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Vol. 10 (2005), Paper no. 27, pages 925-947.
Journal URL
http://www.math.washington.edu/∼ejpecp/
On the Increments of the Principal Value
of Brownian Local Time
Endre Csáki1
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences,
H-1364 Budapest, P.O.B. 127, Hungary
csaki@renyi.hu
and
Yueyun Hu
Département de Mathématiques, Institut Galilée (L.A.G.A. UMR 7539)
Paris XIII, 99 Avenue J-B Clément, 93430 Villetaneuse, France
yueyun@math.univ-paris13.fr
Abstract: Let W be a one-dimensional Brownian motion starting from 0. Define Y (t) =
R t ds
Rt
:= lim²→0 0 1(|W (s)|>²) Wds(s) as Cauchy’s principal value related to local time. We prove
0 W (s)
limsup and liminf results for the increments of Y .
Keywords: Brownian motion, local time, principal value, large increments.
AMS 2000 Subject Classification: 60J65 60J55 60F15
Submitted to EJP on January 10, 2005. Final version accepted on June 1, 2005.
1
Research supported by the Hungarian National Foundation for Scientific Research, Grant No.
T 037886 and T 043037.
-- 925 --
1. Introduction
Let {W (t); t ≥ 0} be a one-dimensional standard Brownian motion with W (0) = 0, and let
{L(t, x); t ≥ 0, x ∈ R} denote its jointly continuous local time process. That is, for any Borel
function f ≥ 0,
Z
t
f (W (s)) ds =
0
Z
∞
f (x)L(t, x) dx,
t ≥ 0.
−∞
We are interested in the process
(1.1)
Y (t) :=
Rigorously speaking, the integral
Rt
0
Z
t
0
ds
,
W (s)
t ≥ 0.
ds/W (s) should be considered in the sense of Cauchy’s prin-
cipal value, i.e., Y (t) is defined by
(1.2)
Y (t) := lim
ε→0+
Z
t
0
ds
1l{|W (s)|≥ε} =
W (s)
Z
∞
0
L(t, x) − L(t, −x)
dx.
x
Since x 7→ L(t, x) is Hölder continuous of order ν, for any ν < 1/2, the integral on the extreme
right in (1.2) is almost surely absolutely convergent for all t > 0. The process {Y (t), t ≥ 0} is
called the principal value of Brownian local time.
It is easily seen that Y (·) inherits a scaling property from Brownian motion, namely, for any
fixed a > 0, t 7→ a−1/2 Y (at) has the same law as t 7→ Y (t). Although some properties distinguish
Y (·) from Brownian motion (in particular, Y (·) is not a semimartingale), it is a kind of folklore
that the asymptotic behaviors of Y are somewhat like that of a Brownian motion. For detailed
studies and surveys on principal value, and relation to Hilbert transform see Biane and Yor [4],
Fitzsimmons and Getoor [13], Bertoin [2], [3], Yamada [20], Boufoussi et al. [5], Ait Ouahra and
Eddahbi [1], Csáki et al. [11] and a collection of papers [22] together with their references. Biane
and Yor [4] presented a detailed study on Y and determined a number of distributions for principal
values and related processes.
Concerning almost sure limit theorems for Y and its increments, we summarize the relevant
results in the literature. It was shown in [17] that the following law of the iterated logarithm holds:
Theorem A. (Hu and Shi [17])
(1.3)
lim sup √
T →∞
√
Y (T )
= 8,
T log log T
a.s.
This was extended in [10] to a Strassen-type [18] functional law of the iterated logarithm.
-- 926 --
Theorem B. (Csáki et al. [10]) With probability one the set
½
(1.4)
√
Y (xT )
,0≤x≤1
8T log log T
¾
T ≥3
is relatively compact in C[0, 1] with limit set equal to
(1.5)
S :=
½
f ∈ C[0, 1] : f (0) = 0, f is absolutely continuous and
Z
1
0
¾
(f 0 (x))2 dx ≤ 1 .
Concerning Chung-type law of the iterated logarithm, we have the following result:
Theorem C. (Hu [16])
(1.6)
lim inf
T →∞
r
log log T
sup |Y (s)| = K1 ,
T
0≤s≤T
a.s.
with some (unknown) constant K1 > 0.
The large increments were studied in [7] and [8]:
Theorem D. (Csáki et al. [7]) Under the conditions

0 < aT ≤ T,






T 7→ aT and T 7→ T /aT are both non-decreasing,



log(T /aT )


 lim
= ∞,
T →∞ log log T
(1.7)
we have
(1.8)
lim
T →∞
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
p
= 2,
aT log(T /aT )
a.s.
Wen [19] studied the lag increments of Y and among others proved the following results.
Theorem E. (Wen [19])
(1.9)
lim sup sup
T →∞ 0≤t≤T
supt≤s≤T |Y (s) − Y (s − t)|
p
= 2,
t(log(T /t) + 2 log log t)
a.s.
Under the conditions 0 < aT ≤ T , aT → ∞ as T → ∞, we have
(1.10)
lim sup
sup
T →∞ 0≤t≤T −aT
p
sup0≤s≤aT |Y (t + s) − Y (t)|
aT (log((t + aT )/aT ) + 2 log log aT )
If aT is onto, then we have equality in (1.10).
-- 927 --
≤ 2,
a.s.
In this note our aim is to investigate further limsup and liminf behaviors of the increments of
Y.
