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On the Kolmogorov Distance for the Maximum Likelihood Estimator in the Explosive

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 28223-0001, USACorrespondence: J.Bishwal@uncc.edu

Abstract. The paper estimates the Kolmogorov distance between the distribution of the normalizedmaximum likelihood estimator of the positive drift parameter in the nonergodic Ornstein-Uhlenbeckprocess and the standard Cauchy distribution and shows exponential error rate for large time limit.
1. Introduction

Estimating the rate in the Kolmogorov distance between two distributions has a long history inprobability and statistics. The estimate could be useful in finding confidence interval and inhypothesis testing, see Bishwal [8, 11]. In the i.i.d. case, the Berry-Esseen bound for minimumcontrast estimators was obtained in Pfanzagl [28] improving that from Michel and Pfanzagl [24].Borokov [14] obtained the rate of convergence for the invariance principle in the i.i.d. case. Hall andHeyde [20] obtained rate of convergence in the central limit theorem for martingales using Skorohodembedding. Uniform rate of weak convergence for the minimum contrast estimator in the Ornstein-Uhlenbeck (O-U) process was studied in Bishwal [5]. The rates of convergence of the conditionalleast squares estimator and an approximate maximum likelihood estimator when the O-U processis observed at discrete time points in [0, T ] has been studied in Bishwal and Bose [13](2001) inthe ergodic case. In a Bayesian framework, the rates of convergence of the posterior distributionsand the Bayes estimators has been studied in Bishwal [6] and Bishwal [10] for the continuousobservation and discrete observations respectively in the ergodic case. In finance, asset price maybehave in nonergodic manner, i.e., efficient market hypotheses may not hold, possibly be due tosocial interaction among consumers among other reasons, see Horst and Wenzelburger [21]. Westudy the nonergodic Ornstein-Uhlenbeck process in this paper and focus on the rate of convergenceof the Kolmogorov distance.
Received: 5 Jul 2023.
Key words and phrases. Itô stochastic differential equation, explosive Ornstein-Uhlenbeck process, maximum likeli-hood estimator, Kolmogorov distance, inefficient market. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 2Let (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 + dWt , t ≥ 0, X0 = 0 (1.1)

where {Wt}t≥0 is a standard Wiener process with respect to the filtration {Ft}t≥0 and θ > 0is the unknown parameter to be estimated on the basis of continuous observation of the process
{Xt}t≥0 on the time interval [0, T ].Let us denote the realization {Xt , 0 ≤ t ≤ T} by XT0 . Let P Tθ be the measure generated on thespace (CT , BT ) of continuous functions on [0, T ] with the associated Borel σ-algebra BT generatedunder the supremum norm by the process XT0 and P T0 be the standard Wiener measure. It is wellknown that when θ is the true value of the parameter P Tθ is absolutely continuous with respect to
P T0 and the Radon-Nikodym derivative (likelihood) of P Tθ with respect to P T0 based on XT0 is givenby

LT (θ) :=
dP Tθ
dP T0

(XT0 ) = exp

{
θ

∫ T

0

XtdXt −
θ2

2

∫ T

0

X2
t dt

}
. (1.2)

Maximizing the log-likelihood with respect to θ provides the maximum likelihood estimate (MLE)
θT :=

∫ T
0 XtdXt∫ T
0 X2

t dt
. (1.3)

In this transient case, we show that this estimator converges to the Cauchy distribution withan error rate O(e−θT ). Note that in the transient case, with random norming, specifically if onenormalizes the MLE by the square root of the observed Fisher information, then the MLE convergesto the normal distribution, see Feigin [16]. Maximum likelihood estimation in non-recurrent casewas studied in Dietz and Kutoyants [15]. Local asymptotic mixed normality for discretely observednon-recurrent Ornstein-Uhlenbeck processes was studied in Shimizu [30].
θT − θ :=

∫ T
0 XtdWt∫ T
0 X2

t dt
=
ZT
IT

(1.4)

where
ZT :=

∫ T

0

XtdWt and IT :=

∫ T

0

X2
t dt. (1.5)

Hence
eθT

2θ
(θT − θ) =

e−θT 2θZT
e−2θT 4θ2IT

=

(
e−2θT 4θ2

)1/2
ZT

e−2θT 4θ2IT
(1.6)In (1.6), the numerator of the normalized MLE is a normalized martingale which converges tothe standard normal variable and the denominator is its corresponding increasing process whichconverges to a chi-square random variable as T → ∞ which is independent of the numerator.Hence the ratio converges to a Cauchy distribution with parameters (0, 1).Let us introduce two Wiener integrals:

