Statistics & Probability Letters 64 (2003) 133 – 145
Precise asymptotics in laws of the iterated logarithm for
Wiener local time
Ji-Wei Wen∗ , Li-Xin Zhang
Department of Mathematics, Zhejiang University, Xixi Campus, Hangzhou 310028, China
Received May 2002; received in revised form November 2002
Abstract
In this paper, we study the asymptotic properties of the upper and lower tail probabilities of the maximum
local time L∗ (t) of Wiener process (Brownian motion), and obtain some precise asymptotics in the law of
the iterated logarithm and the Chungs-type laws of the iterated logarithm for the supremum of Wiener local
time L(x; t); x ∈ R; t ∈ R+ .
c 2003 Elsevier B.V. All rights reserved.
MSC: 60F17; 60G15
Keywords: The law of the iterated logarithm; Local time; Brownian motion; Precise asymptotics
1. Introduction and main results
Let {W (t); t ¿ 0} be a standard one-dimensional Wiener process (Brownian motion) and let
L(x; t); x ∈ R; t ∈ R+ be its jointly continuous local time. Put L(0; t)=L(t) and supx∈R L(x; t)=L∗ (t).
Kesten (1965) proved the iterated logarithm laws
lim sup
L∗ (t)
=1
(2t log log t)1=2
lim inf
L∗ (t)(log log t)1=2
=
t 1=2
t →∞
a:s:
(1.1)
and
t →∞
a:s:;
Research supported by National Natural Science Foundation of China (No. 10071072).
Corresponding author.
E-mail addresses: jiweiwen@163.com (J.-W. Wen), lxzhang@mail.hz.zj.cn (L.-X. Zhang).
∗
c 2003 Elsevier B.V. All rights reserved.
0167-7152/03/$ - see front matter doi:10.1016/S0167-7152(03)00142-1
(1.2)
134
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
where the exact value of was not given by Kesten, he only estimated it from both sides. By using
a result of Borodin (1982) concerning the Laplace transform of the distribution of L∗ (t), CsEaki and
FGoldes (1986) gave the following expression for the exact and asymptotic distributions of L∗ (t).
From the scale change of the Wiener process it follows that t −1=2 L∗ (t) has the same distribution
as L∗ (1).
Theorem A. For any x ¿ 0,
P(L∗ (1) ¡ x) =
∞
k=1
∞ 2j 2
ck 2k 2 2
;
ak exp − 2k +
bk + 2 exp − 2
x
x
x
(1.3)
k=1
where 0 ¡ j1 ¡ j2 ¡ · · · are the positive zeros of the Bessel function
J0 (x) =
∞
(−1)k x 2k
(k!)2 2
k=0
and
ak =
4
sin2 jk
(k = 1; 2; : : :);
J1 (k)
−
bk = 4 −1 +
kJ0 (k)
ck = 16k
J1 (x) =
J1 (k)
J0 (k)
∞
k=0
Furthermore,
J1 (k)
J0 (k)
(k = 1; 2; : : :);
(k = 1; 2; : : :);
(−1)k x 2k+1
:
k!(k + 1)! 2
2j 2
P(L (1) ¡ x) ∼ a1 exp − 21
x
∗
2 as x → 0:
(1.4)
As a consequence
due
√
√ to CsEaki and FGoldes (1986), the exact value in (1.2) turns out to be
equal to 2j1 ≈ 2:4048 2 ≈ 3:4009.
CsEaki (1989) studied an integral test for the supremum of Wiener local time, and gave a large
deviation estimate for the distribution of L∗ (t).
1=2
2
2
∗
P(L (1) ¿ x) ∼ 4
xe−x =2 as x → ∞:
(1.5)
Recently, Zhang (2001a, b) studied the precise asymptotics in the lim sup-type and lim inf-type
laws of the iterated logarithm for partial sums of i.i.d.r.v’s. In those papers, by using the strong
approximation and Feller’s and Einmahl’s truncation method, he showed many sharper results than
that of Gut and SpKataru (2000), obtained the best necessary and suLcient conditions.
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
135
Inspired by Gut and SpKataru (2000) and Zhang (2001a, b), we investigate the corresponding
asymptotic properties for the local time of Brownian motion, and obtain the following results.
Theorem 1.1. Let ¿ − 1 and ¿ −
n () log log n → 3
2
and let n () be a function of such that
as n → ∞ and √
1 + ;
(1.6)
where is a real number. Then
lim
(2 − − 1)+3=2
√
1+
=8
∞
(log n) (log log n)
n
n=3
P{L∗ (1) ¿
2 log log n( + n ())}
+1
3
1=2
exp{−2(1 + ) } +
:
2
(1.7)
Here, (·) is the gamma function.
Theorem 1.2. For any ¿ − 1, we have
lim 2(+1)
∞
(log log n)
0
n=3
n log n
P{L∗ (1) ¿ 2 log log n}
(1.8)
= 2−(+1) ( + 1)−1 E(L∗ (1))2(+1) :
Notice that L∗ (1)= 2 log log n and L∗ (n)= 2n log log n have the same distribution, Theorems 1.1
and 1.2 give the asymptotic properties of lim sup-type law of the iterated logarithm of L∗ (·). As
corresponding properties of lim inf-type (Chung’s type) law of the iterated logarithm of L∗ (·), we
show in Theorems 1.3 and 1.4.
Theorem 1.3. Let ¿ − 1 and ¿ − 1 and let n () be a function of such that
n () log log n → √
as n → ∞ and 1= 1 + ;
(1.9)
where is a real number. Then
lim
√
1= 1+
= a1
√
1
−
1+
1
2(1 + )3=2
+1 ∞
+1
n=3