Theorem 1.1. Assume that T 7→ aT is a function such that 0 < aT ≤ T , and both aT and T /aT
are non-decreasing. Then
(i)
(1.11)
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)| √
r ³
= 8,
´
p
aT log T /aT + log log T
lim sup
T →∞
If aT > T (log T )−α for some α < 2, then
r
log log T
(1.12)
lim inf
sup |Y (t + s) − Y (t)| = K2 ,
sup
T →∞
aT
0≤t≤T −aT 0≤s≤aT
(iia)
(iib)
a.s.
a.s.
If aT ≤ T (log T )−α for some α > 2, then
(1.13)
lim inf
T →∞
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
p
= K3 ,
aT log(T /aT )
a.s.
with some positive constants K2 , K3 . If, moreover,
lim
T →∞
log(T /aT )
= ∞,
log log T
then K3 = 2.
Theorem 1.2. Assume that T 7→ aT is a function such that 0 < aT ≤ T , and both aT and T /aT
are non-decreasing. Then
(i)
(1.14)
lim inf
√
T →∞
T log log T
aT
inf
sup
0≤t≤T −aT 0≤s≤aT
|Y (t + s) − Y (t)| = K4 ,
a.s.
√
with some positive constant K4 . If limT →∞ (aT /T ) = 0, then K4 = 1/ 2.
(iia)
If 0 < limT →∞ (aT /T ) = ρ ≤ 1, then
(1.15)
lim sup
T →∞
(iib)
√
inf 0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
√
= ρ 8,
T log log T
If
a.s.
aT (log log T )2
= 0,
T →∞
T
lim
then
(1.16)
lim sup
T →∞
aT
√
√
T
log log T
inf
sup
0≤t≤T −aT 0≤s≤aT
with some positive constant K5 .
-- 928 --
|Y (t + s) − Y (t)| = K5 ,
a.s.
Remark 1. The exact values of the constants Ki , i = 2, 3, 4, 5 are unknown in general and it
seems difficult to determine them except in certain particular cases. In the proofs we establish
different upper and lower bounds. It follows however by 0-1 law for Brownian motion that the
limsup’s and liminf’s considered here are non-random constants.
Remark 2. Plainly we recover some previous results on the path properties of Y by considering
particular cases of Theorems 1.1 and 1.2. For instance, Theorems A and C follow from (1.11) and
(1.12) respectively by taking aT = T , and (1.8) follows from (1.11) combining with (1.13). However
in Theorem 1.1(ii) and Theorem 1.2(ii) there are still small gaps in aT .
The organization of the paper is as follows: In Section 2 some facts are presented needed in
the proofs. Section 3 contains the necessary probability estimates. Theorem 1.1(i) and Theorem
1.1(iia,b) are proved in Sections 4 and 5, resp., while Theorem 1.2(i) and Theorem 1.2(iia,b) are
proved in Sections 6 and 7, resp.
Throughout the paper, the letter K with subscripts will denote some important but unknown
finite positive constants, while the letter c with subscripts denotes some finite and positive universal
constants not important in our investigations. When the constants depend on a parameter, say δ,
they are denoted by c(δ) with subscripts.
2. Facts
Let {W (t), t ≥ 0} be a standard Brownian motion and define the following objects:
(2.1)
(2.2)
(2.3)
g := sup{t : t ≤ 1, W (t) = 0}
W (sg)
B(s) := √ ,
0 ≤ s ≤ 1,
g
|W (g + s(1 − g))|
√
m(s) :=
,
0 ≤ s ≤ 1.
1−g
Here we summarize some well-known facts needed in our proofs.
Fact 2.1. (Biane and Yor [4])
(2.4)
P(Y (1) ∈ dx)
=
dx
r
µ
¶
∞
(2k + 1)2 x2
2 X
k
(−1) exp −
,
π3
8
k=0
x ∈ R.
Consequently we have the estimate: for δ > 0
(2.5)
µ
c1 exp −
z2
8(1 − δ)
¶
µ 2¶
z
≤ P(Y (1) ≥ z) ≤ exp −
,
8
-- 929 --
z≥1
with some positive constant c1 = c1 (δ). Moreover, g, {B(s), 0 ≤ s ≤ 1} and {m(s), 0 ≤ s ≤ 1} are
independent, g has arcsine distribution, B is a Brownian bridge and m is a Brownian meander.
¶
µZ 1
¯
dv
¯
< z ¯ m(1) = 0
P
0 m(v)
√
¶
µ 2 2¶
µ
(2.6)
∞
∞
X
8π 2 2π X
k z
2k 2 π 2
2 2
=
, z > 0.
=
(1 − k z ) exp −
exp − 2
2
z3
z
k=−∞
k=1
P(m(1) > x) = e−x
(2.7)
2
/2
,
x > 0.
Fact 2.2. (Yor [21, Exercise 3.4 and pp. 44]) Let Qδx→0 be the law of the square of a Bessel bridge
from x to 0 of dimension δ > 0 during time interval [0, 1]. The process (m2 (1 − v), 0 ≤ v ≤ 1)
conditioned on {m2 (1) = x} is distributed as Q3x→0 . Furthermore, we have
Qδx→0 = Qδ0→0 ∗ Q0x→0 ,
(2.8)
∀ δ > 0, x > 0,
where ∗ denotes convolution operator. Consequently, for any x > 0
¶
µZ 1
¶
µZ 1
¯
¯
dv
dv
¯
¯
< z ¯ m(1) = x ≥ P
< z ¯ m(1) = 0 .
(2.9)
P
0 m(v)
0 m(v)
Fact 2.3. (Hu [16]) For 0 < z ≤ 1
³ c ´
³ c ´
3
5
(2.10)
c2 exp − 2 ≤ P( sup |Y (s)| < z) ≤ c4 exp − 2
z
z
0≤s≤1
with some positive constants c2 , c3 , c4 , c5 .