ξt :=

∫ t

0

e−θsdWs , and ηt :=

∫ t

0

eθsdWs , t ≥ 0. (1.7)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 3

and the respective limits ξ = limT→∞
∫ T

0 e−θsdWs :=
∫∞

0 e−θsdWs which has N (0, 1
2θ ) distribu-tion and η := limT→∞ e

−θTηT = limT→∞ e
−θT ∫ T

0 eθsdWs =
∫∞

0 e−θsdWs which has N (0, 1
2θ )distribution.With these notations

ZT :=

∫ T

0

XtdWt =

∫ T

0

eθtξtdWt and IT :=

∫ T

0

X2
t dt =

∫ T

0

e2θtξ2
t dt,

θT − θ =

∫ T
0 eθtξtdWt∫ T
0 e2θtξ2

t dt
, (1.8)

eθT

2θ
(θT − θ) =

(
e−2θT 4θ2

)1/2 ∫ T
0 eθtξtdWt

e−2θT 4θ2
∫ T

0 e2θtξ2
t dt

=

(
e−2θT 4θ2

)1/2 ∫ T
0 eθt(

∫ t
0 e
−θsdWs)dWt

e−2θT 4θ2
∫ T

0 e2θt(
∫ t

0 e
−θsdWs)2dt

=
ξT ξ

2θe−2θT
∫ T

0 e2θtξ2
t dt
×
e−θT

∫ T
0 eθtdWt

ξ
=

ξT ξ

2θe−2θT IT
×
e−θTηT

ξ
=: AθT × BθT . (1.9)

We have
AθT → 1 almost surely as T →∞, (1.10)

BθT
D→ N√

2θξ
as T →∞ (1.11)

where √2θξ = N1, and N1 and N are independent standard normal random variables. Since
N√
2θξ

D
= C(1) as T →∞ (1.12)

where C(1) is the standard Cauchy distribution, by Slutsky’s theorem, we have
AθT × BθT

D→C(1) as T →∞. (1.13)

Note that
ξT
D→ ξ as T →∞. (1.14)Using Borel-Cantelli lemma and stochastic Fubini theorem, it can be shown that ξT → ξ almostsurely and in L2(Ω) as T →∞. By integration by parts we have

e−2θT IT = e−2θT

∫ T

0

X2
s ds = e−2θT

∫ T

0

e2θsξ2
s ds

=
ξ2
T

2θ
−
e−2θT

θ

∫ T

0

e2θsξsdξs −
Te−2θT

θ
=
ξ2
T

2θ
−
e−2θT

θ

∫ T

0

eθsξsdWs −
Te−2θT

θ
. (1.15)

This equality together with
E

(∫ T

0

eθsξsdWs

)2

=

∫ T

0

e2θsE(ξ2
s )ds =

1

2θ

∫ T

0

e2θs(1− e−2θs)ds =
e2θT − 1− 2θT

4θ2
(1.16)

by the CLT for stochastic integrals provides
e−2θT

∫ T

0

X2
s ds

D→ ξ2

2θ
as T →∞ (1.17)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 4

i.e., e−2θT IT
D→ ξ2

2θ
as T →∞.It can be shown that

e−2θT IT →
ξ2

2θ
almost surely as T →∞. (1.18)By Itô formula, we have

ZT =

∫ T

0

XsdWs =

∫ T

0

eθsξsdWs =

∫ T

0

ξsdηs = ξTηT − T −
∫ T

0

ηsdξs

= ξTηT − T −
∫ T

0

ηse
−θsdWs . (1.19)

Hence
e−θTZT = e−θT ξTηT − e−θTT − e−θT

∫ T

0

ηse
−θsdWs . (1.20)

Direct calculation gives
e−θTηT

D→ η ∼ N
(

0,
1

2θ

) as T →∞ (1.21)

and
E(ξη) = lim

T→∞
E(ξTηT e

−θT ) = lim
T→∞

Te−θT = 0. (1.22)Hence
e−θTZT

D→ ξη as T →∞. (1.23)Hence the limit distribution of the pair (ξT , e
−θTηT ) is a Gaussian distribution of two indepen-dent variables. Thus

eθT
ξT
ηT

D→ ζ as T →∞ (1.24)

where ζ is the standard Cauchy variable with probability density function
f (x) =

1

π(1 + x2)
, x ∈ R. (1.25)

and cdf
C(x) =

1

2
+

1

π
arctan x, x ∈ R (1.26)and characteristic function ∫ ∞

−∞
e iλxdC(x) = e−|λ|. (1.27)

Hence
eθT

2θ
(θT − θ)

D→ ζ. (1.28)We estimate the rate of convergence in this phenomenon. We need the following lemma in thesequel.
Lemma 1.1 (Esseen’s Smoothing Lemma)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 5Let F be a non-decreasing function and H be a differentiable function of bounded variation on thereal line with F (±∞) = G(±∞). Denote the corresponding Fourier-Stieltjes transforms by F̂ and
Ĝ, respectively. Then for all Λ > 0,

sup
x∈R
|F (x)− G(x)| ≤

1

π

∫ Λ

−Λ

∣∣∣F̂ (λ)− Ĝ(λ)
∣∣∣

|λ| dλ+
24

πΛ
sup
x∈R
|G′(x)|.

Proof: See Petrov [27] or Feller [18].
Let Φ(·) denote the standard normal distribution function and C(·) denotes the standard Cauchydistribution function. Throughout the paper C denotes a generic constant (perhaps depending on
θ, but not on anything else).We need the following well known inequality.
Lemma 1.2

1√
2π

exp(
−x2

2
)(

1

x
−

1

x3
) ≤ 1−Φ(x) ≤

1√
2πx

exp(
−x2

2
)for x > 0. As x →∞,

1−Φ(x)∼
1√
2πx

exp(
−x2

2
).

Proof: See Feller ( [17], p.166).

