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On the Kolmogorov Distance for the Least Squares Estimator in the Fractional

Ornstein-Uhlenbeck Process

Jaya P. N. Bishwal
Department of Mathematics and Statistics, University of North Carolina at Charlotte,376 Fretwell Bldg, 9201 University City Blvd. Charlotte, NC 28223, USACorrespondence: J.Bishwal@uncc.edu

Abstract. The paper shows that the distribution of the normalized least squares estimator of the driftparameter in the fractional Ornstein-Uhlenbeck process observed over [0,T] converges to the standardnormal distribution with an uniform optimal error bound of the order O(T−1/2) for 0.5 ≤ H ≤ 0.63and of the order O(T4H−3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractionalBrownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squaresestimator, the error bound is of the order O(T−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T4H−3)for 0.69<H < 0.75.
1. IntroductionThe fractional Ornstein-Uhlenbeck process, is an extension of Ornstein-Uhlenbeck process withfractional Brownian motion (fBm) driving term. In finance it is known as fractional Vasicek model,and is being extensively used these days as one-factor short-term interest rate model which takesinto account the long memory effect of the interest rate. The model parameter is usually unknownand must be estimated from data.Parameter estimation in stochastic differential equations is studied in Bishwal [1]. For thestandard Ornstein-Uhlenbeck process, sufficiency and Rao-Blackwellization was studied in Bish-wal [4] where also a time transformation to reduce the general problem to a fixed time case andthe asymptotics were studied in large parameter case. For the fractional Ornstein-Uhlenbeck pro-cess, Berry-Esseen inequalities of minimum contrast estimators based on continuous and discreteobservations was studied in Bishwal [2]. Hu et al. [11] studied parameter estimation for the frac-tional Ornstein-Uhlenbeck process of general Hurst parameter. Bishwal [5] studied Berry-Esseeninequalities for the fractional Black-Karasinski model of term structure of interest rates. Usingfractional Levy process as the driving term which include jumps, maximum quasi-likelihood estima-tion in fractional Levy stochastic volatility model was studied in Bishwal [3]. Parameter estimationin partially observed stochastic differential system was studied in Bishwal [6].

Received: 12 Nov 2022.
Key words and phrases. Itô stochastic differential equation; fractional Brownian motion; fractional Ornstein-Uhlenbeck process; long-memory; least squares estimator; quasi-least squares estimator; rate of weak convergence;Kolmogorov distance; Wiener chaos; Fourier method; analytic continuation.1

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https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 2
Let (Ω,F,{Ft}t≥0,P ) be a stochastic basis on which is defined the Ornstein-Uhlenbeck process

Xt satisfying the Itô stochastic differential equation
dXt = θXtdt + dW

H
t , t ≥ 0, X0 = 0 (1.1)

where {WHt } is a fractional Brownian motion with H > 1/2 with the filtration {Ft}t≥0 and θ < 0is the unknown parameter to be estimated on the basis of continuous observation of the process
{Xt} on the time interval [0,T ].Recall that a fractional Brownian motion (fBM) has the covariance

C̃H(s,t) =
1

2

[
s2H + t2H −|s − t|2H

]
, s,t > 0. (1.2)

For H > 0.5 the process has long range dependence or long memory and the process is self-similar.For H 6= 0.5, the process is neither a Markov process nor a semimartingale. For H = 0.5, theprocess reduces to standard Brownian motion.Note that the solution of the equation (1.1) is given by
Xt =

∫ t
0

eθ(t−s)dWHs . (1.3)

Let the realization {Xt, 0 ≤ t ≤ T} be denoted by XT0 . Let PTθ be the measure generatedon the space (CT ,BT ) of continuous functions on [0,T ] with the associated Borel σ-algebra BTgenerated under the supremum norm by the process XT0 and PT0 be the standard Wiener measure.Applying Girsanov type formula for fBm, when θ is the true value of the parameter, PTθ is absolutelycontinuous with respect to PT0 and the Radon-Nikodym derivative (likelihood) of PTθ with respectto PT0 based on XT0 is given by
LT (θ) :=

dPTθ
dPT0

(XT0 ) = exp

{
θ

∫ T
0

QtdZt −
θ2

2

∫ T
0

Q2tdvt

}
. (1.4)

Consider the score function, the derivative of the log-likelihood function, which is given by
YT (θ) :=

∫ T
0

QtdZt −θ
∫ T
0

Q2tdvt. (1.5)

A solution of YT (θ) = 0 provides the maximum likelihood estimate (MLE)
θT :=

∫T
0
QtdZt∫T

0
Q2tdvt.

(1.6)

Kleptsyna and Le Breton [13] showed that θT is strongly consistent. Using the Fourier method,Bishwal [2] proved a Berry-Esseen type theorem for the estimator θT which gives the rate of weakconvergence in asymptotic normality.Using the fractional Itô formula, the score function YT (θ) can be written as
YT (θ) =

1

2

[
λH

(2 − 2H)
ZT

∫ T
0

t2H−1dZt −T
]
−θ

∫ T
0

Q2tdvt. (1.7)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 3
Consider the contrast function

KT (θ) := −
THΓ(H)

2
−θ

∫ T
0

Q2tdvt (1.8)

and the minimum contrast estimate (MCE)
θ̄T :=

−THΓ(H)
2
∫T
0
Q2tdvt

. (1.9)