Fact 2.4. (Csörgő and Révész [12]) Assume that T 7→ aT is a function such that 0 < aT ≤ T , and
both aT and T /aT are non-decreasing. Then
(2.11)
lim sup
T →∞
sup0≤t≤T −aT sup0≤s≤aT |W (t + s) − W (t)| √
p
= 2,
aT (log(T /aT ) + log log T )
a.s.
Fact 2.5. (Strassen [18]) If f ∈ S defined by (1.5), then for any partition x0 = 0 < x1 < . . . <
xk < xk+1 = 1 we have
k+1
X
(2.12)
i=1
(f (xi ) − f (xi−1 ))2
≤ 1.
xi − xi−1
Fact 2.6. (Chung [6])
(2.13)
lim inf
t→∞
r
log log t
π
sup |W (s)| = √ ,
t
8
0≤s≤t
a.s.
Define g(T ) := max{s ≤ T : W (s) = 0}. A joint lower class result for g(T ) and M (T ) :=
sup0≤s≤T |W (s)| reads as follows.
-- 930 --
Fact 2.7. (Grill [15]) Let β(t), γ(t) be positive functions slowly varying at infinity, such that
√
0 < β(t) ≤ 1, 0 < γ(t) ≤ 1, β(t) is non-increasing, β(t) t ↑ ∞, γ(t) is monotone, γ(t)t ↑ ∞,
γ(t)/β 2 (t) is monotone. Then
³
√
P M (T ) ≤ β(T ) T , g(T ) ≤ γ(T )T
according as I(β, γ) < ∞ or = ∞, where
I(β, γ) =
Z
∞
1
1
2
tβ (t)
µ
β 2 (t)
1+
γ(t)
¶−1/2
´
i.o. = 0 or
1
µ
(4 − 3γ(t))π 2
exp −
8β 2 (t)
¶
dt.
Now define d(T ) := min{s ≥ T : W (s) = 0}. Since {d(T ) > t} = {g(t) < T }, we deduce from
Fact 2.7 the following estimate on d(T ) when T → ∞.
Fact 2.8. With probability 1
d(T ) = O(T (log T )3 ),
T → ∞.
3. Probability estimates
Lemma 3.1. For T ≥ 1, δ, z > 0 we have
µ
¶
P
sup
sup |Y (t + s) − Y (t)| > z
0≤t≤T −1 0≤s≤1
(3.1)
µ
¶
µ
¶¶
µ
√
z2
z2
≤ c6
+ T exp −
T exp −
8(1 + δ)
2(1 + δ)
with some positive constant c6 = c6 (δ).
For the proof see Csáki et al. [7], Lemma 2.8.
Lemma 3.2. For T > 1, 0 < δ < 1/2, z > 1 we have
µ
¶
P
sup (Y (t + 1) − Y (t)) ≥ z
(3.2)
0≤t≤T −1
≥ min
µ
√
¶¶
µ
¡
¢
1 c7 T − 1
z2
− exp −z 2
,
exp −
2
z
8(1 − δ)
with some positive constant c7 = c7 (δ) > 0.
Proof. Let us construct an increasing sequence of stopping times by η0 := 0 and
ηk+1 := inf{t > ηk + 1 : W (t) = 0},
-- 931 --
k = 0, 1, 2, . . .
Let
νt := min{i ≥ 1 : ηi > t}
Zi := Y (ηi−1 + 1) − Y (ηi−1 ),
i = 1, 2, . . .
Then (Zi , ηi − ηi−1 )i≥1 are i.i.d. random vectors with
law
law
ηi − ηi−1 = 1 + τ 2 ,
Zi = Y (1),
where τ has Cauchy distribution. Clearly, for t > 0,
sup (Y (s + 1) − Y (s)) ≥ max Zi = Z νt ,
1≤i≤νt
0≤s≤t
with Z k := max1≤i≤k Zi . First consider the Laplace transform (λ > 0):
Z ∞
¡
¢
e−λu P Z νu < z du
λ
0
=λ
=
=
=
∞
X
k=1
∞
X
E
k=1
∞ ³
X
k=1
∞ ³
X
k=1
=1−
=1−
=1−
=1−
i.e.,
(3.3)
E
Z
³h
∞
0
e−λu 1{ηk−1 ≤u<ηk } 1{Z k <z} du
´
i
e−ληk−1 − e−ληk 1{Z k <z}
h
i
h
i´
E 1{Z k <z} e−ληk−1 − E 1{Z k <z} e−ληk
i
h
i
h
i´
h
E 1{Z k−1 <z} e−ληk−1 − E 1{Z k−1 <z, Zk ≥z} e−ληk−1 − E 1{Z k <z} e−ληk
∞
h
i
X
E 1{Z k−1 <z, Zk ≥z} e−ληk−1
k=1
∞
X
i
h
E 1{Z k−1 <z} e−ληk−1 P(Y (1) ≥ z)
k=1
∞ ³
X
k=1
h
i ´k−1
E 1{Z1 <z} e−λη1
P(Y (1) ≥ z)
P(Y (1) ≥ z)
h
i,
1 − E 1{Z1 <z} e−λη1
λ
Z
∞
0
¡
¢
e−λu P Z νu ≥ z du =
But (recalling that Z1 = Y (1))
P(Y (1) ≥ z)
h
i.
