L²-optimal MA-fBM approximation error · Type I · T = 1
γ endpoints scale per-point as
exp(±α√K / ((1 − H or H)·A(H))), with α = 1.06418 and A(I)(H) = √(1/H + 1/(1 − H)) — same formula for all H ∈ (0, 1), since the history term from the stationary initial condition dominates the head in both regimes. See proof below.
Error vs K
0.3
empirical: \(\mathcal{E}^{(I)}(\omega^*)\)
predicted rate: \(E_0\cdot\exp(-(2\alpha/A^{(I)})(\sqrt K - \sqrt{K_0}))\)
Type I convergence rate
Let \(\omega^*\) minimize the time-integrated MSE \(I(\omega) = \int_0^T \mathbb{E}\!\left[\,(B^H_t - \hat B^H_t)^2\right] dt\) on geometric nodes \(\gamma_k = \gamma_{\min}^{1-z_k}\,\gamma_{\max}^{z_k}\), \(z_k = (k-1)/(K-1)\), \(k=1,\dots,K\). With endpoints scaled per-point as
\[ \gamma_{\min} \asymp \exp\!\Big(\!-\frac{\alpha\sqrt K}{(d - H)\,A^{(I)}}\Big), \qquad \gamma_{\max} \asymp \exp\!\Big(\!+\frac{\alpha\sqrt K}{H\,A^{(I)}}\Big), \]where \(\alpha = 1.06418\) (Bayer & Breneis 2023) and
\[ A^{(I)}(H) = \sqrt{\frac{1}{H} + \frac{1}{1 - H}}, \qquad \text{(}d = 1\text{ throughout, all } H \in (0, 1)\text{)} \]the error obeys
\[ \mathcal{E}^{(I)}(\omega^*) \;\le\; C(H, T)\,\exp\!\Big(\!-\frac{2\alpha}{A^{(I)}}\sqrt K\Big). \]
Proof sketch (all H ∈ (0, 1)).
Same quasi-optimality argument as Type II: \(\omega^*\) minimizes \(I^{(I)}(\omega)\), so it suffices to exhibit good comparison weights on the geometric nodes. The Type I error functional gains a history term over the infinite past, from the stationary initial condition (thesis Eqs. 3.15, 3.18): \(\int_0^\infty [\Delta G(t+r) - \Delta G(r)]^2 dr\). The integrand asymptotic at large \(r\) is \(\sim t^2(H-1/2)^2 r^{2H-3}\) — same form for both \(H \lessgtr 1/2\) (the kernel's tail decay is \(r^{H-3/2}\) throughout, and the difference structure of stationary increments keeps the integral convergent for any \(H \in (0, 1)\)). Integrating \(r^{2H-3}\) over \([1/\gamma_{\min}, \infty)\) gives \(\gamma_{\min}^{2-2H}\), so
\[\text{head}_{I} \sim \gamma_{\min}^{2 - 2H}, \qquad \text{tail}_{I} \sim \gamma_{\max}^{-2H}.\]
For H > 1/2 the [0, t] piece of the integral picks up an additional \(\gamma_{\min}^{5-2H}\) contribution (Type II's smooth-regime head), but since \(5 - 2H > 2 - 2H\), it's strictly sub-dominant. Balancing head against tail yields a single uniform constant
\[A^{(I)}(H) = \sqrt{1/H + 1/(1 - H)}, \qquad \forall\, H \in (0, 1).\]
Same superpolynomial rate class as Type II, modestly slower constant since \(A^{(I)}(H) > A^{(II)}(H)\) for all \(H\) (numerically check at \(H = 1/2\): \(A^{(I)} = 2\) vs \(A^{(II,<)} = \sqrt{3}\) or \(A^{(II,>)} = \sqrt{5/2}\)).
At H = 0.3: \(A^{(I)}\) = —, slope \(2\alpha/A^{(I)}\) = —.