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 6

2. Main Results

We start with the Dambis-Dubins-Schwarz (DDS) theorem, see Protter [29]. Since ZT is a contin-uous time martingale, due to time change (Skorohod embedding), ZT = BIT where B is a Brownianmotion independent of W, we have
E(exp(iuZT )) = E exp(−

u2

2
IT ), (2.1)

E(exp(iue−θT
√

2θZT )) = E exp(−
u2

2
e−2θT 2θIT ), (2.2)

θT − θ =
BIT
IT
, (2.3)

eθT

2θ
(θT − θ) =

e−θTBIT
e−2θT 2θIT

=

(
e−2θT 4θ2

)1/2
BIT

e−2θT 4θ2IT
=: YT (2.4)

where
YT =

e−θTBIT
e−2θT 2θIT

. (2.5)

Our main claim in the paper is to show that
|E(e iuYT )− e−|u|| ≤ C|u|e−|u|/2e−θT . (2.6)

This is done through several lemmas. Once it is shown, let
F (x) = P (YT ≤ x), (2.7)

C(x) =
1

2
+

1

π
arctan x, x ∈ R, (2.8)

Take Λ = eθT . Then
sup
x∈R
|F (x)− C(x)| ≤

1

π
J +

24

πeθT
sup C′(x) (2.9)

where
J :=

1

π

∫
|λ|≤eθT

∣∣∣F̂ (λ)− Ĉ(λ)
∣∣∣

|λ| dλ. (2.10)

Clearly
sup C′(x) <∞and

J ≤
C

eθT

∫
|λ|≤eθT

e−|λ|/2dλ ≤
C

eθT

∫ ∞
−∞

e−|λ|/2dλ = O(e−θT ). (2.11)

which would ultimately give
sup
x∈R
|F (x)− C(x)| = O(e−θT ). (2.12)

First we start with Kolmogorov distance for Wiener chaos and its relative:

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 7

Lemma 2.1 We have the following rate of convergence for the double-stochastic integral or thesecond Wiener chaos ∫ T0 eθt
(∫ t

0 e
−θsdWs

)
dWt :

(a) sup
x∈R

∣∣∣∣P {eθT (2θe−2θT

∫ T

0

eθt
(∫ t

0

e−θsdWs

)
dWt

)
≤ x

}
−Φ(x)

∣∣∣∣ ≤ Ce−θT .
(b) sup

x∈R

∣∣∣∣P {eθT (2θe−2θT

∫ T

0

e2θtξ2
t dt − ξ2

)
≤ x

}
−Φ(x)

∣∣∣∣ ≤ Ce−θT .
Proof. Observe that

ZT =

∫ T

0

XtdWt =

∫ T

0

eθt
(∫ t

0

e−θsdWs

)
dWt

The integral ∫ T

0

eθt
(∫ t

0

e−θsdWs

)
dWtis second Wiener chaos. One can use the Stein-Malliavin method (see Nourdin and Peccati( [25], [26]) and estimate the Kolmogorov distance for ZT . However, part (a) follows as a conse-quence of Lemma 2.4(c) below along with Lemma 1.1 above. Part (b) follows as a consequence ofLemma 2.2 below along with Lemma 1.1 above.

Note that ξ2 ∼ χ2
1. The next theorem gives an exponential estimate on the rate of convergenceto the chi-square distribution for energy IT of the O-U process.

Theorem 2.1
sup
x∈R

∣∣P {e−2θT 2θIT ≤ x
}
− P

{
ξ2 ≤ x

}∣∣ = O(e−θT ).

The above theorem is a consequence of the following lemma and the Esseen’s smoothing lemma
1.1

Lemma 2.2 For |u| ≤ eθT ε, ε sufficiently small, we have∣∣∣∣∣E exp
(
iue−2θT 2θIT

)
−

1

(1− 2iu)
1
2

∣∣∣∣∣ ≤ C(|u|+ |u|3)e−θT .

Proof. From Liptser and Shiryayev [23], we have
E exp

(
iue−2θT 2θIT

)
= exp

(
θT

2

)[
2γ

(γ − θ)e−γT + (γ + θ)eγT

]1/2

(2.12)

where
γ :=

(
θ2 − 2iue−2θT 2θ

)1/2
. (2.13)The lemma is an easy consequence of this result.

Lemma 2.3 For every δ > 0,
P
{∣∣e−2θT 2θIT − ξ2

∣∣ ≥ δ} ≤ Ce−2θT δ−2.

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 8

Proof : It is clear that
XT =

∫ T

0

e−θ(T−s)dWs . (2.14)

Further, Itô formula (see Friedman [19]), we have∫ T

0

eθ(T−s)dWs = WT − θ
∫ T

0

eθ(T−s)Wsds,

ξt =

∫ t

0

e−θsdWs = e−θtWt − θ
∫ t

0

e−θsdWs , ηt =

∫ t

0

eθsdWs = eθtWt + θ

∫ t

0

eθsdWs .Note that
E(X2

T ) =
1− e−2θT

2θ
, E(X4

T ) =
3(1− e−2θT )2

4θ
and E(IT ) =

2θT − 1 + e−2θT

4θ2
. (2.15)