1 − E 1{Z1 <z} e−λη1
h
i
1 − E 1{Z1 <z} e−λη1 ≤ 1 − E(e−λη1 ) + P(Y (1) ≥ z)
-- 932 --
and (cf. [14], 3.466/1)
1 − Ee
−λη1
hence
λ
Z
1
=1−
π
∞
0
Z
∞
−∞
2
e−λ(1+x
1 + x2
)
2
dx = √
π
Z
√
0
λ
√
2
e−x dx ≤ 2 λ,
¢
¡
P(Y (1) ≥ z)
.
e−λu P Z νu ≥ z du ≥ √
2 λ + P(Y (1) ≥ z)
On the other hand, for any u0 > 0 we have
λ
Z
∞
e
−λu
0
¡
¢
P Z νu ≥ z du = λ
u0
e
−λu
0
¡
≤ P Z νu0
It turns out that
(3.4)
Z
¡
P Z νu ≥ z
¢
≥ z + e−λu0 .
¢
du + λ
¡
¢
P(Y (1) ≥ z)
P Z νu0 ≥ z ≥ √
− e−λu0 ≥ min
2 λ + P(Y (1) ≥ z)
where the inequality
x
≥ min
y+x
µ
1 x
,
2 2y
¶
,
µ
Z
∞
u0
¡
¢
e−λu P Z νu ≥ z du
1 P(Y (1) ≥ z)
√
,
2
4 λ
¶
− e−λu0 ,
x > 0, y > 0
was used. Choosing u0 = T − 1, λ = z 2 /u0 , and applying (2.5) of Fact 2.1, we finally get
P
(3.5)
µ
sup
0≤t≤T −1
≥ min
µ
(Y (t + 1) − Y (t)) ≥ z
¶
√
¶¶
µ
¡
¢
1 c8 (δ) T − 1
z2
− exp −z 2 .
,
exp −
2
z
8(1 − δ)
This proves Lemma 3.2.
u
t
Lemma 3.3. For T ≥ 2, 0 ≤ κ < 1 and δ, z > 0 we have
(3.6)
P
µ
sup
0≤t≤T −1
(Y (t + 1) − Y (t)) < z
¶
≤
5
T κ/2
´
³
2
+ exp −c9 T (1−κ)/2 e−(1+δ)z /8
with some positive constant c9 = c9 (δ).
See Csáki et al. [7], Lemma 3.1.
Lemma 3.4. For T ≥ 1, 0 < z ≤ 1/2 we have
(3.7)
P
µ
sup
sup |Y (t + s) − Y (t)| < z
0≤t≤T −1 0≤s≤1
-- 933 --
¶
³ c ´
c10
11
≥ √ exp − 2
z
T
with some positive constants c10 , c11 .
Proof. Define the events
A :=
½
4
2
z
, W (1) ≥ , inf W (u) ≥
4
z 1≤u≤T
z
sup |Y (s)| <
0≤s≤1
and
e :=
A
½
sup
¾
¾
sup |Y (t + s) − Y (t)| < z .
0≤t≤T −1 0≤s≤1
e since if A occurs and t < 1, t + s ≤ 1, then
Then A ⊂ A,
|Y (t + s) − Y (t)| ≤ 2 sup |Y (s)| ≤
0≤s≤1
z
< z.
2
If A occurs and t < 1, s ≤ 1, 1 < t + s ≤ T , then
|Y (t + s) − Y (t)| ≤ Y (t + s) − Y (1) + |Y (t) − Y (1)| ≤
Z
t+s
1
z
du
+ < z.
W (u) 2
Moreover, if A occurs and 1 ≤ t, s ≤ 1, t + s ≤ T , then
|Y (t + s) − Y (t)| =
Z
t+s
t
du
z
≤ < z.
W (u)
2
e as claimed. But by the Markov property of W ,
Hence A ⊂ A
(3.8)
P(A) =
Z
∞
4/z
P
µ
sup |Y (s)| <
0≤s≤1
¶ µ
¶
2 ¯¯
z ¯¯
W
(1)
=
x
P
inf
W
(u)
≥
W
(1)
=
x
ϕ(x) dx,
¯
¯
1≤u≤T
4
z
where ϕ denotes the standard normal density function.
Using reflection principle and x ≥ 4/z, z ≤ 1/2, we get
(3.9)
µ
¶
¶
µ
2 ¯¯
x − 2/z
√
P
inf W (u) ≥ ¯ W (1) = x = 2Φ
−1
1≤u≤T
z
T −1
¶
µ
µ
¶
c12
2
4
√
≥ 2Φ
− 1 ≥ 2Φ √
−1≥ √ ,
z T −1
T
T
with some constant c12 > 0, where Φ(·) is the standard normal distribution function. Hence
(3.10)
c12
e ≥ P(A) ≥ √
P(A)
P
T
µ
sup |Y (s)| ≤
0≤s≤1
4
z
, W (1) ≥
4
z
¶
.
To get a lower bound of the probability on the right-hand side, define g, (m(v), 0 ≤ v ≤ 1),
(B(u), 0 ≤ u ≤ 1) by (2.1), (2.2) and (2.3), respectively. Recall (see Fact 2.1 ) that these three
objects are independent, g has arc sine distribution, m is a Brownian meander and B is a Brownian
-- 934 --
bridge. Moreover, (g, m, B) are independent of sgn(W (1)) which is a Bernoulli variable. Observe
that
¯
du ¯¯
sup |Y (s)| =
¯,
0≤s≤g
0 B(u)
Z
p
sup |Y (s)| = |Y (1) − Y (g)| = 1 − g
√
g≤s≤1
|W (1)| =
Then
¯Z
¯
g sup ¯¯
0≤s≤1
p
s
1
0
dv
,
m(v)
1 − g m(1).