The least squares estimator (LSE) of θ minimizes∫ T
0

|Ẋt −θXt|2dt (1.10)

and is given by
θ̂T :=

∫T
0
XtdXt∫T

0
X2t dt.

= θ−
∫T
0
XtdW

H
t∫T

0
X2t dt.

(1.11)

Based on ergodicity, quasi least squares estimate (QLSE)
θ̃T :=

(
−THΓ(2H)∫T
0
X2t dt

) 1
2H

(1.12)

The LSE and the QLSE are strongly consistent and asymptotically norma as T →∞
√
T (θ̂T −θ) →D N(0,θσ2H),

√
T (θ̃T −θ) →D N(0,

θσ2H
4H2

) (1.13)

where
σ2H := (4H − 1)

(
1 +

Γ(3 − 4H)Γ(4H − 1)
Γ(2 − 2H)Γ(2H)

)
. (1.14)

Observe that H = 1/2, σ2H = 2. In this case the LSE and the MLE are identical. Since θ̃T is aconsistent estimator of θ, we can derive the self normalized limit distributions immediately:
(
T

σ2
H
θ̃T

)1/2(θ̂T −θ) →D N(0, 1), 2H(
T

σ2
H
θ̃T

)1/2(θ̃T −θ) →D N(0, 1). (1.15)

Define
MT :=

∫ T
0

Xt dW
H
t and IT := ∫ T

0

X2t dt, NT := θ
2HIT −THΓ(2H). (1.16)

VH,θ := θ
−2HHΓ(2H). (1.17)

Observe that (
T

−σ2
H
θ

)1/2
(θ̂T −θ) =

(
−σ2Hθ
T

)1/2
MT(

σ2
H
θ

T

)
IT

(1.18)

Applying Taylor’s formula to the function x− 12H at the point VH,θ, we have(
IT
T

)− 1
2H

= V
− 1
2H

H,θ
−

1

2H
V
−1+2H
2H

H,θ

(
IT
T
−VH,θ

)
+

1 + 2H

8H2
$
−1+4H
2H

T

(
IT
T
−VH,θ

)2
(1.19)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 4
where $T is a random point between VH,θ and ITT . Further
θ̃T −θ = −

θ1+2H

2H2Γ(2H)

(
IT
T
−VH,θ

)
+

(1 + 2H)(HΓ(2H))
1
2H

8H2
$
−1+4H
2H

T

(
IT
T
−VH,θ

)2
. (1.20)

Thus
2H

(
T

−σ2
H
θ

)1/2
(θ̃T −θ) =

(
−σ2Hθ
4TH2

)1/2
NT(

σ2
H
θ

4TH2

)
IT

. (1.21)

We study the large deviations, moderate deviations and Berry-Esseen bounds of the LSE andthe QLSE in this paper. We will use the following optimal fourth moment theorem from Nourdinand Peccati [14] in the sequel. See also Douissi et al. [15].
Theorem 1.1 (Skewness Kurtosis Inequality) Let (Xn)n≥1 be a sequence of random variables in
fixed Wiener chaos of order q ≥ 2 such that V ar(Xn) = 1. Assume Xn converges to normal
distribution which is equivalent to limn E(Xn)4 = 3, which is also known as the Fourth Moment
Theorem. Then we have the following optimal rate for dTV (Xn,N) known as the Optimal Fourth
Moment Theorem: There exist two constants c,C > 0 depending only on the sequence (Xn)n≥1
but not on n, such that

c max{E(X4n ) − 3, |E(X
3
n )|}≤ dTV (Xn,N) ≤ C max{E(X

4
n ) − 3, |E(X

3
n )|}. (1.22)

Let Φ(·) denote the standard normal distribution function. Throughout the paper, C denotes ageneric constant (which does not depend on T and x). We have not tried to estimate the constantin the bound on normal approximation.Hu et al. [11] obtained limiting normal distribution of the LSE and the QLSE for the memoryrange 1
2
< H ≤ 3

4
with the rate √T for 1

2
< H < 3

4
and √T (log T )−1/2 for the case H = 3

4
, andlimiting Rosenblatt distribution for the memory range 3

4
< H < 1.We only consider the memory range 1

2
< H < 3

4
. Jiang et al. [12] used self-normalization alongwith the splitting method for the LSE and the QLSE in fractional Ornstein-Uhlenbeck process andobtained the rate T−1/2 log T for the range 1

2
≤ H ≤ 5

8
for the LSE and T−1/4 log T for the range

1
2
≤ H ≤ 11

16
for the QLSE. They obtained the rate T4H−3 for the range 5

8
< H < 3

4
for the LSEand the same rate T4H−3 for the range 11

16
< H < 3

4
for the QLSE.In this paper we improve the first rate to T−1/2 for the MLE for the range 1

2
≤ H ≤ 5

8
and

T−1/4 for the range 1
2
≤ H ≤ 11

16
for the QLSE using the squeezing method as in Chapter 1 inBishwal [1]. The main contribution of the paper is thus improvement in the rate my removing the

log T term.
Note the critical points:
1
2

= 0.50, 5
8

= 0.63, 2
3

= 0.67, 11
16

= 0.69, 3
4

= 0.75. Also 0.63 + 0.06 = 0.69, 0.69 + 0.06 = 0.75.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 5
Remark on the Critical point 5

8
: For the discrete observations case, Es-Sebaiy and Viens [7]pointed out that if 0 < H < 5

8
, then the fourth moment is of the order n−1 and if 5

8
< H < 3

4
,then the fourth moment is of the order n2(4H−3) where n is the number of observations. TheBerry-Esseen rate for θ̂ is shown to be of the order n−1/4 for 0 < H < 5

8
and of the order

n−(4H−3)/2 if 5
8
< H < 3

4
. for H = 3

4
, the rate is (log n)−1/4.

