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Elect. Comm. in Probab. 15 (2010), 396–410

ELECTRONIC
COMMUNICATIONS
in PROBABILITY

CENTRAL LIMIT THEOREM FOR THE THIRD MOMENT IN SPACE
OF THE BROWNIAN LOCAL TIME INCREMENTS
YAOZHONG HU
Department of Mathematics, University of Kansas, Lawrence, Kansas
66045 USA
email: hu@math.ku.edu
DAVID NUALART1
Department of Mathematics, University of Kansas, Lawrence, Kansas
66045 USA
email: nualart@math.ku.edu
Submitted February 19, 2010, accepted in final form September 13, 2010
AMS 2000 Subject classification: 60H07, 60H05, 60F05
Keywords: Brownian motion, local time, Clark-Ocone formula
Abstract
The purpose of this note is to prove a central limit theorem for the third integrated moment of the

Brownian local time increments using techniques of stochastic analysis. The main ingredients of
the proof are an asymptotic version of Knight’s theorem and the Clark-Ocone formula for the third
integrated moment of the Brownian local time increments.

1

Introduction

Let {B t , t ≥ 0} be a standard one-dimensional Brownian motion. We denote by {L tx , t ≥ 0, x ∈ R}
a continuous version of its local time. The following central limit theorem for the L 2 modulus of
continuity of the local time has been recently proved:
Œ

‚Z
− 32

(L tx+h

h

R

−

L tx )2 d x

− 4th

8
−→ p
3

Œ1

‚Z

2

L

(L tx )2 d x

η,

(1)

R

where η is a N (0, 1) random variable independent of B and L denotes the convergence in law.
This result has been first proved in [3] by using the method of moments. In [4] we gave a simple
proof based on Clark-Ocone formula and an asymptotic version of Knight’s theorem (see Revuz
and Yor [9], Theorem (2.3), page 524). Another simple proof of this result with the techniques of
stochastic analysis has been given in [11].
The following extension of this result to the case of the third integrated moment has been proved
recently by Rosen in [12] using the method of moments.
1

D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0904538.

396

CLT for the third moment of Brownian local time increments

397

Theorem 1. For each fixed t > 0
1
h2

Z
(L tx+h
R

−

L tx )3 d x

p
−→ 8 3

Œ1

‚Z

2

L

(L tx )3 d x

η

R

as h tends to zero, where η is a normal random variable with mean zero and variance one that is
independent of B.
The purpose of this paper is to provide a proof of Theorem 1 using the same ideas as in [4]. The
main ingredient is to use Clark-Ocone stochastic integral representation formula which allows us
to express the random variable
Z

F th =
R

(L tx+h − L tx )3 d x

(2)

as a stochastic integral. In comparison with the L 2 modulus of continuity, the situation is here more
complicated and we require some new and different techniques. First, there are four different
terms (instead of two) in the stochastic integral representation, and two of them are martingales.
Surprisingly, some of the terms of this representation converge in L 2 (Ω) to the derivative of the
self-intersection local time and the limits cancel out. Finally, there is a remaining martingale term
to which we can apply the asymptotic version of Knight’s theorem. As in the proof of (1), to show
the convergence of the quadratic variation of this martingale and other asymptotic results we
make use of Tanaka’s formula for the time-reversed Brownian motion and backward Itô stochastic
integrals.
We believe that a similar result could be established for the integrated moment of order p for
an integer p ≥ 4 using Clark-Ocone representation formula, but the proof would be much more
involved.
These results are related to the behavior of the Brownian local time in the space variable. It was

proved by Perkins [7] that for any fixed t > 0, {L tx , x ∈ R} is a semimartingale with quadratic
Rb
variation 〈L t 〉 b − 〈L t 〉a = 4 a L tx d x. This property provides an heuristic explanation of the central
limit theorems presented above. The proof of these theorems, however, requires more complicated
tools.
The paper is organized as follows. In the next section we recall some preliminaries on Malliavin
calculus. In Section 3 we establish a stochastic integral representation for the derivative of the
self-intersection local time, which has its own interest, and for the random variable F th defined
in (2). Section 4 is devoted to the proof of Theorem 1, and the Appendix contains two technical
lemmas.