µ
¶
z
4
P sup |Y (s)| ≤ , W (1) ≥
4
z
0≤s≤1
µ
¶
z
z
4
≥ P sup |Y (s)| ≤ , Y (1) − Y (g) ≤ , W (1) ≥
8
8
z
0≤s≤g
¯Z s
¯
µ
¶
Z
1
¯
du ¯¯ z p
z p
4
dv
√
2
¯
g sup ¯
≥P
≤ , 1 − g m(1) ≥ , W (1) > 0, g < z
¯ ≤ 8, 1 − g
8
z
0≤s≤1
0 B(u)
0 m(v)
¯
¯Z s
¶
µ
Z 1
¯
dv
z
4
du ¯¯ 1
2
√
,
≤
,
m(1)
≥
≤
,
W
(1)
>
0,
g
<
z
≥ P sup ¯¯
¯ 8
8
z 1 − z2
0≤s≤1
0 m(v)
0 B(u)
¯Z s
¯
¶ µZ 1
µ
¶
¯
dv
du ¯¯ 1
z
4
¯
= P sup ¯
≤ , m(1) ≥ √
P(W (1) > 0)P(g < z 2 )
¯≤ 8 P
8
z 1 − z2
0≤s≤1
0 m(v)
0 B(u)
¶
µZ 1
z
4
dv
≤ , m(1) ≥ √
≥ c13 zP
8
z 1 − z2
0 m(v)
¶
µZ 1
Z ∞
z ¯¯
dv
≤ ¯ m(1) = x P(m(1) ∈ dx).
= c13 z
P
√
8
4/(z 1−z 2 )
0 m(v)
It follows from Facts 2.1 and 2.2 that for x > 0, z > 0
(3.11)
P
µZ
1
0
and
z ¯¯
dv
≤ ¯ m(1) = x
m(v)
8
¶
≥P
µZ
1
0
dv
z ¯¯
≤ ¯ m(1) = 0
m(v)
8
¶
≥
³ c ´
c14
15
exp
− 2
z3
z
µ
¶
µ
¶
4
8
P m(1) > √
= exp − 2
.
z (1 − z 2 )
z 1 − z2
(3.12)
Putting (3.10), (3.11), (3.12) together, we get (3.7).
u
t
Lemma 3.5. For T > 1, 0 < z ≤ 1/2, 0 < δ ≤ 1/2 we have
P
(3.13)
µ
inf
sup |Y (t + s) − Y (t)| < z
0≤t≤T −1 0≤s≤1
≤ c16
µ
µ
exp −
(1 − δ)2
2(1 + δ)2 z 2 T
¶
¶
µ
+ exp −
c5 δ
4(1 + δ)2 z 2
-- 935 --
¶
+ exp
µ
c17
c18 z 2 c19 /z2
−
e
z2
T
¶¶
with some positive constants c16 , c17 = c17 (δ), c18 = c18 (δ), c19 = c19 (δ).
Proof. Consider a positive integer N to be given later, h = (T − 1)/N , tk = kh, k = 0, 1, 2, . . . , N .
Then for 0 < δ ≤ 1/2 we have
µ
¶
P
inf
sup |Y (t + s) − Y (t)| < z
0≤t≤T −1 0≤s≤1
≤P
µ
inf
sup |Y (tk + s) − Y (tk )| ≤ (1 + δ)z
0≤k≤N 0≤s≤1
=: P1 + P2 .
¶
+P
µ
sup
sup |Y (t + s) − Y (t)| > δz
0≤t≤T −1 0≤s≤h
¶
By scaling and Lemma 3.1
Ã
!
δz
P2 = P
sup
sup |Y (t + s) − Y (t)| > √
h
0≤t≤(T −1)/h 0≤s≤1
Ãr
µ
¶
µ
¶
µ
¶!
δ2 z2
δ2 z2
T −1
T −1
≤ c6
+ 1 exp −
+ 1 exp −
+
h
8h(1 + δ)
h
2h(1 + δ)
¶
µ
δ2 z2
.
≤ 2c6 (N + 1) exp −
8h(1 + δ)
To bound P1 , we denote by d(t) := inf{s ≥ t : W (s) = 0} the first zero of W after t. Consider
those k for which sup0≤s≤1 |Y (tk + s) − Y (tk )| ≤ (1 + δ)z. If, moreover, d(tk ) ≥ tk + 1 − δ, which
means that the Brownian motion W does not change sign over [tk , tk + 1 − δ), then
Z 1−δ
ds
1−δ
(1 + δ)z ≥ |Y (tk + 1 − δ) − Y (tk )| =
≥
,
|W (tk + s)|
sup0≤s≤T |W (s)|
0
and it follows that
¶
(1 − δ)
P1 ≤ P sup |W (s)| >
z(1 + δ)
0≤s≤T
µ
¶
+ P ∃k ≤ N : sup |Y (tk + s) − Y (tk )| ≤ (1 + δ)z; d(tk ) < tk + 1 − δ
µ
0≤s≤1
¶
(1 − δ)2
≤ 4 exp −
2(1 + δ)2 z 2 T
µ
¶
N
X
+
P sup |Y (tk + s) − Y (tk )| ≤ (1 + δ)z; d(tk ) < tk + 1 − δ .
µ
k=0
0≤s≤1
c (s) = W (d(tk ) + s) for s ≥ 0 and Yb (s) be the associated principal values. Observe
Let W
that on {sup0≤s≤1 |Y (tk + s) − Y (tk )| ≤ (1 + δ)z; d(tk ) < tk + 1 − δ}, we have sup0≤u≤δ |Yb (u) +
(Y (d(tk )) − Y (tk ))| < (1 + δ)z, and |Y (d(tk )) − Y (tk )| ≤ (1 + δ)z which imply that
sup |Yb (u)| < 2(1 + δ)z.