The proofs also need large deviation results for the stochastic integral and the energy integral.These integrals can be represented by multiple Wiener integrals. Then their expectations andvariances as well as the fourth moment of their Malliavin derivatives can be estimated.First we calculate bounds on the moments. Let
ϕT (s,t) := e

−θ|t−s|, ψT (s,t) := e
−2θT+θ(s+t), gT (s,t) := e

−θ(t−s)I[0,t](s), (1.23)

VH,θ := θ
−2HHΓ(2H), CH,θ := θ

1−4H(4H − 1)H2
(

Γ2(2H) +
Γ(2H)Γ(3 − 4H)Γ(4H − 1)

Γ(2 − 2H)

)
. (1.24)

Observe that
Xt = I1(gT (·,t)), (1.25)

MT =

∫ T
0

XtdW
H
t =

∫ T
0

∫ t
0

eθ(t−s)dWHs dW
H
t =

1

2
eθ|t−s|dWHs dW

H
t =

1

2
I2(ϕT ), (1.26)

IT =

∫ T
0

X2t dt =
1

2θ
I2(ϕT ) +

1

2θ
I2(ψT ) +

∫ T
0

‖gT (·,t)‖2Hdt (1.27)where I1 and I2 are first and second Wiener chaos respectively. Furthermore,∫ T
0

‖gT (·,t)‖2Hdt = VH,θT + o(T ). (1.28)

For 1
2
< H < 3

4
,

E(XtXs) ≤ C|t − s|2H−2, (1.29)

‖ϕT‖2H = 2T (cH,T + (o(1)), ‖ψT‖
2
H = O(1). (1.30)For H = 1

2
, by the isometry of the Itô integral, we obtain
‖ϕT‖2H = 2

∫ T
0

∫ t
0

e2θ(t−s)dtds =
T

θ
+
e2θT − 1

2θ2
= 2T (CH,T + (o(1)). (1.31)

‖ψT‖2H = e
−4θT

∫ T
0

∫ T
0

e−2θ(t+s)dtds =
(e−2θT − 1)2

4θ2
= O(1). (1.32)

For 1
2
< H < 3

4
,, using Lemma 5.3 in Hu and Nualart [10], we have

‖ψT‖2H ≤
Γ2(2H)

(2H − 1)2
θ−4H. (1.33)

Let ΥT = T for H = 12 and ΥT = T8H−4 for 12 < H < 34.
We obtain the variances bounds on the Malliavin derivative of MT and IT .

E(‖DMT‖2H −E‖DMT‖
2
H) ≤ CΥT , (1.34)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 6
E(‖DIT‖2H −E‖DIT‖

2
H) ≤ CΥT (1.35)where D is the Malliavin derivative operator.We have the bound on the fourth moment

E(‖DI2(ϕT )‖2H −E‖DI2(ϕT )‖
2
H)
2 ≤ CΥT . (1.36)

For 1
2
< H < 3

4
,, we have the bound on the fourth moment

E(‖DI2(ϕT )‖2H −E‖DI2(ϕT )‖
2
H)
2 ≤ CT8H−4. (1.38)

We have the bound on the fourth moment
E(‖DI2(ψT )‖2H −E‖DI2(ψT )‖

2
H)
2 ≤ C. (1.39)

DsI2(ψT ) = −2e−2θT+θs
∫ T
0

eθtdWHt . (1.40)

E‖DsI2(ψT )‖4H = 16e
−8θT

(∫ T
0

eθtdWHt

)4(∫ T
0

e2θtdt

)2
= 48e−8θT

(∫ T
0

e2θtdt

)4
. (1.41)

E‖DsI2(ψT )‖2H = 4e
−4θT

(∫ T
0

e2θtdt

)2
. (1.42)

Therefore
E(‖DI2(ψT )‖2H −E‖DI2(ψT )‖

2
H)
2 = E‖DI2(ψT )|4H − (E‖DI2(ψT )‖

2
H)
2 =

2(1 −e−2θT )4

θ4
.

(1.43)Similarly for the case 1
2
< H < 3

4
, it can be shown that

E(‖DI2(ψT )‖2H −E‖DI2(ψT )‖
2
H)
2 ≤

32Γ4(2H)

(2H − 1)4
θ−8H. (1.44)

First we have the Berry-Esseen bounds for the stochastic integral and adjusted energy integral. Byusing the Optimal Fourth Moment theorem (Skewness-Kurtosis Inequality) from Stein-Malliavintheory, we have:For 1
2
≤ H ≤ 5/8, we have

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2
MT ≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2
MT‖2H −E‖D

(
C−1
H,θ

T

)1/2
MT‖2H

)2
1/2

≤ CT−1/2.

