1 Correction of Theorem 2.2 Let Ba,bbe a weighted fractional Brownian motion with parameters a＞?1,|b|＜1,|b|＜a+1.Consider the parameter estimation problem for the non-ergodic case Ornstein-Uhlenbeck process with θ＞0.Using the LSE(see,for example,Hu and Nualart[2])def i ned by where the integralis interpreted as a Young integral(see Young[4]). For the SDE(1.1),the explicit solution is given by where the stochastic integralis a Young integral.De fi ne By using the equation(1.1)and(1.3),we can rewrite the LSE bθtdef i ned in(1.2)as follows The proof of Theorem 2.2 in Shen et al.[3]is based on Lemma 3.4,but Lemma 3.4 is incorrect,so Theorem 2.2 is incomplete.Here is the corrected result. Theorem 1.1 Assume thatas t tends to in fi nity,where C(1)denotes the standard Cauchy distribution. In order to prove Theorem 1.1,we need the following lemmas. Lemma 1.2 Assume that＜a＜0,?a＜b＜1+a.Let F be any σ(Ba,b)-measurable random variable such that P(F＜∞)=1.Then, when t tends to inf i nity,where N～N(0,1)is independent of Ba,b. Proof For any d≥1,s1···sd∈[0,∞),observe that F is σ(Ba,b)-measurable it is enough to prove that To get(1.5),it is suffi cient to check the convergence of its covariance matrix,since the left-hand side in the previous convergence is a Gaussian vector(see,Es-Sebaiy and Nourdin[1]).Let us fi rst prove that the limiting variance ofexists as t→∞.We have Proof First,we prove the convergence(1.6).In fact, [1]Es-Sebaiy K,Nourdin I.Parameter estimation for α fractional bridges.Springer Proc Math Statist,2013, 34:385–412 [2]Hu Y,Nualart D.Parameter estimation for fractional Ornstein-Uhlenbeck process.Stat Probab Lett,2010, 80:1030–1038 [3]Shen G,Yin X,Yan L.Least squares estimation for Ornstein-Uhlenbeck processes driven by the weighted fractional Brownian motion.Acta Math Sci,2016,36B(2):394–408 [4]Young L C.An inequality of the H¨older type connected with Stieltjes integration.Acta Math,1936,67: 251–282 1 Correction of Theorem 2.2 Let Ba,bbe a weighted fractional Brownian motion with parameters a＞?1,|b|＜1,|b|＜a+1.Consider the parameter estimation problem for the non-ergodic case Ornstein-Uhlenbeck process with θ＞0.Using the LSE(see,for example,Hu and Nualart[2])def i ned by where the integralis interpreted as a Young integral(see Young[4]). For the SDE(1.1),the explicit solution is given by where the stochastic integralis a Young integral.De fi ne By using the equation(1.1)and(1.3),we can rewrite the LSE bθtdef i ned in(1.2)as follows The proof of Theorem 2.2 in Shen et al.[3]is based on Lemma 3.4,but Lemma 3.4 is incorrect,so Theorem 2.2 is incomplete.Here is the corrected result. Theorem 1.1 Assume thatas t tends to in fi nity,where C(1)denotes the standard Cauchy distribution. In order to prove Theorem 1.1,we need the following lemmas. Lemma 1.2 Assume that＜a＜0,?a＜b＜1+a.Let F be any σ(Ba,b)-measurable random variable such that P(F＜∞)=1.Then, when t tends to inf i nity,where N～N(0,1)is independent of Ba,b. Proof For any d≥1,s1···sd∈[0,∞),observe that F is σ(Ba,b)-measurable it is enough to prove that To get(1.5),it is suffi cient to check the convergence of its covariance matrix,since the left-hand side in the previous convergence is a Gaussian vector(see,Es-Sebaiy and Nourdin[1]).Let us fi rst prove that the limiting variance ofexists as t→∞.We have Proof First,we prove the convergence(1.6).In fact, [1]Es-Sebaiy K,Nourdin I.Parameter estimation for α fractional bridges.Springer Proc Math Statist,2013, 34:385–412 [2]Hu Y,Nualart D.Parameter estimation for fractional Ornstein-Uhlenbeck process.Stat Probab Lett,2010, 80:1030–1038 [3]Shen G,Yin X,Yan L.Least squares estimation for Ornstein-Uhlenbeck processes driven by the weighted fractional Brownian motion.Acta Math Sci,2016,36B(2):394–408 [4]Young L C.An inequality of the H¨older type connected with Stieltjes integration.Acta Math,1936,67: 251–282

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