By Itô formula, we have
IT =

X2
T

2θ
−
T

2θ
−
ZT
θ
. (2.16)By Chebyshev inequality, we have

P
{∣∣e−2θT 2θIT − ξ2

∣∣ ≥ δ} ≤ 1

δ2
E
∣∣e−2θT 2θIT − ξ2

∣∣2
=

1

δ2
E

∣∣∣∣e−2θT 2θ

∫ T

0

e2θtξ2
t dt − ξ2

∣∣∣∣2 =
1

δ2
E

∣∣∣∣e−2θT 2θ

∫ T

0

e2θtξ2
t dt − ξ2

T + ξ2
T − ξ2

∣∣∣∣2
≤

2

δ2

[
E|e−2θT 2θ

∫ T

0

e2θtξ2
t dt − ξ2

T |2 + E|ξ2
T − ξ2|2

]
≤

2

δ2

[
E|e−2θT 2θ

∫ T

0

e2θtξ2
t dt − ξ2

T |2 + E|ξT − ξ|2E|ξT + ξ|2
]

≤
2

δ2

[
E|e−2θT 2θ

∫ T

0

e2θtξ2
t dt − ξ2

T |2 +
e−2θT

√
2θ

]
≤ Ce−2θT δ−2 (2.17)

since E|ξT + ξ|2 ≤ 2E|ξT |2 + 2E|ξ|2 <∞.Since
E(ξT − ξ)2 =

∫ ∞
T

∫ ∞
T

e−θre−θs |r − s|−1drds =
e−2θT

√
2θ

(2.18)

hence
E(ξT − ξ)2 =

e−2θT

√
2θ

(2.19)

gives the L2 convergence rate. Recall that
IT =

∫ T

0

e2θtξ2
t dt, (2.20)

E(ξT − ξ)2 → 0 as T →∞, (2.21)

E(ξt − ξs)2 ≤ C(t − s). (2.22)We have
E
(

2θe−2θT IT − ξ2
)2

= E

(
2θe−2θT

∫ T

0

e2θtξ2
t dt − ξ2

)2

= E

(
2θ

e2θT

∫ T

0

e2θtξ2
t dt − ξ2

)2

. (2.23)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 9Further, by Toeplitz’s lemma
lim
T→∞

2θe−2θT

∫ T

0

e2θtξ2
t dt = lim

T→∞
ξ2
T = ξ2 almost surely. (2.24)

E(ξ2) <∞ which implies that P (ξ = 0) = 0. We have
lim
T→∞

[
2θe−2θT

∫ T

0

e2θtξ2
t dt − ξ2

T

]
= 0 almost surely. (2.25)

Because of the continuity of ξt , for every t ≥ 0,∫ T

0

e2θtξ2
t dt ≥

∫ T

T
2

e2θtξ2
t dt ≥

T

2
eθT

(
inf

T
2
<t<T

ξ2
t

) almost surely. (2.26)

Furthermore the continuity of ξt , gives
lim
T→∞

(
inf

T
2
<t<T

ξ2
t

)
= ξ2 almost surely. (2.27)

lim
T→∞

∫ T

0

e2θtξ2
t dt =∞ almost surely. (2.28)

By L’Hopital rule,
lim
T→∞

∫ T
0 e2θtξ2

t dt

e2θT
= lim
T→∞

ξ2
T

2θ
=
ξ2

2θ
almost surely. (2.29)

θT − θ =

∫ T
0 eθtξtdWt∫ T
0 e2θtξ2

t dt
=

ξ2
T

2e−2θT
∫ T

0 e2θtξ2
t dt
− θ. (2.30)

θT − θ → 0 almost surely. (2.31)

θ̂T − θ =
ξ2
T − 2θe−2θT

∫ T
0 e2θtξ2

t dt

2e−2θT
∫ T

0 e2θtξ2
t dt

. (2.32)

It is easy to verify that
E

[
ξ2
T − 2θe−2θT

∫ T

0

e2θtξ2
t dt

]2

≤ Ce−2θT . (2.33)

This completes the proof of the lemma.
The following lemma (Cameron-Martin Type Theorem) gives the bound on the joint characteristicfunctions of the sufficient statistics defining the MLE:

Lemma 2.4 (a) Let φT (z1, z2) := E exp(z1IT + z2X
2
T ), z1, z2 ∈ C. Then φT (z1, z2) exists for

|zi | ≤ δ, 1 = 1,2 for some δ > 0 and is given by
φT (z1, z2) = exp

(
θT

2

)[
2γ

(γ − θ + 2z2)e−γT + (γ + θ − 2z2)eγT

]1/2

where γ = (θ2 − 2z1)1/2 and we choose the principal branch of the square root.

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 10

(b) Let HT,x :=
(
e−2θT 4θ2

)1/2
ZT −

(
e−2θT 4θ2IT − ξ2

)
x. Then for |x | ≤ 2(log e2θT )1/2 and for

|u| ≤ εeθT , where ε is sufficiently small∣∣∣∣E exp(iuHT,x)− exp(−
u2

2
)

∣∣∣∣ ≤ C exp(
−|u|

2
)(|u|+ |u|3)e−θT .

(c) For |u| ≤ ε1e
θT , where ε1 is sufficiently small, we have as T →∞,∣∣∣∣E exp
{
iu
(
e−θT 2θ

)
ZT
}
− exp(−

u2

2
)

∣∣∣∣ ≤ C exp(−
|u|
2

)(|u|+ |u|3)e−θT .