(1.45)

For 5
8
< H < 3

4
,

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2
MT ≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2
MT‖2H −E‖D

(
C−1
H,θ

T

)1/2
MT‖2H

)2
1/2

≤ C T4H−3.

(1.46)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 7
For 1

2
≤ H ≤ 5/8, we have

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2(
θ̃TIT −

T

−σ2
H

)
≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2(
θ̃TIT − T−σ2

H

)
‖2H −E‖D

(
C−1
H,θ

T

)1/2(
θ̃TIT − T−σ2

H

)
‖2H

)2
1/2

≤ CT−1/2.
(1.47)For 5

8
< H < 3

4
,

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2(
θ̃TIT −

T

−σ2
H

)
≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2(
θ̃TIT − T−σ2

H

)
‖2H −E‖D

(
C−1
H,θ

T

)1/2(
θ̃TIT − T−σ2

H

)
‖2H

)2
1/2

≤ C T4H−3.
(1.48)For 1

2
≤ H ≤ 5

8
, we have for |x| ≤ 2(log T )1/2,

sup
y∈R

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
MT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x ≤ y

− Φ(y)
∣∣∣∣∣∣ ≤ CT−1/2. (1.49)

For 5
8
< H < 3

4
, we have for |x| ≤ 2(log T )1/2,

sup
y∈R

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
MT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x ≤ y

− Φ(y)
∣∣∣∣∣∣ ≤ CT4H−3. (1.50)

For 1
2
≤ H ≤ 11

16
, we have

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2
NT ≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2
NT‖2H −E‖D

(
C−1
H,θ

T

)1/2
NT‖2H

)2
1/2

≤ CT−1/2.

(1.51)

For 11
16
< H < 3

4
,

sup
x∈R

∣∣∣∣∣∣P

(
C−1
H,θ

T

)1/2
NT ≤ x

− Φ(x)
∣∣∣∣∣∣

≤ C

E
(
‖D
(
C−1
H,θ

T

)1/2
NT‖2H −E‖D

(
C−1
H,θ

T

)1/2
NT‖2H

)2
1/2

≤ C T4H−3.

(1.52)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 8
For 1

2
≤ H ≤ 11

16
, we have for |x| ≤ 2(log T )1/2,

sup
y∈R

∣∣∣∣∣∣P
2H

(
−σ2Hθ̃T
T

)1/2
NT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x ≤ y

− Φ(y)
∣∣∣∣∣∣ ≤ CT−1/4. (1.53)

For 11
16
< H < 3

4
, we have for |x| ≤ 2(log T )1/2,

sup
y∈R

∣∣∣∣∣∣P
2H

(
−σ2Hθ̃T
T

)1/2
NT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x ≤ y

− Φ(y)
∣∣∣∣∣∣ ≤ CT4H−3. (1.54)

2. Main Results

We need the next two lemmas from Jiang et al. [12] on large deviations to obtain bounds onthe tail probabilities of the estimators. The first lemma is on large deviations for stochastic integral.
Lemma 2.1 For every δ > 0,

P

{∣∣∣∣MTT
∣∣∣∣ ≥ δ} ≤ C exp

(
−
T1/2δ

4C
1/2
H,θ

)
.

Remark For the case H = 0.5, there is a long history of work:
For every δ > 0,

P

{∣∣∣∣MTT
∣∣∣∣ ≥ δ} ≤ C0 exp (−C1Tδ2) .

See Gao and Jiang [9].For any 0 ≤ α ≤ θ2/4, there exist constants C3 and C4 such that
E(eαIT ) ≤ C3eC4αT . (2.1)

See Gao and Jiang [9]. By Chebyshev inequality, we have
P (|XT −E(XT )| ≥ δ) ≤ 2 exp(−θδ2). (2.2)

The second lemma is on large deviations in the ergodic theorem.
Lemma 2.2 For every δ > 0,

P

{∣∣∣∣ITT −VH,θ
∣∣∣∣ ≥ δ} ≤ C exp

(
−
T1/2δ

4C
1/2
H,θ

)
.

Observe that by (1.11)
θ̂T = θ−

MT
IT
.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 9
Using the elementary inequality

P (|
ξ

η
| ≥ u) ≤ P (|ξ| ≥ uv) + P (η − 2v| ≥ v), (2.3)

we have
P (|θ̂T −θ| ≥ δ)

≤ P (|IT −VH,θT | ≥ 12VH,θT ) + P (|θ̂T −θ| ≥ δ, |IT −VH,θT | <
1
2
VH,θT )

≤ P (|IT −VH,θT | ≥ 12VH,θT ) + P (|MT | ≥
1
2
VH,θTδ).

(2.4)

Combining Lemma 2.1 and Lemma 2.2, we obtain
Lemma 2.3 For every δ > 0 and large T > 0, we have

a) P (|θ̂T −θ| ≥ δ) ≤ C0 exp(−C1T1/2δ)

b) P (|θ̃T −θ| ≥ δ) ≤ C0 exp(−C1T1/2δ1/2).