0≤u≤δ
-- 936 --
By scaling and Fact 2.3 we have
µ
¶
µ
b
P sup |Y (u)| < 2(1 + δ)z ≤ c4 exp −
0≤u≤δ
c5 δ
4(1 + δ)2 z 2
¶
.
Therefore, we obtain:
µ
(1 − δ)2
P1 ≤ 4 exp −
2(1 + δ)2 z 2 T
Hence
µ
P1 + P2 ≤ 4 exp −
¶
µ
c5 δ
+ c4 (N + 1) exp −
4(1 + δ)2 z 2
(1 − δ)2
2(1 + δ)2 z 2 T
¶
µ
+ c4 (N + 1) exp −
µ
+2c6 (N + 1) exp −
By taking N = [ec5 δ/(4(1+δ)
P1 + P 2
µ
µ
≤ c16 exp −
2 2
z )
δ2 z2
8h(1 + δ)
¶
¶
.
c5 δ
4(1 + δ)2 z 2
¶
.
] + 1, we get
(1 − δ)2
2(1 + δ)2 z 2 T
¶
µ
+ exp −
c5 δ
4(1 + δ)2 z 2
¶
+ exp
µ
c17
c18 z 2 c19 /z2
−
e
z2
T
with relevant constants c16 , c17 , c18 , c19 , proving (3.13).
¶¶
u
t
4. Proof of Theorem 1.1(i)
The upper estimation, i.e.
(4.1)
lim sup
T →∞
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
r
≤ 1,
³ p
´
8aT log T /aT + log log T
a.s.
follows easily from Wen’s Theorem E.
Now we prove the lower bound, i.e.
(4.2)
lim sup
T →∞
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
r
≥ 1,
³ p
´
8aT log T /aT + log log T
a.s.
In the case when aT = T , (4.2) follows from the law of the iterated logarithm (1.3) of Theorem
A. Now we assume that aT /T ≤ ρ < 1, with some constant ρ for all T > 0.
By scaling, (3.2) of Lemma 3.2 is equivalent to
µ
¶
√
P
sup (Y (t + a) − Y (t)) ≥ z a
(4.3)
0≤t≤T −a
≥ min
Ã
1 c7
,
2
p
¶!
µ
¡
¢
T /a − 1
z2
exp −
− exp −z 2
z
8(1 − δ)
-- 937 --
for 0 < a < T , 0 < δ < 1/2, z > 1.
Define the sequences
tk := e7k log k ,
(4.4)
k = 1, 2, . . .
and θ0 := 0,
(4.5)
θk := inf{t > Tk : W (t) = 0},
k = 1, 2, . . . ,
where Tk := θk−1 + tk . For 0 < δ < min(1/2, 1 − ρ) define the events
Ak :=
(
sup
0≤t≤tk (1−δ)−atk
(Y (θk−1 + t + atk ) − Y (θk−1 + t)) ≥ (1 − δ)βk
with
)
,
k = 1, 2, . . .
v
à s
!
u
u
t
k
βk := t8atk log
+ log log tk .
a tk
q
p
Applying (4.3) with T = tk (1 − δ), a = atk , z = (1 − δ) 8(log tk /atk + log log tk ), we have
for k large
P(Ak ) = P
≥ min
with
Ã
µ
sup
0≤t≤tk (1−δ)−atk
bk
1
,
2 (log tk )1−δ
¶
(Y (t + atk ) − Y (t)) ≥ (1 − δ)βk
−
!
1
(log tk )8(1−δ)2
p
tk (1 − δ)/atk − 1
c20
q p
bk =
.
≥√
log
k
(1−δ)/2
(tk /atk )
log tk /atk + log log tk
c7
Hence
P
k
P(Ak ) = ∞ and since Ak are independent, Borel-Cantelli lemma yields
P(Ak i.o.) = 1.
It follows that
(4.6)
lim sup
k→∞
sup0≤t≤tk (1−δ)−at (Y (θk−1 + t + atk ) − Y (θk−1 + t))
k
r
≥ 1 − δ,
³ q
´
tk
8atk log at + log log tk
k
It can be seen (cf. [9]) that we have almost surely for large enough k
tk ≤ T k ≤ t k
µ
1
1+
k
-- 938 --
¶
,
a.s.
consequently
(4.7)
lim
k→∞
tk
= 1,
Tk
a.s.
Since by our assumptions
tk
at
≤ k ≤ 1,
Tk
a Tk
we have also
a tk
= 1,
k→∞ aTk
(4.8)
lim
a.s.
On the other hand, for any δ > 0 small enough we have almost surely for large k
aTk ≤ (1 + δ)atk ≤ tk δ + atk ,
thus
T k − a Tk ≥ T k − t k δ − a tk ,
consequently
sup
sup
0≤t≤Tk −aTk 0≤s≤aTk
(4.9)
≥
sup
0≤t≤tk (1−δ)−atk
|Y (t + s) − Y (t)|
(Y (θk−1 + t + atk ) − Y (θk−1 + t)),
hence we have also
(4.10)
lim sup
k→∞
sup0≤t≤Tk −aT sup0≤s≤aT |Y (t + s) − Y (t)|
k
k
r
≥ 1 − δ,
´
³ q
tk
8atk log at + log log tk
a.s.
k
and since δ > 0 can be arbitrary small, (4.2) follows by combining (4.7), (4.8), (4.9) and (4.10). u
t
5. Proof of Theorem 1.1(ii)
First assume that
(5.1)
aT >
T
(log T )α
for some
α < 2.