Part (a) is from Bishwal [5].We shall prove part (b) in details. Proof of part (c) is very similar to part (b) and will be omitted.
Proof : By Itô formula,

ZT = θIT +
X2
T

2
−
T

2
.Note that

E exp(iuHT,x) = E exp
[
−iu

(
e−2θT 4θ2

)1/2
ZT − iu

((
e−2θT 4θ2

)
IT − ξ2

)
x
]

= E exp

[
−iu

(
e−2θT 4θ2

)1/2
{
θIT +

X2
T

2
−
T

2

}
− i t

((
e−2θT 4θ2

)
IT − ξ2

)
x

]
= E exp(z1IT + z2X

2
T + z3) = exp(z3)φT (z1, z2)

where
z1 = −iuθδT,x , z2 = −

iu

2

(
e−2θT 4θ2

)1/2
, z3 =

iuT

2
δT,x , δT,x =

(
e−2θT 4θ2

)1/2
+

2x

T
.

Note that (z1, z2) satisfies the conditions of (a) by choosing ε sufficiently small. Let
α1,T (u), α2,T (u),

α3,T (u) and α4,T (u) be functions which are of the orders O(|u|e−θT/2), O(|u|2e−θT/2),

O(|u|3e−3θT/2) and O(|u|3e−θT/2) respectively. Note that for the given range of values of xand u, the conditions on zi for part (a) of Lemma are satisfied. Note also that z2 = α1,T (u).Further, with
βT (t) = 1 + iu

δT,x
θ

+
u2δ2

T,x

2θ2
,

γ = (θ2 − 2z1)1/2 = θ

[
1−

z1

θ2
−
z2

1

2θ4
+
z3

1

2θ8
+ · · ·

]
= θ

[
1 + iu

δT,x
θ

+
u2δ2

T,x

2θ2
+
iu3δ3

T,x

2θ3
+ · · ·

]
= θ[1 + α1,T (u) + α2,T (u) + α3,T (u)] = θβT (u) + α3,T (u) = θ[1 + α1,T (u)].

Thus γ − θ = α1,T , γ + θ = 2θ + α1,T . Hence the above expectation equals
exp

(
z3 +

θT

2

)[
2θβT (u) + α3,T (u)

α1,T exp{−θTβT (u) + α4,T (u)}+ (2θ + α1,T (u)) exp{θTβT (u) + α4,T (u)}

]1/2

=

[
1 + α1,T (u)

α1,T exp(χT (u)) + (1 + α1,T (u)) exp(ψT (u))

]1/2

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 11where
χT (u) := −θTβT (u) + α4,T (u)− 2z3 − θT = −2θT + α1,T (u) + t2α1,T (u),

ψT (u) := θTβT (u) + α4,T (u)− 2z3 − θeθT = θT

[
1 + iu

δT,x
θ

+
u2δ2

T,x

2θ2

]
+ α4,T (u)− i teθT δT,x − θeθT

=
u2eθT

2θ

[(
4θ2

e2θT

)1/2

+
2x

eθT

]2

= u2 + u2α1,T (u).

Hence, for the given range of values of u, χT (u)−ψT (u) ≤ −θeθT . Hence the above expectationequals
exp(−

t2

2
)(1 + α1,T )1/2

[
α1,T exp{−2θeθT + α1,T + u2α1,T }+ (1 + α1,T (u)) exp{t2α1,T (u)}

]−1/2

= exp(−
u2

2
)
[
1 + α1,T )(1 + α1,T (1 + α1,T ) exp{−θeθT + α1,T + t2α1,T }

]
exp(u2α1,T (u)).

Lemma 2.4 (c) and Lemma 2.2 respectively give the Berry-Esseen rate for ZT and IT immediatelyby using the Esseen’s lemma 1.1.
Corollary 2.1

(a) sup
xεR

∣∣∣∣∣P
{(

4θ2

e2θT

)1/2

ZT ≤ x

}
−Φ(x)

∣∣∣∣∣ ≤ Ce−θT .
(b) sup

x∈R

∣∣∣∣∣P
{(

4θ2

e2θT

)1/2(
θIT − ξ2 e

θT

2

)
≤ x

}
−Φ(x)

∣∣∣∣∣ ≤ Ce−θT .
Remark Though this was basically shown in Lemma 2.1, here we obtain Kolmogorov distance fora martingale and Kolmogorov distance for its quadratic variation through Cameron-Matin typeresults which are generalization of Levy area formula. In Lemma 2.1, one could go directly to theStein-Malliavin way through Wiener chaos expansion which does not depend on any martingalecharacteristics.Before we prove the results on the Berry-Esseen bound on the Kolmogorov distance for theMLE with random norming we need the following large deviation result for the MLE. This canbe obtained as a consequence of Lemma 3.1 of Bercu et al. [3] or Bercu and Richou [4] who usethe Gartner-Ellis’s theorem and the contraction principle. However we give a direct proof usingFeller’s approach.
Lemma 2.5

P

{(
e2θT

4θ2

)1/2

|θT − θ| ≥ 2(2θT )1/2

}
≤ Ce−θT .