To obtain the rate of normal approximation for the LSE and the QLSE, we need the followingtail probability estimate of the estimators.
Lemma 2.4

(a) P


(

T

−σ2
H
θ̃T

)1/2
|θ̂T −θ| ≥ 2(log T )1/2

 ≤ CT−1/2.
(b) P

2H
(

T

−σ2
H
θ̃T

)1/2
|θ̃T −θ| ≥ 2(log T )1/2

 ≤ CT−1/4.
Proof : Observe that

P


(

T

−σ2
H
θ̃T

)1/2
|θ̂T −θ| ≥ 2(log T )1/2


= P


∣∣∣∣∣∣∣∣∣
(
−σ2Hθ̃T
T

)1/2
MT

(
−σ2

H
θ̃T

T
)IT

∣∣∣∣∣∣∣∣∣
≥ 2(log T )1/2


≤ P


∣∣∣∣∣∣
(
−σ2Hθ̃T
T

)1/2
MT

∣∣∣∣∣∣ ≥ (log T )1/2
 + P

{∣∣∣∣∣−σ2Hθ̃TT IT
∣∣∣∣∣ ≤ 12

}

≤

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
|MT | ≥ (log T )1/2

− 2Φ(−(log T )1/2)
∣∣∣∣∣∣

+2Φ(−(log T )1/2) + P

{∣∣∣∣∣σ2Hθ̃TT IT − 1
∣∣∣∣∣ ≥ 12

}

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 10
≤ sup

x∈R

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
|MT | ≥ x

− 2Φ(−x)
∣∣∣∣∣∣

+2Φ(−(log T )1/2) + P

{∣∣∣∣∣
(
−σ2Hθ̃T
T

)
IT − 1

∣∣∣∣∣ ≥ 12
}

≤ CT−1/2 + C(T log T )−1/2 + C exp

(
−
T1/2

8C
1/2
H,θ

)
≤ CT−1/2.The bounds for the first and the third terms come from Lemma 2.2 and Lemma 2.1 respectively andthat for the middle term comes from Feller [8] (p. 166). Proof of (b) is similar.

Now we are ready to obtain the uniform rate of normal approximation of the distribution of theLSE and the QLSE.Recall that
σ2H := (4H − 1)

(
1 +

Γ(3 − 4H)Γ(4H − 1)
Γ(2 − 2H)Γ(2H)

)
. (2.5)

Theorem 2.5a) If 1
2
≤ H ≤ 5

8

sup
x∈R

∣∣∣∣∣∣P

(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT−1/2.

b) If 5
8
< H < 3

4

sup
x∈R

∣∣∣∣∣∣P

(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT4H−3.

c) If 1
2
≤ H ≤ 11

16

sup
x∈R

∣∣∣∣∣∣P
2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT−1/4.

d) If 11
16
< H < 3

4

sup
x∈R

∣∣∣∣∣∣P
2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT4H−3.

Proof : First we prove (a). We shall consider two possibilities (i) and (ii).
(i) |x| > 2(log T )1/2.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 11
We shall give a proof for the case x > 2(log T )1/2. The proof for the case x < −2(log T )1/2 runssimilarly. Note that∣∣∣∣∣∣P


(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ P


(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≥ x

+Φ(−x). (2.6)
But from Feller [8] (p. 166) we have

Φ(−x) ≤ Φ(−2(log T )1/2) ≤ CT−1. (2.7)

Moreover, by Lemma 2.4 (a), we have
P


(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≥ 2(log T )1/2

 ≤ CT−1/2. (2.8)
Hence ∣∣∣∣∣∣P


(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT−1/2. (2.9)

(ii) |x| ≤ 2(log T )1/2.
Let AT :=


(

T

−σ2
H
θ̃T

)1/2
|θ̂T −θ| ≤ 2(log T )1/2

 and BT :=
{
IT
T
> c0

}
(2.10)

where 0 < c0 < 1−σ2
H
θ
. By Lemma 2.4, we have

P (AcT ) ≤ CT
−1/2. (2.11)

By Lemma 2.1, we have
P (BcT ) = P

{(
−σ2Hθ̃T
T

)
IT − 1 < σ2Hθc0 − 1

}

< P

{∣∣∣∣∣
(
−σ2Hθ̃T
T

)
IT − 1

∣∣∣∣∣ > 1 −σ2Hθc0
}
≤ C exp

(
−
T1/2(1 −σ2Hθc0)

4C
1/2
H,θ

)
. (2.12)

Let b0 be some positive number. On the set AT∩BT for all T > T0 with 4b0(log T0)1/2(σ2HθT )1/2 ≤
c0, we have (

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

⇒ IT + b0T (θ̂T −θ) < IT +

(
T

−σ2
H
θ̃T

)1/2
σ2Hb0θx

⇒

(
T

−σ2
H
θ̃T

)1/2
(θ̂T −θ)[IT + b0T (θT −θ)] < x[IT +

(
T

−σ2
H
θ̃T

)1/2
σ2Hb0θx]

⇒ (θ̂T −θ)IT + b0T (θT −θ)2 <

(
−σ2Hθ̃T
T

)1/2
ITx + σ

2
Hb0θx

2

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 12
⇒ −MT + (θ̂T −θ)IT + b0T (θ̂T −θ)2 < −MT +

(
σ2Hθ̃T

T

)1/2
ITx + σ

2
Hb0θx

2

⇒ 0 < −MT +

(
−σ2Hθ̃T
T

)1/2
ITx + σ

2
Hb0θx

2

since
IT + b0T (θ̂T −θ) > Tc0 + b0T (θ̂T −θ)

> 2σ2Hb0(log T )
1/2

(
−σ2Hθ̃T
T

)1/2
−σ2Hb0(log T )

1−H

(
−σ2Hθ̃T
T

)1/2

= σ2Hb0(log T )
1/2

(
−σ2Hθ̃T
T

)1/2
> 0.