2

Preliminaries on Malliavin Calculus

Let us recall some basic facts on the Malliavin calculus with respect the Brownian motion B =
{B t , t ≥ 0}. We refer to [5] for a complete presentation of these notions. We assume that B is
defined on a complete probability space (Ω, F , P) €such that F Šis generated by B. Consider the set
S of smooth random variables of the form F = f B t 1 , . . . , B t n ,, where t 1 , . . . , t n ≥ 0, n ∈ N and
f is bounded and infinitely differentiable with bounded derivatives of all orders. The derivative

operator D on a smooth random variable of this is defined by
Dt F =

n
X
∂f €
i=1

∂ xi

Š
B t 1 , . . . , B t n 1[0,t i ] (t).

398

Electronic Communications in Probability

We denote by D1,2 the completion of S with respect to the norm kF k1,2 given by
‚Z ∞
Œ

” —

2
2
2
kF k1,2 = E F + E
Dt F d t .
0

The classical Itô representation theorem asserts that any square integrable random variable can
be expressed as
Z∞
F = E[F ] +

ut d Bt ,

0

where u = {u t , t ≥ 0} is a unique adapted process such that E

R ∞

0

u2t d t

1,2


< ∞. If F belongs to

D , then u t = E[D t F |F t ], where {F t , t ≥ 0} is the filtration generated by B, and we obtain the
Clark-Ocone formula (see [6])
Z∞
F = E[F ]+

0

3

E[D t F |F t ]d B t .

(3)

Stochastic integral representations

Consider the random variable γ t defined rigorously as the limit in L 2 (Ω)
Z tZ

u

γ t = lim

ǫ→0

0

0

pǫ′ (Bu − Bs )dsdu,

(4)

1

where pǫ (x) = (2πǫ)− 2 exp(−x 2 /2ǫ). The process γ t coincides with the derivative − ddy α t ( y)| y=0
of the self-intersection local time
Z tZ u
α t ( y) =
0

0

δ y (Bu − Bs )dsdu.

The derivative of the self-intersection local time has been studied by Rogers and Walsh in [10] and
by Rosen in [11]. We are going to use Clark-Ocone formula to show that the limit (4) exists and
to provide an integral representation for this random variable.
Rt Ru
Lemma 2. Set γεt = 0 0 pǫ′ (Bu − Bs )dsdu. Then, γεt converges in L 2 (Ω) as ǫ tends to zero to the
random variable
Œ
Z t ‚Z r
p t−r (B r − Bs )ds − L rBr

γt = 2
0

0

d Br .

Proof. By Clark-Ocone formula applied to γεt we obtain the integral representation
Z
γεt

1

=
0

E(D r γεt |F r )d B r ,

where {F t , t ≥ 0} denotes the filtration generated by the Brownian motion. Then,
Z tZ

D r γεt

u

=
0

0

pε′′ (Bu − Bs )1[s,u] (r)dsdu,

CLT for the third moment of Brownian local time increments

and for any r ≤ t

Z tZ

E(D r γεt |F r )

Z tZ

r
′′
pε+u−r
(B r

=
0

r

Z

=

399

− Bs )dsdu = 2

r

r

∂ pε+u−r
∂u

0

(B r − Bs )dsdu

r

(pε+t−r (B r − Bs ) − pε (B r − Bs ))ds.

2
0

As ǫ tends to zero this expression converges in L 2 (Ω × [0, t]) to
Œ
‚Z r
p t−r (B r − Bs )ds − L rBr

2
0

,

which completes the proof.
Let us now obtain a stochastic integral representation for the third integrated moment F th =
R
(L tx+h − L tx )3 d x. Notice first that E(F th ) = 0 because F th is an odd functional of the BrownR
ian motion.
Rt
P4
Proposition 3. We have F th = 0 Φ r d B r , where Φ r = i=1 Φ(i)
r , and
Z
Φ(1)
r

=

€

6
R

L z+h
− L zr
r

Z Z
Φ(2)
r

=

Φ(3)
r

=

Φ(4)
r

=

−6

€

0
r

R

12h
p
2π

h

Z Z
0

12h
−p
2π

Z

r

1[0,h] (B r − z)dz

L z+h
− L zr
r
Z

h

∞
h2
t−r

−h

0

Š2

Š2

p t−r (B r − z − y)d y dz
3

z

p t−r− h2 (B r − Bs + y)z − 2 (1 − e− 2 )dzd y ds
z

Z

1[−h,h] (B r − Bs )ds

∞
h2
t−r

3

z

z − 2 (1 − e− 2 )dz.

