may not be the same in each occurrence.

Proof of Theorem 2.1
In order to prove the main result of this paper, we need a technical lemma. Before we state this technical lemma, we first introduce the following notation and The technical lemma follows.
Before we prove it, we need the following lemma which is due to Maejima and Mason [14].
If λ D > 0 and r > 0, then for any δ > 0, there exist positive constants K 1 and K 2 such that Next, we give the detailed proof of Lemma 3.1. Proof of Lemma 3.1: In order to simplify the discussion, we split the proof into two steps.
Step 1. We claim that for any t ∈ [0, 1], Therefore, we have where we have used the Cauchy-Schwartz inequality and by (2.6). Therefore, On the other hand, by (2.2) and Lemma 3.2, since λ D − δ > 1 2 . Therefore, {G 2 n (t, u)} is uniformly integrable. On the other hand, we have for any u ∈ (0, 1], Step 2. We prove the original claim. In order to simplify the discussion, we let t n q = ⌊ntq⌋ n and t n l = ⌊nt l ⌋ n . By (3.5), we can get for t l , t q ∈ [0, 1], as n → ∞.
In fact, it follows from (3.5) that   as n → ∞.

6
For the left-hand side of (3.15), we have On the other hand, using the same method as in the proof of the inequality (3.52) below, By the condition (2.6) and (3.16), (3.15) can be bounded bỹ    for any t l , t q ∈ (0, 1].
Next, we prove the main result of this paper. Before we give the details, we first introduce a technical tool.

Lemma 3.3 can be found in Shiryaev
Proof of Theorem 2.1: We will prove this theorem by two steps.
Step 1: First, we have to show that the finite-dimensional distributions of X n converge to those of X. It suffices to prove that for any q ∈ N, a 1 , · · · , a q ∈ R and t 1 , · · · , t q ∈ [0, 1], q l=1 a l X n (t l ) D → q l=1 a l X(t l ).
By some calculations, we can get that (3.26) is equivalent to In order to simplify the discussion, we definē Hence (3.27) can be rewrote as follows. We will prove (3.29) by Lemma 3.3. We first prove the Lindeberg condition holds in our case. For convenience, define: We have (3.37) Combining (3.33) and (3.37), we have Combining (3.36) and (3.41), one can easily prove that, as n approaches ∞, Hence the Lindeberg condition holds. Next, we show the condition (3.23) holds. We first study the right-hand side of (3.29). We have (t j , s)K(t l , s)ds. In order to simplify the discussion, let t = ⌊nt⌋ n , ands = ⌊ns⌋ n .
Next, we show that where H = λ D − δ.
In fact,