On the other hand, on the set AT ∩BT for all T > T0 with 4b0(log T0)1/2(−σ2Hθ̃TT0
)1/2

≤ c0, wehave (
T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) > x

⇒ IT −b0T (θ̂T −θ) < IT −

(
T

σ2
H
θ̃T

)1/2
2b0θx

⇒

(
T

−σ2
H
θ̃T

)1/2
(θ̂T −θ)[IT −b0T (θ̂T −θ)] > x[IT −

(
T

−σ2
H
θ̃T

)1/2
σ2Hb0θx]

⇒ (θ̂T −θ)IT −b0T (θ̂T −θ)2 >

(
T

−σ2
H
θ̃T

)−1/2
ITx −σ2Hb0θx

2

⇒ −MT + (θ̂T −θ)IT −b0T (θ̂T −θ)2 > −MT +

(
T

−σ2
H
θ̃T

)−1/2
ITx −σ2Hb0θx

2

⇒ 0 > −MT +

(
−σ2Hθ̃T
T

)1/2
ITx −σ2Hb0θx

2

since
IT −b0T (θ̂T −θ) > Tc0 −b0T (θ̂T −θ)

> 2σ2Hb0(log T )
1/2

(
−σ2Hθ̃T
T

)1/2
−σ2Hb0(log T )

1/2

(
−σ2Hθ̃T
T

)1/2

= σ2Hb0(log T )
1/2

(
−σ2Hθ̃T
T

)1/2
> 0.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 13
Hence

0 < −MT +

(
T

−σ2
H
θ̃T

)1/2
ITx −σ2Hb0θx

2 ⇒

(
T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x.

Letting
D±
T,x

:=

−MT +
(
σ2Hθ̃T

T

)1/2
ITx ±σ2Hb0θx

2 > 0

 ,
we obtain

D−
T,x
∩AT ∩BT ⊆ AT ∩BT ∩


(

T

−σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

 ⊆ D+T,x ∩AT ∩BT . (2.13)
If it is shown that ∣∣P {D±

T,x

}
− Φ(x)

∣∣ ≤ CT−1/2 (2.14)
for all T > T0 and |x| ≤ 2(log T )1/2, then the theorem would follow from (2.11) - (2.14).We shall prove (2.4) for D+

T,x
. The proof for D−

T,x
is analogous. Observe that∣∣P {D+T,x}− Φ(x)∣∣

=

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
MT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x < x + σ2H

(
−σ2Hθ̃T
T

)1/2
b0θx

2

− Φ(x)
∣∣∣∣∣∣

≤ sup
y∈R

∣∣∣∣∣∣P

(
−σ2Hθ̃T
T

)1/2
MT −

((
−σ2Hθ̃T
T

)
IT − 1

)
x ≤ y

− Φ(y)
∣∣∣∣∣∣

+

∣∣∣∣∣∣Φ
x + (−σ2Hθ̃T

T

)1/2
b0θx

2

− Φ(x)
∣∣∣∣∣∣

=: ∆1 + ∆2.

(2.15)(1.50) immediately yields
∆1 ≤ CT−1/2. (2.16)

On the other hand, for all T > T0,
∆2 ≤ 2

(
−σ2Hθ̃T
T

)1/2
b0θx

2(2π)−1/2 exp(−x2/2)

where
|x −x| ≤ 2

(
−σ2Hθ̃T
T

)1/2
b0θx

2.

Since |x| ≤ 2(log T )1/2, it follows that |x̄| > |x|/2 for all T > T0 and consequently
∆2 ≤ 2

(
−σ2Hθ̃T
T

)1/2
b0θx

2(2π)−1/2x2 exp(−x2/8)

≤ CT−1/2.
(2.17)

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 14
From (2.15) - (2.17), we obtain ∣∣P {D+

T,x

}
− Φ(x)

∣∣ ≤ CT−1/2.
This completes the proof of part (a) of the theorem.
Next we prove (c). Again we shall consider two possibilities (i) and (ii).
(i) |x| > 2(log T )1−/2.

We shall give a proof for the case x > 2(log T )1/2. The proof for the case x < −2(log T )1/2runs similarly. Note that∣∣∣∣∣∣P
2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ P

2H
(

T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≥ x

 + Φ(−x).
By (2.7) and Lemma 2.4 (b), we have

P

2H
(

T

σ2
H
θ̃T

)1/2
(θ̃T −θ) ≥ 2(log T )1/2

 ≤ CT−1/4.
Hence ∣∣∣∣∣∣P

2H
(

T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CT−1/4.