Proof : Observe that
P

{(
e2θT

4θ2

)1/2

|θT − θ| ≥ 2(2θT )1/2

}
= P


∣∣∣∣∣∣∣
(

4θ2

e2θT

)1/2

ZT

( 2θ
e2θT )IT

∣∣∣∣∣∣∣ ≥ 2(2θT )1/2



https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 12

≤ P

{∣∣∣∣∣
(

4θ2

e2θT

)1/2

ZT

∣∣∣∣∣ ≥ (θT )1/2

}
+ P

{∣∣∣∣ 2θ

e2θT
IT

∣∣∣∣ ≤ 1

2

}
≤

∣∣∣∣∣P
{(

4θ2

e2θT

)1/2

|ZT | ≥ (θT )1/2

}
− 2Φ(−(2θT )1/2)

∣∣∣∣∣+ 2Φ(−(2θT )1/2) + P

{∣∣∣∣ 2θ

e2θT
IT − ξ2

∣∣∣∣ ≥ 1

2

}
≤ sup

x∈R

∣∣∣∣∣P
{(

4θ2

e2θT

)1/2

|ZT | ≥ x

}
− 2Φ(−x)

∣∣∣∣∣+ 2Φ(−(2θT )1/2) + P

{∣∣∣∣( 4θ2

e2θT

)
IT − ξ2

∣∣∣∣ ≥ 1

2

}
≤ sup

x∈R

∣∣∣∣∣P
{(

4θ2

e2θT

)1/2

|ZT | ≥ x

}
− 2Φ(−x)

∣∣∣∣∣+ 2Φ(−(2θT )1/2) + P

{∣∣∣∣( 4θ2

e2θT

)
IT − ξ2

∣∣∣∣ ≥ 1

2

}
≤ Ce−θT + C(e2θT 2θT )−1/2 + C(e2θT )−1 ≤ Ce−θT .

The bounds for the first and the third terms come from Corollary 2.1 (a) and Lemma 2.3 respectivelyand that for the middle term comes from Feller ( [17], p. 166).
We are now in a position to obtain the Berry-Esseen bound of the order O(e−θT ) on theKolmogorov distance for the MLE.
Theorem 2.2

sup
x∈R

∣∣∣∣P {(eθT2θ

)
(θT − θ) ≤ x

}
− C(x)

∣∣∣∣ = O(e−θT ).

Proof : We shall consider two possibilities: (i) |x | > 2(θT )1/2 and (ii) |x | ≤ 2(θT )1/2.(i) We shall give a proof for the case x > 2(θT )1/2. The proof for the case x < −2(θT )1/2 runssimilarly. Note that∣∣∣∣P {(eθT2θ

)
(θT − θ) ≤ x

}
− C(x)

∣∣∣∣ ≤ P {(eθT2θ

)
(θT − θ) ≥ x

}
+ C(−x)

But C(−x) ≤ C(−2(θT )1/2) ≤ Ce−2θT . Moreover by Lemma 2.5, we have
P

{(
eθT

2θ

)
(θT − θ) ≥ 2(θT )1/2

}
≤ Ce−θT/2.

Hence ∣∣∣∣∣P
{(

eθT

2θ

)1/2

(θT − θ) ≤ x

}
− C(x)

∣∣∣∣∣ ≤ Ce−θT/2.

(ii)
Let AT :=

{(
eθT

2θ

)
|θT − θ| ≤ 2(θT )1/2

} and BT :=

{
IT
eθT

> c0

}
where 0 < c0 <

1
2θ . By Lemma 2.5, we have

P (AcT ) ≤ Ce−θT . (2.34)

By Lemma 2.3, we have
P (BcT ) = P

{
2θ

eθT
IT − ξ2 < 2θc0 − ξ2

}
< P

{∣∣∣∣ 2θ

eθT
IT − ξ2

∣∣∣∣ > ξ2 − 2θc0

}
≤ Ce−θT . (2.35)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 13Let b0 be some positive number. For ω ∈ AT ∩BT and for all T > T0 with 4b0(2θT0)1/2( 2θ
eθT0

)1/2 ≤
c0, we have (

eθT

2θ

)
(θT − θ) ≤ x

⇒ IT + b0e
θT (θT − θ) < IT +

(
eθT

2θ

)
2b0θx

⇒
(
eθT

2θ

)
(θT − θ)[IT + b0e

θT (θT − θ)] < x [IT +

(
eθT

2θ

)
2b0θx ]

⇒ (θT − θ)IT + b0T (θT − θ)2 <

(
2θ

eθT

)
IT x + 2b0θx

2

⇒ ZT + (θT − θ)IT + b0e
θT (θT − θ)2 < ZT +

(
2θ

eθT

)
IT x + 2b0θx

2

⇒ 0 < ZT +

(
2θ

eθT

)
IT x + 2b0θx

2

since
IT + b0e

θT (θT − θ) > eθT c0 + b0e
θT (θT − θ)

> 4b0(θT )1/2

(
2θ

eθT

)
− 2b0(θT )1/2(

2θ

eθT
) = 2b0(θT )1/2

(
2θ

eθT

)
> 0.

Hence, for ω ∈ AT ∩ BT ,(
eθT

2θ

)
(θT − θ) ≤ x ⇒ ZT +

(
2θ

eθT

)
IT x + 2b0θx

2 > 0.