By Theorem C,
log log T
sup
sup |Y (t + s) − Y (t)|
aT
0≤t≤T −aT 0≤s≤aT
r
log log aT
a.s.,
sup |Y (s)| ≥ K1 ,
≥ lim inf
T →∞
aT
0≤s≤aT
lim inf
(5.2)
r
T →∞
-- 939 --
proving the lower bound in (1.12).
To get an upper bound, note that by scaling, (3.7) of Lemma 3.4 is equivalent to
r
µ
¶
³ c ´
√
a
11
(5.3)
P
sup
sup |Y (s + t) − Y (t)| < z a ≥ c10
exp − 2
T
z
0≤t≤T −a 0≤s≤a
for T ≥ a, 0 < z ≤ 1/2.
Let tk and θk be defined by (4.4) and (4.5), resp., Tk = θk−1 + tk as in the proof of Theorem
1.1(i). Let c11 be the constant as in (5.3) and choose δ > 0 such that α/2 + c11 /δ 2 < 1. For ε > 0
define the events
(
Ek :=
sup
sup
0≤t≤(1+ε)tk −atk (1+ε) 0≤s≤atk (1+ε)
|Y (θk−1 + t + s) − Y (θk−1 + t)| ≤ δ
r
a tk
log log tk
)
.
√
Then putting T = (1 + ε)tk , a = a(1+ε)tk , z = δ/ log log tk into (5.3), we get
P(Ek ) = P
hence
P
≥ c10
k
r
Ã
sup
sup
0≤t≤(1+ε)tk −atk (1+ε) 0≤s≤atk (1+ε)
|Y (t + s) − Y (t)| ≤ δ
r
a tk
log log tk
!
c10
c10
a tk
,
exp(−(c11 /δ 2 ) log log(tk ) ≥
2 =
α/2+c
/δ
11
tk
(log tk )
(7k log k)α/2+c11 /δ2
P(Ek ) = ∞, and since Ek are independent, we have P(Ek i.o.) = 1, i.e.
(5.4) lim inf
k→∞
s
log log tk
a tk
sup
sup
0≤t≤(1+ε)tk −atk (1+ε) 0≤s≤atk (1+ε)
|Y (θk−1 + t + s) − Y (θk−1 + t)| ≤ δ,
a.s.
for any ε > 0. For large enough k by (4.7) and (4.8) we have aTk ≤ (1 + ε)atk , a.s. and Tk − aTk ≤
θk−1 + (1 + ε)tk − (1 + ε)atk , a.s. Thus given any ε > 0, we have for large k
sup
(5.5)
sup
0≤t≤Tk −aTk 0≤s≤aTk
≤2
sup
0≤t≤θk−1
|Y (t + s) − Y (t)|
|Y (t)| +
sup
sup
0≤t≤(1+ε)tk −atk (1+ε) 0≤s≤atk (1+ε)
|Y (θk−1 + t + s) − Y (θk−1 + t)|.
By Theorem A, Fact 2.8, (4.7), (5.1) and simple calculation,
sup
0≤t≤θk−1
|Y (t)| = O(θk−1 log log θk−1 )1/2
(5.6)
3
= O(tk−1 (log tk−1 ) log log tk−1 )
1/2
=o
µ
a tk
log log tk
¶1/2
,
as k → ∞. Assembling (5.4), (5.5) and (5.6), we get
s
log log tk
sup
lim inf
sup |Y (t + s) − Y (t)|
k→∞
a tk
0≤t≤Tk −aTk 0≤s≤aTk
-- 940 --
a.s.
= lim inf
k→∞
s
log log Tk
a Tk
sup
sup
0≤t≤Tk −aTk 0≤s≤aTk
|Y (t + s) − Y (t)| ≤ δ,
a.s.
which together with (5.2) yields (1.12).
Now assume that
(5.7)
aT ≤
T
(log T )α
for some
α > 2.
By Theorem 1.1(i),
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
p
T →∞
aT log(T /aT )
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
p
≤ lim sup
aT log(T /aT )
T →∞
lim inf
(5.8)
sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)|
r
≤2
≤ lim sup
³ p
´
T →∞
2αaT
log T /aT + log log T
α+2
r
α+2
,
α
i.e., an upper bound in (1.13) follows.
To get a lower bound under (5.7), observe that by scaling, (3.6) of Lemma 3.3 is equivalent to
!
Ã
µ
¶
µ ¶(1−κ)/2
³ a ´κ/2
√
T
−(1+δ)z 2 /8
P
sup (Y (t + a) − Y (t)) < z a ≤ 5
e
+ exp −c9
T
a
0≤t≤T −a
for a ≤ T , 0 ≤ κ < 1, 0 < δ, 0 < z. Using (5.7) we get further
µ
¶
√
P
sup (Y (t + a) − Y (t)) < z a
0≤t≤T −a
(5.9)
´
³
5
α(1−κ)/2 −(1+δ)z 2 /8
.
≤
+
exp
−c
(log
T
)
e
9
(log T )ακ/2
In the case when (1.7) holds, (1.13) was proved in [7]. In other cases the proof is similar. Let
Tk = ek and define the events
(
Fk =
sup
0≤t≤Tk −aTk
(Y (t + aTk ) − Y (t)) ≤ C1
with
C1 = 2
s
s
Tk
aTk log
a Tk
)
α − 2 − 2εα
.
(1 + δ)α
By (5.9) with κ = 2/α + ε,
P(Fk ) ≤
5
k ακ/2
´
³
2
+ exp −c9 k α((1−κ)/2−(1+δ)C1 /8) ≤
-- 941 --
5
k 1+αε/2
´
³
+ exp −c9 k αε/2 .