2
(log n) (log log n)
2j1
P L∗ (1) 6
( + n ())


n
log log n
exp {2(1 + )3=2 }( + 1):
Here a1 and j1 are de;ned in Theorem A.
(1.10)
136
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
Theorem 1.4. For any ¿ − 12 , we have


∞


2
(log
log
n)
2j
1
P L∗ (1) 6 = ¡ ∞;
lim −2(+1)

∞
n log n
log log n 
(1.11)
n=3
where
=
∞
k=1
2(+1)
2(+1)
j1
j1
( + 1)ck j1 2(+1)
ak
( + 1):
+ bk
+
jk
k
2k 2 2
k
(1.12)
Here, jk (k = 1; 2; : : :) are de;ned in Theorem A.
2. Proofs of the theorems
In this section, we prove Theorems 1.1–1.4.
Proof of Theorem 1.1. First, note that the limit in (1.7) does not depend on any Mnite terms of the
inMnite series. Secondly, by (1.5) and condition (1.6) we have
P{L∗ (1) ¿ 2 log log n( + n ())}
1=2
2
∼4
( + n ()) 2 log log n exp{−( + n ())2 log log n}
8 ∼√
log log n exp{−2 log log n} exp{−2n () log log n}
√
√
as n → ∞, uniformly in ∈ ( 1 + ; 1 + + ) √
for some√ ¿ 0. So, for any 0 ¡ ¡ 1, there
exist ¿ 0 and n0 such that for all n ¿ n0 and ∈ ( 1 + ; 1 + + ),
√
1 + log log n exp{−2 log log n} exp{−2 1 + − }
8
6 P{L∗ (1) ¿ 2 log log n( + n ())}
√
1 + 68
log log n exp{−2 log log n} exp{−2 1 + + }
(2.1)
by condition (1.6) again. And
lim
(2 − − 1)+3=2
√
1+
∞
(log n) (log log n)+1=2
n=3
n
exp{−2 log log n}
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
2
= lim
( − − 1)
√
+3=2
1+
1+
=
∞
y
0
+1=2 −y
e
∞
e
(2 − − 1)+3=2
= lim
√
∞
0
137
(log x) (log log x)+1=2
exp{−2 log log x} d x
x
y+1=2 exp{−y(2 − − 1)} dy
3
dy = +
2
:
(2.2)
So, combining (2.1) and (2.2), we have
8
√
1+
3
exp{−2 1 + − } +
2
6 lim
(2 − − 1)+3=2
√
1+
68
∞
(log n) (log log n)
n=3
n
P{L∗ (1) ¿
2 log log n( + n ())}
√
1+
3
:
exp{−2 1 + + } +
2
Now, let → 0, (1.7) is proved.
Proof of Theorem 1.2. We have
lim 2(+1)
0
∞
(log log n)
n log n
n=3
= lim 0
= 2− 2(+1)
0
e
∞
∞
P{L∗ (1) ¿ 2 log log n}
(log log x)
P{L∗ (1) ¿ 2 log log x} d x
x log x
y2+1 P{L∗ (1) ¿ y} dy = 2−(+1) ( + 1)−1 E(L∗ (1))2(+1) :
The theorem is now proved.
Now, we turn to prove Theorem 1.3.
Proof of Theorem 1.3. As in the proof of Theorem 1.1, we note that the limit in (1.10) does not
depend on any Mnite terms of the inMnite series. By (1.4) and condition (1.9) we have




