A Gladyshev theorem for trifractional Brownian motion and $n$-th order fractional Brownian motion

We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and $n$-th order fractional Brownian motion. Furthermore, a sufficient condition for the $L^P$-convergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvai\v{s}a.

The well-known fractional Brownian motion is governed by the Hurst parameter H between 0 and 1. However, many observations reveal that the Hurst parameter could be larger than one in real life. Motivated by this fact, Perrin et al. [11] introduced the n-th order fractional Brownian motion as an extension to the fractional Brownian motion. An n-th order fractional Brownian motion B H,n := {B H,n (t), t ∈ [0, ∞)} is a centered Gaussian process with the following covariance function, where n ∈ N, H ∈ (n − 1, n), and C n H = Γ(2H + 1)| sin(πH)| −1 . The class of n-th order fractional Brownian motions allows a wider range of the Hurst parameter H. In other words, the class of n-th order fractional Brownian motions extends the Hurst parameter beyond the constraint H ∈ (0, 1) and includes the case of fractional Brownian motion for n = 1. Besides, an n-th order fractional Brownian motion is still H-self-similar and n-stationary. For a real-valued process X and α ∈ R, we study the limit of as n ↑ ∞, provided it exists in some sense. In particular, our study concerns the convergence results of (1.1) for the trifractional Brownian motion and n-th order fractional Brownian motion for various α.
Our results also lead to a statistical estimator for the self-similarity index for the trifractional Brownian motion.

Statement of main results
For a real-valued stochastic process X = {X(t), t ∈ [0, ∞)}, if there exists γ ∈ R such that lim n↑∞ 2 αn almost surely, we then say γ is the critical exponent of the weighted quadratic variation of the process X. This notion of the critical exponent measures the roughness of the process X; the rougher a process is, the smaller its critical exponent will be. At the critical case α = γ, the left-hand side of (2.1) may be infinite, finite or nonexistent. We refer to [3,4] for a discussion of what can happen for deterministic fractal functions. For Gaussian processes, their critical exponents can be computed by Gladyshev's theorem and its succeeding extensions. For instance, in the very recent complement [10], Norvaiša considers Gaussian processes X with structure functions ψ X of the following form, where d > 0, λ ∈ (0, 2), and b(·, ·) is a symmetric function such that for all ǫ > 0 Under these conditions, Norvaiša shows that the Gaussian process X admits the critical exponent γ = 1 − λ, and the weighted quadratic variation converges to d almost surely at the critical case α = γ. For instance, the results in [10] directly apply to the bifractional Brownian motion, which has a critical exponent 2HK − 1. However, as stated in previous paragraphs, these existing methods fail to calculate the critical exponent for the trifractional Brownian motion and the n-th order fractional Brownian motion. For instance, the trifractional Brownian motion Z H,K has the following structure function, for t, s ∈ [0, ∞). It is clear that d = 0 in this case. Moreover, for the n-th fractional Brownian motion B H,n , one has for t, s ∈ [0, ∞). The constant d = (−1) n+1 C n H could possibly take negative values. The rest of this paper is organized as follows. We establish (2.1) and compute the critical exponent for the trifractional Brownian motion and the n-th order fractional Brownian motion in Theorem 2.1 and Theorem 2.2, respectively. Proposition 2.3 studies the L P -convergence of the weighted quadratic variation, which is needed in the proofs of subsequent theorems. We then prove the limiting theorems at the critical cases for each process in Theorem 2.4 and Theorem 2.5. Finally, Corollary 2.6 constructs a consistent estimator for the self-similarity index of the trifractional Brownian motion. Proofs are given in Section 3. Now we state our main results, let us start with the trifractional Brownian motion.
The following proposition is needed to study limit theorems at the critical cases, i.e., α = 1 in part (b) of Theorem 2.1 and α = 1 in Theorem 2.2. In Proposition 2.3, we provide a sufficient condition for the L P -convergence of (1.1) for α = 1 as n ↑ ∞.
converges to some non-constant random variable in L p as n ↑ ∞ for all p ∈ (1, ∞).
For the convenience of exposition in subsequent proofs, we introduce the following notation: For a centered Gaussian process X, we denote for m, n ∈ N, 1 ≤ j ≤ 2 m and 1 ≤ k ≤ 2 n . In particular, when the arguments m = n, we denote φ in short. In Section 3, we assign the Gaussian process X to be either the trifractional Brownian motion or the n-th order fractional Brownian motion. Therefore, By the condition (2.9), the double sequence (a m,n ) m,n∈N is uniformly bounded. Thus, there exists M > 0 such that sup m,n a m,n ≤ M. Next, for any m, n ∈ N, we have Therefore, one has a m,n = 2 n+m By an analogous argument, one also gets a m,n ≤ a m+1,n . Therefore, for each fixed m ∈ N, there exists a positive non-decreasing sequence (b m ) m∈N such that lim n↑∞ a m,n = lim n↑∞ a n, Next, we will show the double limit lim m,n↑∞ a m,n exists and is equal to b. For any ǫ > 0, there The first inequality holds as the double sequence a m,n is increasing in each argument. Therefore, the double limit lim m,n↑∞ a m,n exists and lim m,n↑∞ a m,n = b > 0. Thus, by virtue of [13, Theorem 2.1], the weighted sum 2 n 2 n k=1 X k 2 n − X k−1 2 n 2 converges to some non-constant random variable in L p for all p ∈ (1, ∞). This completes the proof. Now, we state the convergence results at the critical cases for the trifractional Brownian motion and the n-th order fractional Brownian motion.
converges to some non-constant random variable in L p for all p ∈ (1, ∞) as n ↑ ∞. As observed in Theorem 2.4 and Theorem 2.5, the weighted quadratic variation of the trifractional Brownian motion and the n-th order fractional Brownian motion behaves essentially different to the Gaussian processes that Gladyshev's theorem apply to. To be more specific, if a Gaussian process fulfills conditions in Gladyshev [2] or Norvaiša [10], its weighted quadratic sum (1.1) converges to some positive constant at the critical case α = γ almost surely. However, for the trifractional Brownian motion Z H,K , its weighted quadratic variation converges to some non-constant random variable in L P and in probability for HK > 1/2 and does not converge in probability for HK < 1/2 at each critical case. Furthermore, as in Theorem 2.5, the weighted sum of the n-th order fractional Brownian motion converges to some non-constant random variable at the critical case. These facts further refute the applicability of the existing literature [2,10] to the trifractional Brownian motion and the n-th order fractional Brownian motion.
As previously stated, for many self-similar Gaussian processes, their critical exponents of weighted quadratic variation directly relate to their self-similarity indices. However, a counterexample to this relation is established in part (b) of Theorem 2.1, where the critical exponent of the weighted quadratic variation does not yield the self-similarity index of the trifractional Brownian motion. The same is true for the n-th order fractional Brownian motion when n ≥ 3. Nevertheless, an estimator for the selfsimilarity index HK of the trifractional Brownian motion Z H,K can be constructed for HK ≤ 1 2 , which is the content of the following corollary. This directly implies the result.