Proof. Let us write
Z ‚Z
F th

=

lim

ǫ→0

R

Z Z

=

t

6 lim

ǫ→0

D

0





pǫ (Bs − x − h) − pǫ (Bs − x) ds

3 ”
Y
R i=1

Œ3
dx

—
pǫ (Bsi − x − h) − pǫ (Bsi − x) d x ds,

where D = {(s1 , s2 , s3 ) ∈ [0, t]3 : s1 < s2 < s3 }. We can pass to the limit as ε tends to zero the first
factor pǫ (Bs1 − x − h) − pǫ (Bs1 − x), and we obtain
Z n
”
—”
—
h
pǫ (Bs2 − Bs1 ) − pǫ (Bs2 − Bs1 + h) pǫ (Bs3 − Bs1 ) − pǫ (Bs3 − Bs1 + h)
F t = lim 6
ǫ→0

”

D

− pǫ (Bs2 − Bs1 − h) − pǫ (Bs2 − Bs1 )
Z
=

Φǫ (s)ds.

lim 6

ǫ→0

D

—”

pǫ (Bs3 − Bs1 − h) − pǫ (Bs3 − Bs1 )

—o

ds

400

Electronic Communications in Probability

R
We are going to apply the Clark-Ocone formula to the random variable D Φǫ (s)ds. Fix r ∈ [0, t].
R




We need to compute D E D r Φǫ (s) |F r ds. To do this we decompose the region D, up to a set
of zero Lebesgue measure, as D = D0 ∪ D1 ∪ D2 , where
=

D1

=

{s : 0 ≤ s1 < s2 < r < s3 ≤ t},

{s : 0 ≤ s1 < r < s2 < s3 ≤ t},


and D0 = {s : 0 ≤ r ≤ s1 } ∪ {s : s3 ≤ r ≤ t}. Notice that on D0 , D r Φǫ (s) = 0.
Step 1 For the region D1 we obtain
”
—




E D r Φǫ (s) |F r
=
pǫ (Bs2 − Bs1 ) − pǫ (Bs2 − Bs1 + h)
h
i
′
′
× pǫ+s
(B
−
B
)
−
p
(B
−
B
+
h)
r
s1
r
s1
ǫ+s3 −r
3 −r
”
—
− pǫ (Bs2 − Bs1 − h) − pǫ (Bs2 − Bs1 )
h
i
′
′
× pǫ+s
(B
−
B
−
h)
−
p
(B
−
B
)
.
r
s
r
s
ǫ+s3 −r
1
1
3 −r
D2

We can write this in the following form
Zh
”
—




′′
(B r − Bs1 + y)d y
E D r Φǫ (s) |F r
=
pǫ (Bs2 − Bs1 + h) − pǫ (Bs2 − Bs1 ) pǫ+s
3 −r
0

Z

−

0
h

Z
=

h

2

”

”

0

Z

−2

h

0

—
′′
pǫ (Bs2 − Bs1 ) − pǫ (Bs2 − Bs1 − h) pǫ+s
(B r − Bs1 − y)d y
3 −r

pǫ (Bs2 − Bs1 + h) − pǫ (Bs2 − Bs1 )
”

— ∂ pǫ+s3 −r

pǫ (Bs2 − Bs1 ) − pǫ (Bs2 − Bs1 − h)

∂ s3

(B r − Bs1 + y)d y

— ∂ pǫ+s3 −r
∂ s3

(B r − Bs1 − y)d y.

Integrating with respect to the variable s3 yields
Z
Z
Zh
”
—




E D r Φǫ (s) |F r ds = 2
pǫ (Bs2 − Bs1 + h) − pǫ (Bs2 − Bs1 )
D

0≤s