(ii) |x| ≤ 2(log T )1/2.
Let A1,T :=

2H
(

T

−σ2
H
θ̃T

)1/2
|θ̃T −θ| ≤ 2(log T )1/2

 and B1,T :=
{
IT
T
> c0

}
where 0 < c0 < 1−σ2

H
θ
. By Lemma 2.4, we have

P (Ac1,T ) ≤ CT
−1/4. (2.18)

By Lemma 2.1, we have
P (Bc1,T ) = P

{(
−σ2Hθ
4TH2

)
IT − 1 < σ2Hθc0 − 1

}
< P

{∣∣∣∣(−σ2Hθ4TH2
)
IT − 1

∣∣∣∣ > 1 −σ2Hθc0} ≤ CT−1.
(2.19)Let b0 be some positive number. On the set A1,T ∩ B1,T for all T > T0 with

4b0(log T0)
1/2
(
−σ2Hθ
4T0H2

)1/2
≤ c0, we have

2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

⇒ IT + b0T (θ̃T −θ) < IT + 2H

(
T

−σ2
H
θ̃T

)1/2
σ2Hb0θx

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 15
⇒ 2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ)[IT + b0T (θT −θ)] < x

IT + 2H
(

T

−σ2
H
θ̃T

)1/2
σ2Hb0θx


⇒ (θ̃T −θ)IT + b0T (θT −θ)2 <

(
−σ2Hθ
4TH2

)1/2
ITx + σ

2
Hb0θx

2

⇒ −NT + (θ̃T −θ)IT + b0T (θ̃T −θ)2 < −NT +
(
−σ2Hθ
4TH2

)1/2
ITx + σ

2
Hb0θx

2

⇒ 0 < −NT +
(
−σ2Hθ
4TH2

)1/2
ITx + σ

2
Hb0θx

2

since
IT + b0T (θT −θ) > Tc0 + b0T (θT −θ)

> 4b0(log T )
1/2

(
−σ2Hθ
4TH2

)1/2
−σ2Hb0(log T )

1−H
(
−σ2Hθ
4TH2

)1/2
= σ2Hb0(log T )

1/2

(
−σ2Hθ
4TH2

)1/2
> 0.

On the other hand, on the set A1,T ∩B1,T for all T > T0 with 4b0(log T0)1/2(−σ2Hθ4T0H2)1/2 ≤ c0,we have
2H

(
T

σ2
H
θ̃T

)1/2
(θT −θ) > x

⇒ IT −b0T (θ̃T −θ) < IT − 2H

(
T

−σ2
H
θ̃T

)1/2
σ2Hb0θx

⇒ 2H

(
T

−σ2
H
θ̃T

)1/2
(θ̃T −θ)[IT −b0T (θT −θ)] > x

IT − 2H
(

T

−σ2
H
θ̃T

)1/2
2b0θx


⇒ (θ̃T −θ)IT −b0T (θ̃T −θ)2 >

(
−σ2Hθ
4TH2

)1/2
ITx −σ2Hb0θx

2

⇒ −NT + (θ̃T −θ)IT −b0T (θT −θ)2 > −NT +
(
−σ2Hθ
4TH2

)1/2
ITx −σ2Hb0θx

2

⇒ 0 > −NT +
(
−σ2Hθ
4TH2

)1/2
ITx −σ2Hb0θx

2

since
IT −b0T (θ̃T −θ) > Tc0 −b0T (θ̃T −θ)

> 2σ2Hb0(log T )
1/2

(
−σ2Hθ
4TH2

)1/2
−σ2Hb0(log T )

1/2

(
−σ2Hθ
4TH2

)1/2
= σ2Hb0(log T )

1/2

(
−σ2Hθ
4TH2

)1/2
> 0.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 16
Hence

0 < −NT +
(
−σ2Hθ
4TH2

)1/2
ITx −σ2Hb0θx

2 ⇒ 2H(
T

−σ2
H
θ̃T

)1/2(θT −θ) ≤ x.

Letting
D±1,T,x :=

{
−NT +

(
−σ2Hθ
4TH2

)1/2
ITx ±σ2Hb0θx

2 > 0

}
,

we obtain
D−1,T,x ∩A1,T ∩B1,T ⊆ A1,T ∩B1,T ∩

2H
(

T

−σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

 ⊆ D+1,T,x ∩A1,T ∩B1,T .
(2.20)If it is shown that ∣∣P {D±1,T,x}− Φ(x)∣∣ ≤ CT−1/4 (2.21)for all T > T0 and |x| ≤ 2(log T )1/2, then the theorem would follow from (2.18) - (2.21).We shall prove (2.21) for D+

1,T,x
. The proof for D−

1,T,x
is analogous.Observe that∣∣∣P {D+1,T,x}− Φ(x)∣∣∣

=

∣∣∣∣∣P
{(
−σ2Hθ
4TH2

)1/2
NT −

((
−σ2Hθ
4TH2

)
IT − 1

)
x < x + 2

(
−σ2Hθ
4TH2

)1/2
b0θx

2

}
− Φ(x)

∣∣∣∣∣
≤ sup

y∈R

∣∣∣∣∣P
{(
−σ2Hθ
4TH2

)1/2
NT −

((
−σ2Hθ
4TH2

)
IT − 1

)
x ≤ y

}
− Φ(y)

∣∣∣∣∣
+

∣∣∣∣∣Φ
(
x +

(
−σ2Hθ
4TH2

)1/2
b0θx

2

)
− Φ(x)

∣∣∣∣∣
=: ∆11 + ∆12.