2
log log n
2j1
P L∗ (1) 6
( + n ()) ∼ a1 exp −


log log n
( + n ())2
138
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
log log n
2 + 2n () + n2 ()
2
log log n
exp 3 n () log log n
∼ a1 exp −
2
= a1 exp −
√
√
for any
as n → ∞, uniformly in ∈ (1= 1 + − ; 1= 1 + ) for some ¿
√0. So, by (1.9),
√
0 ¡ ¡ 1, there exist ¿ 0 and n0 such that for all n ¿ n0 and ∈ (1= 1 + − ; 1= 1 + ),
log log n
exp{2(1 + )3=2 − }
a1 exp −
2




2
2j1
( + n ())
6 P L∗ (1) 6


log log n
log log n
6 a1 exp −
exp{2(1 + )3=2 + }
2
(2.3)
by condition (1.9) again. And
lim
√
1= 1+
1
√
−
1+
+1 ∞
log log n
(log n) (log log n)
exp −
n
2
n=3
log log x
(log x) (log log x)
exp −
= lim
dx
√
x
2
1= 1+
e
+1 ∞
1
1
√
−
y exp −y 2 − 1 − dy
= lim
√
1+
1= 1+
0
+1 −−1 ∞
1
1
√
−
−1−
y e−y dy
= lim
√
2
1+
1= 1+
0
+1 ∞
+1
1
1
−y
y e dy =
( + 1):
=
2(1 + )3=2
2(1 + )3=2
0
1
√
−
1+
+1 ∞
(2.4)
Hence, combining (2.3) and (2.4), we have
a1
1
2(1 + )3=2
+1
6
lim
√
1= 1+
√
exp{2(1 + )3=2 − }( + 1)
1
−
1+
+1 ∞
n=3




2
(log n) (log log n)
2j1
P L∗ (1) 6
( + n ())