On the other hand, for ω ∈ AT ∩ BT and for all T > T0 with 4b0(2θT0)1/2( 2θ
eθT0

) ≤ c0, we have
(
eθT

2θ

)
(θT − θ) > x

⇒ IT − b0e
θT (θT − θ) < IT −

(
eθT

2θ

)
2b0θx

⇒
(
eθT

2θ

)
(θT − θ)[IT − b0e

θT (θT − θ)] > x [IT −
(
eθT

2θ

)
2b0θx ]

⇒ (θT − θ)IT − b0e
θT (θT − θ)2 >

(
2θ

eθT

)
IT x − 2b0θx

2

⇒ ZT + (θT − θ)IT − b0e
θT (θT − θ)2 > ZT +

(
2θ

eθT

)
IT x − 2b0θx

2

⇒ 0 > ZT +

(
2θ

eθT

)
IT x − 2b0θx

2

since
IT − b0e

θT (θT − θ) > eθT c0 − b0e
θT (θT − θ)

> 4b0(θT )1/2

(
2θ

eθT

)
− 2b0(θT )1/2

(
2θ

eθT

)
= 2b0(θT )1/2

(
2θ

eθT

)
> 0.

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 14Hence, for ω ∈ AT ∩ BT ,
0 < ZT +

(
2θ

eθT

)
IT x − 2b0θx

2 ⇒
(
eθT

2θ

)
(θT − θ) ≤ x.

We use the squeezing method developed in Pfanzagl [28] for the i.i.d. case instead of the splittingmethod of Michel and Pfanzagl [28]. Let us introduce the piecewise quadratic random functionsinvolving the martingale and quadratic variation part of θT − θ:
g±(x) := ZT + (

2θ

eθT
)IT x ± 2b0θx

2.

Let us introduce the events
D±T,x :=

{
ZT + (

2θ

eθT
)IT x ± 2b0θx

2 > 0

}
.

Thus we have
D−T,x ∩ AT ∩ BT ⊆ AT ∩ BT ∩

{(
eθT

2θ

)
(θT − θ) ≤ x

}
⊆ D+

T,x ∩ AT ∩ BT . (2.36)

This gives
P (D−T,x ∩ AT ∩ BT ) ≤ P

(
AT ∩ BT ∩

{(
eθT

2θ

)
(θT − θ) ≤ x

})
≤ P (D+

T,x ∩ AT ∩ BT )

so that ∣∣∣∣P (AT ∩ BT ∩{(eθT2θ

)
(θT − θ) ≤ x

})
− C(x)

∣∣∣∣
≤ max

{
|P (D−T,x ∩ AT ∩ BT )− C(x)|, |P (D+

T,x ∩ AT ∩ BT )− C(x)|
}

≤ max
{
|P (D−T,x)− C(x)|, |P (D+

T,x)− C(x)|
}

+ P (AT ∩ BT )c .From (2.34) and (2.35),
P (AT ∩ BT )c ≤ Ce−θTfor all T > T0 and |x | ≤ 2(θT )1/2. If it is shown that∣∣P {D±T,x}− C(x)

∣∣ ≤ Ce−θT (2.37)

for all T > T0 and |x | ≤ 2(θT )1/2, then the theorem would follow from (2.34) – (2.37).We shall prove (2.37) for D+
T,x . The proof for D−T,x is analogous.Note that ∣∣P {D+

T,x

}
− C(x)

∣∣ =

∣∣∣∣P {−(
2θ

eθT
)ZT −

(
2θ

eθT
IT − ξ2

)
x < x + 2

(
2θ

eθT

)
b0θx

2

}
− C(x)

∣∣∣∣
≤ sup

y∈R

∣∣∣∣P {−( 2θ

eθT

)
ZT −

(
2θ

eθT
IT − ξ2

)
x ≤ y

}
− C(y)

∣∣∣∣+

∣∣∣∣C (x +

(
2θ

eθT

)
b0θx

2

)
− C(x)

∣∣∣∣
=: ∆1 + ∆2.

(2.38)Lemma 2.4 (b) and Esseen’s Smoothing Lemma 1.1 immediately yield
∆1 ≤ Ce−θT . (2.39)

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 15On the other hand, for all T > T0,
∆2 ≤ 2(

2θ

eθT
)b0θx

2(2π)−1/2 exp(−x2/2)

where
|x − x | ≤ 2

(
2θ

eθT

)
b0θx

2.

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

(
2θ

eθT

)
b0θx

2(2π)−1/2x2 exp(−x2/8) ≤ Ce−θT . (2.40)

From (2.38) - (2.40), we obtain ∣∣P {D+
T,x

}
− C(x)

∣∣ ≤ Ce−θT .
This completes the proof of the theorem.
Concluding Remarks

(1) The bound in Theorem 2.2 is uniform over compact subsets of the parameter space Θ.(2) The bound in Theorem 2.2 is optimal and cannot be improved further.(3) Note that in the critical case, i.e., when θ = 0, the MlE has a distribution concentratedon a half line, precisely the distribution of the ratio of a noncentral chisquare to the to the sumof chisquares. Note that the behaviour of the O-U process depends on both the initial condition
X0 = X0 and the parameter space. Classically it has been assumed that X0 is either has a normaldistribution or a nonzero constant and θ < 0 which makes the process stationary with Gaussianinvariant distribution. If X0 = 0 is with θ < 0, then the process is asymptotically stationary andergodic. In above two cases the model satisfies the LAN (local asymptotic normality) property.With X0 a nonzero constant and θ > 0 the process is transient and satisfies the LAMN (localasymptotic mixed normality) property. With θ = 0, the process is nonstationary and satisfies theLABF (local asymptotic Brownian functional) property. For all θ ∈ R, the model satisfies the LABFproperty, see Bishwal [9] for the definitions of these LAN, LAMN and LABF properties. Bishwal [9]has shown that sequential sampling based on a stopping rule unifies the three properties andmakes them LAN.(4) It remains to study the Kolmogorov distnace for Bayes estimator from both continuous andsiscrete observations and approximate maximum likelihood estimator from discrete observations inthe nonergodic case.(5) Extension to multidimensional process and to multiparameter case remains to be investigated.(6) It remains to investigate the nonuniform rates of convergence to Cauchy distribution whichare more useful.