Proofs
The proofs of the above theorems intensively rely on proper upper bounds of φ (m,n) j,k for both processes.
In the following lemma, we derive upper bounds of φ (m,n) j,k for the trifractional Brownian motion, which are needed in the proofs of main theorems. Moreover, it suffices to prove all convergence results in the above theorems with T = 1. For arbitrary T > 0, these results hold due to the homogeneity of covariance functions.
Now, let us proceed to prove (2.4). There exists a (dependent) sequence of standard normally distributed random variables {Y n , n ≥ 1}, such that Therefore, for HK < 1 2 , α > 2HK and any given M > 0, we get where γ(k, x) = x 0 t k−1 e −t dt is the lower incomplete gamma function. By change of variables and Taylor expansion, we have as n ↑ ∞. As 2HK − α < 0, the Borel-Cantelli lemma implies that S α As m ≥ 2 and H ∈ (m − 1, m), we have 2H − 2 > 0, and this gives As the m-th order fractional Brownian motion is H-self-similar, then Therefore, E(S α n ) = O(2 (α−1)n ) as n ↑ ∞. Now, let us recall that the Cauchy-Schwarz inequality gives (φ This then gives the fast L 2 -convergence for the case α < 1, which guarantees the almost sure convergence. Hence, the result in (2.7) is then verified. The proof of (2.8) follows analogously as the arguments in the proof of (2.6) in Theorem 2.  j,k ) 2 . Next, we show the limit lim n↑∞ a n,n exists and is strictly positive. By the Cauchy-Schwarz inequality, we have (φ (m,n) k,k , then a n,m ≤ 2 2HK(n+m) From (3.9), if HK < 1/2, the sequence ( 2 m j=1 φ (m) j,j ) m∈N is uniformly bounded. Hence, there exists M > 0, such that sup m,n a m,n ≤ M. As the trifractional Brownian motion is HK-self-similar, then a n,n = Therefore, (a n,n ) n∈N forms a non-decreasing uniformly bounded sequence. Hence, the limit lim n↑∞ a n,n exists and lim n↑∞ a n,n ≥ a 0,0 = E([Z H,K (1)] 2 ) = (2 − 2 K ) > 0. Now, we show that for each fixed n ∈ N, lim m↑∞ a m,n = lim m↑∞ a n,m = 0. For k ≥ 2, inequality If H < 1/4, the sequence (j 4H−2 ) j∈N ∈ ℓ 1 , and 2 2HK(m+n) 2 m j=1 (φ The above inequalities in together imply that for each n ∈ N and 1 ≤ k ≤ 2 n , we have Hence, we have lim m↑∞ a m,n = lim m↑∞ a n,m = 0. However, as previously proved, lim n↑∞ a n,n > 0. Therefore, the double limit (3.16) does not exists, and we then complete the proof of assertion (a). We now prove assertion (b). Following