n
log log n
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
6 a1
1
2(1 + )3=2
+1
139
exp{2(1 + )3=2 + }( + 1):
Let → 0, (1.10) is now proved.
To prove Theorem 1.4, we need some lemmas. The Mrst two of them can be found in Zwillinger
(1996, Section 6.18).
Lemma 2.1. Let 0 ¡ j1 ¡ j2 ¡ · · · be the positive zeros of the Bessel function J0 (x)=
(k!)2 )(x=2)2k . Then
1
; n → ∞;
j n = !n + O
n
∞
k=0 ((−1)
k
=
(2.5)
where !n = (n − 14 ).
Lemma 2.2. Let Jv (x) (v = 0; 1) be de;ned as in Theorem A. Then
1
2
1
1
cos x − v − + O
; x → ∞:
Jv (x) =
x
2
4
x
Lemma 2.3. We have
sin jn = (−1)
n+1
(2.6)
√
2
1
;
+O
2
n
n → ∞:
(2.7)
Proof. By Lemma 2.1, write jn = !n + n . Then
sin jn = sin !n cos n + cos !n sin n
√
1
n+1 2
= (−1)
cos n + O
2
n
√
√
1
n+1 2
n+1 2
= (−1)
+ (−1)
(cos n − 1) + O
2
2
n
√
√
1
2
n
= (−1)n+1
+ (−1)n 2 sin2
+O
2
2
n
√
1
n+1 2
= (−1)
+O
; n → ∞:
2
n
Lemma 2.4. Let ak ; bk ; ck (k = 1; 2; : : :) be de;ned as in Theorem A. Then
ck
= −16:
lim ak = 8;
lim bk = −8;
lim
k →∞
k →∞
k →∞ k
(2.8)
140
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
Proof. By Lemma 2.3,
√
sin jk = (−1)
k+1
we have
4
ak =
=
sin2 jk
1
2
1
2
+O
2
k
(k → ∞);
4
1
;
=8+O
k
+ O(1=k)
k→∞
and by Lemma 2.2
√
J1 (k) cos (k − 34 ) + O(1=k) (−1)k+1 ( 2=2) + O(1=k)
1
√
=
;
= −1 + O
=
1
J0 (k) cos (k − 4 ) + O(1=k)
k
(−1)k ( 2=2) + O(1=k)
k → ∞:
So,
ck = 16k J1 (k)=J0 (k) ∼ −16k; k → ∞;
J1 (k) 2
1
J1 (k)
−
;
= −8 + O
bk = 4 −1 +
kJ0 (k)
J0 (k)
k
k → ∞:
Lemma 2.5. For any x ¿ 0; ¿ − 12 , the following series are all absolutely convergent:
∞
∞
∞
2jk2
2k 2 2
2k 2 2
;
;
ak exp − 2 ;
ak exp − 2
bk exp − 2
x
x
x
k=1
∞
k=1
2k 2 2
;
ck exp − 2
x
k=1
k=1
∞
−2(+1)
a k jk
;
k=1
∞
ak k
−2(+1)
;
k=1
∞
ck −2(+1)
k
:
k 2 2
k=1
Proof. By Lemmas 2.1 and 2.4, it is obvious.
Lemma 2.6. For 0 ¡ x 6 x0 , we have
∞
∞
2
2 2 2j
c
2k
k
ak exp − 2k +
bk + 2 exp − 2
x
x
x
k=k0 +1
k=k0 +1
1
22
2j12
#(k0 ; x0 );
6 exp − 2 + 1 + 2 exp − 2
x
x
x
where
#(k0 ; x0 ) = exp
2j12
x02
+ exp
∞
2j 2
|ak | exp − 2k
x0
k=k +1
0
∞
22
x02
(2.9)
2k 2 2
(|bk | + |ck |) exp − 2
x0
k=k +1
0
(2.10)
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
141
and
#(k0 ; x0 ) → 0
as k0 → ∞ for fixed x0 ¿ 0:
Proof. Note that, for 0 ¡ x 6 x0 ,
2jk2
2j12
2(jk2 − j12 )
exp − 2 = exp − 2 exp −
x
x
x2
2j12
2(jk2 − j12 )
6 exp − 2 exp −
x
x02
2
2j1
2jk2
2j12
exp − 2 ;
= exp − 2 exp
x
x02
x0
22
2(k 2 − 1)2
2k 2 2
= exp − 2 exp −
exp − 2
x
x
x2
22
2(k 2 − 1)2
6 exp − 2 exp −
x
x02
2
2
2k 2 2
22
exp − 2
:
6 exp − 2 exp
x
x02
x0
The proof is completed by Lemma 2.5.
Proof of Theorem 1.4. We recall which is deMned in (1.12). By Lemma 2.5, it is obvious that
¡ ∞. Now, let M be a positive real number. For any ¿ 0, put B() = exp(exp(2 =M )); k0 ¿ 1,
then


