The paper by Rainer Dahlhaus, titled "Efficient Parameter Estimation for Self-Similar Processes," published in *The Annals of Statistics* in 1989, claims to establish the asymptotic normality and efficiency of the Gaussian maximum likelihood estimator (MLE) for long-range dependent processes, with a focus on self-similar fractional Brownian motion. The conditions (A0)–(A9) are used to derive these results, but condition (A9) is noted to be too restrictive for long-range dependent processes. Specifically, it is shown that (A9) does not hold for certain sequences of parameters as $n$ approaches infinity. To address this issue, the condition (A9) is relaxed to (A9'), which only requires the continuity of $\alpha$. This relaxation allows the results of the original paper to be extended. The proof of consistency for the MLE, particularly the relation on page 1756, line 7, is crucial and is shown to follow from the equicontinuity of a quadratic form. The proof of this equicontinuity is detailed in an extended correction note, which also addresses a gap in the proof of Lemma 6.2. The changes required in the original paper include removing the need for (A9), adjusting the proof of Lemma 5.5, and addressing a specific case in the proof of Lemma 6.2. The author acknowledges Professor Kung-Sik Chan, Professor Hira Koul, and anonymous referees for their contributions and helpful comments.The paper by Rainer Dahlhaus, titled "Efficient Parameter Estimation for Self-Similar Processes," published in *The Annals of Statistics* in 1989, claims to establish the asymptotic normality and efficiency of the Gaussian maximum likelihood estimator (MLE) for long-range dependent processes, with a focus on self-similar fractional Brownian motion. The conditions (A0)–(A9) are used to derive these results, but condition (A9) is noted to be too restrictive for long-range dependent processes. Specifically, it is shown that (A9) does not hold for certain sequences of parameters as $n$ approaches infinity. To address this issue, the condition (A9) is relaxed to (A9'), which only requires the continuity of $\alpha$. This relaxation allows the results of the original paper to be extended. The proof of consistency for the MLE, particularly the relation on page 1756, line 7, is crucial and is shown to follow from the equicontinuity of a quadratic form. The proof of this equicontinuity is detailed in an extended correction note, which also addresses a gap in the proof of Lemma 6.2. The changes required in the original paper include removing the need for (A9), adjusting the proof of Lemma 5.5, and addressing a specific case in the proof of Lemma 6.2. The author acknowledges Professor Kung-Sik Chan, Professor Hira Koul, and anonymous referees for their contributions and helpful comments.