(2.22)(1.53) immediately yields
∆11 ≤ CT−1/4. (2.23)On the other hand, for all T > T0,

∆12 ≤ 2
(
−σ2Hθ
4TH2

)1/2
b0θx

2(2π)−1/2 exp(−x2/2)

where
|x −x| ≤ 2

(
−σ2Hθ
4TH2

)1/2
b0θx

2.

Since |x| ≤ 2(log T )1/2, it follows that |x̄| > |x|/2 for all T > T0 and consequently
∆12 ≤ 2

(
−σ2Hθ
4TH2

)1/2
b0θx

2(2π)−1/2x2 exp(−x2/8) ≤ CT−1/4. (2.24)

From (2.12) - (2.14), we obtain ∣∣P {D+1,T,x}− Φ(x)∣∣ ≤ CT−1/4.

https://doi.org/10.28924/ada/ma.3.14


Eur. J. Math. Anal. 10.28924/ada/ma.3.14 17
This completes the proof of part (c) of the theorem. Next we demonstrate the proof of (b) and (d).If 5

8
< H < 3

4
by following similar steps, one can show that

sup
x∈R

∣∣∣∣∣∣P

(

T

σ2
H
θ̃T

)1/2
(θ̂T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CθT4H−3.

If 11
16
< H < 3

4
by following similar steps, one can show that

sup
x∈R

∣∣∣∣∣∣P
2H

(
T

σ2
H
θ̃T

)1/2
(θ̃T −θ) ≤ x

− Φ(x)
∣∣∣∣∣∣ ≤ CθT4H−3.

This completes the proof of the theorem.
Concluding Remark For the case 1

2
≤ H ≤ 5

8
, our rate is O(T−1/2) is optimal.

References
[1] J.P.N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Springer-Verlag, Berlin, (2008).[2] J.P.N. Bishwal, Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discretesampling, Fract. Calc. Appl. Anal. 14 (2011) 375–410. https://doi.org/10.2478/s13540-011-0024-6.[3] J.P.N. Bishwal, Maximum quasi-likelihood estimation in fractional levy stochastic volatility model, J. Math. Finance.1 (2011) 58–62. https://doi.org/10.4236/jmf.2011.13008.[4] J.P.N. Bishwal, Sufficiency and Rao-Blackwellization of Vasicek model, Theory Stoch. Processes. 17 (2011) 12-15.[5] J.P.N. Bishwal, Berry–Esseen inequalities for the fractional Black–Karasinski model of term structure of interestrates, Monte Carlo Methods Appl. 28 (2022) 111–124. https://doi.org/10.1515/mcma-2022-2111.[6] J.P.N. Bishwal, Parameter estimation in stochastic volatility models, Springer Nature, Cham. (2022).[7] K. Es-Sebaiy, F.G. Viens, Optimal rates for parameter estimation of stationary Gaussian processes, Stoch. ProcessesAppl. 129 (2019) 3018–3054. https://doi.org/10.1016/j.spa.2018.08.010.[8] W. Feller, An Introduction to Probability Theory and its Applications, Vol. I, Wiley, New York, (1957).[9] F. Gao, H. Jiang, Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein-Uhlenbeck process with linear drift, Electron. Commun. Prob. 14 (2009) 210-220. https://doi.org/10.1214/

ecp.v14-1466.[10] Y. Hu, D. Nualart, Parameter estimation for fractional Ornstein-Uhlenbeck processes, Stat. Prob. Lett. 80 (2010)1030-1083.[11] Y. Hu, D. Nualart, H. Zhou, Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurstparameter, Stat. Inference Stoch. Process. 22 (2017) 111–142. https://doi.org/10.1007/s11203-017-9168-2.[12] H. Jiang, J. Liu, S. Wang, Self-normalized asymptotic properties for the parameter estimation in fractional Orn-stein–Uhlenbeck process, Stoch. Dyn. 19 (2019) 1950018. https://doi.org/10.1142/s0219493719500187.[13] M.L. Kleptsyna, A. Le Breton, Statistical inference for stochastic processes, Stat. Inference Stoch. Processes. 5(2002) 229–248. https://doi.org/10.1023/a:1021220818545.[14] I. Nourdin, G. Peccati, The optimal fourth moment theorem, Proc. Amer. Math Soc. 143 (2015) 3123-3133.[15] S. Douissi, K. Es-Sebaiy, F. G. Viens, Berry-Esseen bounds for parameter estimation of general Gaussian processes,ALEA. 16 (2019) 633. https://doi.org/10.30757/alea.v16-23.

https://doi.org/10.28924/ada/ma.3.14
https://doi.org/10.2478/s13540-011-0024-6
https://doi.org/10.4236/jmf.2011.13008
https://doi.org/10.1515/mcma-2022-2111
https://doi.org/10.1016/j.spa.2018.08.010
https://doi.org/10.1214/ecp.v14-1466
https://doi.org/10.1214/ecp.v14-1466
https://doi.org/10.1007/s11203-017-9168-2
https://doi.org/10.1142/s0219493719500187
https://doi.org/10.1023/a:1021220818545
https://doi.org/10.30757/alea.v16-23

	References