∞


2
−2(+1)
(log log n)
2j1
∗
− (2.11)
I () =: P L (1) 6 
n log n
log log n 
n=3
6 I1 (; M ) + I2 (; M; k0 ) + I3 (M; k0 ) + I4 (; M; k0 ) + I5 (k0 );
where
I1 (; M ) = −2(+1)


2
(log log n) 
2j1 
P L∗ (1) 6 ;

n log n
log log n 
n¡B()
2
k0 (log log n) j log log n
I2 (; M; k0 ) = −2(+1)
ak exp − k 2 2
n log n
j1
n¿B()
k=1
+
ck log log n
bk +
22 j12
k 2 2
exp − 2 2 log log n
j1 (2.12)
142
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
k0
2(+1)
2(+1)
∞
j1
j1
y e−y dy +
bk
y e−y dy
2 2
2
j
k
2
2
k
jk =j1 M
k =j1 M
k=1
2(+1)
∞
j1
ck
+1 −y
y
e
dy
+
;
2 2
2
2k
k
2
2
k =j1 M
−
∞
ak
k 2(+1) ∞
0
j1 2(+1) ∞
j1
−y
ak
y e dy + bk
y e−y dy
I3 (M; k0 ) = 2 2
2
jk
k
2
2
jk =j1 M
k =j1 M
k=1
ck
+ 2 2
2k −
k0
j1
k
2(+1) ∞
k 2 2 =j12 M
y+1 e−y dy
2(+1)
2(+1)
j1
j1
( + 1)ck j1 2(+1)
ak
(
+
1)
+ bk
+
;
jk
k
2k 2 2
k
k=1
I4 (; M; k0 ) = −2(+1)
(log log n)
n log n
n¿B()
2
∞ jk log log n
×
ak exp −
2 j12
k=k0 +1
+
I5 (k0 ) =
ck log log n
bk +
22 j12
;
k 2 2
exp − 2 2 log log n
j1 ∞
k=k0 +1
2(+1)
2(+1)
j1
j1
( + 1)|ck | j1 2(+1)
|ak |
( + 1):
+ |bk |
+
jk
k
2k 2 2
k
Notice that for each k ¿ 1,
lim −2(+1)
∞
2
(log log n) j log log n
ak exp − k 2 2
n log n
j1
n¿B()
2 2
k ck log log n
exp − 2 2 log log n
+ bk +
22 j12
j1 ∞
(log log x)
jk2
−2(+1)
ak exp − 2 2 log log x
= lim ∞
x log x
j1
B()
2 2
k ck log log x
exp − 2 2 log log x
dx
+ bk +
22 j12
j1
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
143
2(+1)
2(+1)
∞
j1
j1
−y
=
ak
y e dy +
bk
y e−y dy
jk
k
jk2 =j12 M
k 2 2 =j12 M
2(+2)
∞
j1
ck
+
y+1 e−y dy:
2
2
k
k 2 2 =j12 M 2k ∞
It follows that for any M ¿ 1; k0 ¿ 1:
lim I2 (; M; k0 ) = 0:
(2.13)
∞
For I1 (; M ), we have
(log log n)
n log n
I1 (; M ) 6 −2(+1)
n6B()
6 +1 =
−2(+1)
+1
+1
B()
(log log x)
+1
d x 6 −2(+1)
x log x
+1
e
1
M
+1
2
M
+1
;
(2.14)
where +1 is an absolute constant.
For I4 (; M; k0 ), since x= 2j12 =log log n 6 x0 = : 2j12 =log log B()= 2Mj12 , whenever n ¿ B().
So by Lemma 2.6,
I4 (; M; k0 )
6
−2(+1)
(log log n) 2j12
1
22
exp − 2 + 1 + 2 exp − 2
#(k0 ; 2Mj12 )
n log n
x
x
x
n¿B()
6 −2(+1)
(log log n)
n log n
n¿B()
log log n
× exp −
2
log log n
+ 1+
22 j12
2 log log n
exp −
2 j12
#(k0 ;
It follows that
lim sup I4 (; M; k0 )