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 16References
[1] B. Bercu, On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases,Bernoulli. 7 (2001) 299-316.[2] B. Bercu, A. Roualt, Sharp large deviation for the Ornstein-Uhlenbeck process, Theory Prob. Appl. 46 (2002) 1-19.[3] B. Bercu, L. Coutin, N. Savy, Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process, Stoch.Process Appl. 122 (2012) 3393-3424.[4] B. Bercu, A. Richou, Large deviations for the Ornstein-Uhlenbeck process without tears, Stat. Prob. Lett. 123 (2017)45-55.[5] J.P.N. Bishwal, Sharp Berry-Esseen bound for the maximum likelihood estimator in the Ornstein-Uhlenbeck process,Sankhya, Ser. A. 62 (2000) 1-10.[6] J.P.N. Bishwal, Rates of convergence of the posterior distributions and the Bayes estimators in the Ornstein-Uhlenbeck process, Rand. Oper. Stoch. Equ. 8 (2000b) 51-70.[7] J.P.N. Bishwal, Accuracy of normal approximation for the maximum likelihood and the Bayes estimators in theOrnstein-Uhlenbeck process using random norming, Stat. Prob. Lett. 52 (2001) 427-439.[8] J.P.N. Bishwal, Parameter estimation in stochastic differential equations, Lecture Notes in Mathematics, 1923,Springer-Verlag, (2008).[9] J.P.N. Bishwal, Sequential maximum likelihood estimation in nonlinear non-Markov diffusion type processes, Dyn.Syst. Appl. 27 (2018) 107-124.[10] J.P.N. Bishwal, Berry-Esseen bounds of approximate Bayes estimators for the discretely observed Ornstein-Uhlenbeck process, Asian J. Stat. Sci. 1 (2021) 83-122.[11] J.P.N. Bishwal, Parameter estimation in stochastic volatility models, Springer Nature, Cham, (2022).[12] J.P.N. Bishwal, A. Bose, Speed of convergence of the maximum likelihood estimator in the Ornstein-Uhlenbeckprocess, Calcutta Stat. Assoc. Bull. 45 (1995), 245-251.[13] J.P.N. Bishwal, A. Bose, Rates of convergence of approximate maximum likelihood estimators in the Ornstein-Uhlenbeck process, Comp. Math. Appl. 42 (2001) 23-38.[14] A.A. Borokov, On the rate of convergence for the invariance principle, Theory Prob. Appl. 18 (1973) 217-234.[15] H.M. Dietz, Y.A. Kutoyants, Parameter estimation in some non-recurrent solutions of SDE, Stat. Decis. 21 (2003)29-45.[16] P.D. Feigin, Maximum likelihood estimation for continuous time stochastic processes, Adv. Appl. Prob. 8 (1976)712-736.[17] W. Feller, An introduction to probability theory and its applications, Vol. I, Wiley, New York, (1957).[18] W. Feller, An introduction to probability theory and its applications, Vol. II, Wiley, New York, (1971).[19] A. Friedman, Stochastic differential equations, Vol. I, Academic Press, New York, (1975).[20] P. Hall, C.C. Heyde, Martingale limit theory and its applications, Academic Press, New York, (1980).[21] U. Horst, J. Wenzelburger, On non-ergodic asset prices, Econ. Theory. 34 (2008) 207-234.[22] A.V. Ivanov, The Berry-Esseen inequality for the distribution of the least squares estimae, Math. Notes. 20 (1976)721-727.[23] R.S. Liptser, A.N. Shiryayev, Statistics of random processes I : general theory, Springer-Verlag, Berlin, (1977).[24] R. Michel, J. Pfanzagl, The accuracy of the normal approximation for minimum contrast estimate, Zeit Wahr. Verw.Gebiete. 18 (1971) 73-84.[25] I. Nourdin, G. Peccati, Stein’s method on Wiener chaos, Prob. Theory Related Fields. 145 (2009) 75-118.[26] I. Nourdin, G. Peccati, Normal approximation with Malliavin calculus: from Stein’s method to universality, CambridgeUniversity Press, Cambridge, (2012).[27] V.V. Petrov, Limit theorems of probability theory, Oxford University Press, Oxford, (1995).[28] J. Pfanzagl, The Berry-Esseen bound for minimum contrast estimators, Metrika. 17 (1971) 82-91.

https://doi.org/10.28924/ada/ma.3.25


Eur. J. Math. Anal. 10.28924/ada/ma.3.25 17

[29] P. Protter, Stochastic integration and differential equations: a new approach, Springer-Verlag, Berlin, (1990).[30] Y. Shimizu, Local asymptotic mixed normality for discretely observed non-recurrent Ornstein-Uhlenbeck processes,Ann. Inst. Stat. Math. 64 (2012) 193-211.

https://doi.org/10.28924/ada/ma.3.25

	References