∞
−2(+1)
2
6 #(k0 ; 2Mj1 ) lim ∞
(log log x)
∞
x log x
e
2
log log x
log log x
log log x
× exp −
+ 1+
exp −
dx
2
22 j12
2 j12
2Mj12 ):
144
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
= #(k0 ;
2Mj12 )
∞
0
6 +2 #(k0 ; 2Mj12 );
y
2 2
y e−y + 1 + 2 e−( =j1 )y dy
2j1
(2.15)
where +2 = +2 () = (1 + (j1 =)2(+1) + [( + 1)=22 ](j1 =)2(+1) )( + 1) ¡ ∞.
So, combining (2.11) – (2.15), we obtain
+1
lim sup I () 6
+1
∞
1
M
+1
+ +2 #(k0 ; 2Mj12 ) + I3 (M; k0 ) + I5 (k0 ):
Notice that by Lemmas 2.5 and 2.6, limk0 →∞ #(k0 ;
show that
I3 (M; k0 ) → 0
It is obvious that
sup I3 (M; k0 ) 6
k0
2Mj12 ) = 0; limk0 →∞ I5 (k0 ) = 0. It remains to
as M → ∞ uniformly in k0 :
∞
0
(2.16)
y e−y dy = ( + 1); ( + 2) = ( + 1)( + 1). Hence, we have
∞
k=1
2(+1) j2 =j2 M
k 1
j1
|ak |
y e−y dy
jk
0
2 2 2
j1 2(+1) (k )=( j1 M ) −y
y e dy
+ |bk |
k
0
2(+1) (k 2 2 )=( j2 M )
1
|ck |
j1
+1 −y
+ 2 2
y
e dy :
2k k
0
By Lemma 2.5, it follows that
lim sup I3 (M; k0 ) 6
M →∞ k0
∞
k=1
+
2(+1)
jk2 =j12 M
j1
|ak |
lim
y e−y dy
M →∞ 0
jk
∞
k=1
|bk |
j1
k
2(+1)
lim
M →∞
0
(k 2 2 )=( j12 M )
y e−y dy
2(+1)
(k 2 2 )=( j12 M )
∞
j1
|ck |
+
lim
y+1 e−y dy
M →∞ 0
2k 2 2 k
k=1
= 0:
The proof is now completed.
(2.17)
J.-W. Wen, L.-X. Zhang / Statistics & Probability Letters 64 (2003) 133 – 145
145
References
Borodin, A.N., 1982. Distribution of integral functionals of Brownian motion. Zap. NauScn. Sem. Leningrad. Otdel. Mat.
Inst. Steklova 119, 19–38.
CsEaki, E., 1989. An integral test for the supremum of Wiener local time. Probab. Theory Related Fields 83, 207–217.
CsEaki, E., FGoldes, A., 1986. How small are the increments of the local time of a Wiener process? Ann. Probab. 14,
533–546.
Gut, A., SpKataru, A., 2000. Precise asymptotics in the law of the iterated logarithm. Ann. Probab. 28, 1870–1883.
Kesten, H., 1965. An iterated logarithm law for local time. Duke Math. J. 32, 447–456.
Zhang, L.X., 2001a. Precise rates in the law of the iterated logarithm. Preprint.
Zhang, L.X., 2001b. Precise asymptotics in Chung’s law of the iterated logarithm. Preprint.
Zwillinger, D., 1996. CRC Standard Mathematical Tables and Formulae, 30th Edition. CRC Press, Boca Raton, FL.