One can easily see that with these choices
lim inf
k→∞
P
k
P(Fk ) < ∞, consequently
sup0≤t≤Tk −aT (Y (t + aTk ) − Y (t))
qk
≥ C1 ,
aTk log aTTk
a.s.,
k
implying also
sup0≤t≤Tk −aT sup0≤s≤aT |Y (t + s) − Y (t)|
k
k
q
lim inf
≥2
k→∞
aTk log aTTk
k
r
α−2
,
α
a.s.,
for ε can be choosen arbitrary small.
Since sup0≤t≤T −aT sup0≤s≤aT |Y (t + s) − Y (t)| is increasing in T , we obtain a lower bound
in (1.13). This together with the 0-1 law for Brownian motion complete the proof of Theorem
1.1(ii).
u
t
6. Proof of Theorem 1.2(i)
If aT = T , then (1.14) is equivalent to Theorem C. Now assume that ρ := limT →∞ aT /T < 1.
First we prove the lower bound, i.e.
√
T log log T
inf
(6.1)
lim inf
T →∞
0≤t≤T −aT
aT
sup
0≤s≤aT
|Y (t + s) − Y (t)| ≥ c,
a.s.
By scaling, (3.13) of Lemma 3.5 is equivalent to
¶
µ
√
P
inf
sup |Y (t + s) − Y (t)| < z a
0≤t≤T −a 0≤s≤a
(6.2)
µ
µ
¶
µ
¶
µ
¶¶
a(1 − δ)2
c5 δ
c17
c18 az 2 c19 /z2
≤ c16 exp −
+ exp −
+ exp
−
e
2(1 + δ)2 z 2 T
4(1 + δ)2 z 2
z2
T
for a < T , 0 < z ≤ 1/2, 0 < δ ≤ 1/2.
Define the events
(
Gk =
inf
sup
0≤t≤Tk+1 −aTk 0≤s≤aT
k
√
|Y (t + s) − Y (t)| < zk aTk
)
k = 1, 2, . . .
Let Tk = ek and put T = Tk+1 , a = aTk ,
z = z k = C2
r
a Tk
Tk+1 log log Tk+1
into (6.2). The constant C2 will be choosen later. Denoting the terms on the right-hand side of
(6.2) by I1 , I2 , I3 , resp., we have
(k)
P(Gk ) ≤ c16 (I1
(k)
+ I2
-- 942 --
(k)
+ I3 ),
where
µ
¶
c21
= exp − 2 log log Tk+1 ,
C2
µ
¶
c22 Tk
= exp − 2
log log Tk+1 ,
C2 a Tk
(k)
I1
(k)
I2
(k)
I3
= exp
Ã
c25 Tk
c24 C22 a2Tk
c23 Tk log log Tk+1
C 2 aT
2
k
(log
T
)
−
k+1
C22 aTk
Tk2 log log Tk+1
!
with some constants c21 = c21 (δ), c22 = c22 (δ), c23 , c24 , c25 .
One can see easily that for any choice of positive C2 and for all possible aT (satisfying our
P (k)
conditions) we have k I3 < ∞. So we show that for appropriate choice of C2 we have also
P (k)
< ∞, j = 1, 2.
k Ij
³√
q ´
c21 , cρ22
First consider the case 0 < ρ. Choosing a positive δ, one can select C2 < min
P
P (k)
and it is easy to verify that k Ij < ∞, j = 1, 2, hence also k P(Gk ) < ∞.
√
P (k)
In the case ρ = 0 choose C2 < (1 − δ)/((1 + δ) 2). With this choice we have k I1 < ∞
P (k)
P
for arbitrary δ > 0. Since limk→∞ (Tk /aTk ) = ∞, we have also k I2 < ∞ and k P(Gk ) < ∞.
The Borel-Cantelli lemma and interpolation between Tk ’s finish the proof of (6.1). We have also
√
verified that in the case ρ = 0 one can choose c = 1/ 2 in (6.1), since δ > 0 can be choosen
arbitrary small.
Now we turn to the proof of the upper bound, i.e.
(6.3)
lim inf
√
T →∞
T log log T
aT
inf
sup
0≤t≤T −aT 0≤s≤aT
|Y (t + s) − Y (t)| ≤ C3 ,
a.s.
with some constant C3 .
If ρ > 0, then
inf
sup
0≤t≤T −aT 0≤s≤aT
|Y (t + s) − Y (t)| ≤
sup |Y (s)| ≤ sup |Y (s)|
0≤s≤T
0≤s≤aT
and hence (6.3) with some positive constant C3 follows from Theorem C.
If ρ = 0, then let for any ε > 0
(6.4)
λT := inf{t : |W (t)| =
sup
0≤s≤T (1−ε)
|W (s)|}.
According to the law of the iterated logarithm, with probability one there exists a sequence {Ti , i ≥
1} such that limi→∞ Ti = ∞ and
(6.5)
|W (λTi )| ≥
p
2Ti (1 − ε) log log Ti .
-- 943 --
But Fact 2.4 implies that for ε > 0
(6.6)
|W (λTi ) − W (s)| ≤
p
2(1 + ε)εTi log log Ti ,
λTi ≤ s ≤ λTi + εTi ,
i ≥ 1.
Now assume that W (λTi ) > 0. The case when W (λTi ) < 0 is similar. Then (6.5) and (6.6) imply
(6.7)
W (s) ≥
´p
³√
p
1 − ε − ε(1 + ε)
2Ti log log Ti ,
λTi ≤ s ≤ λTi + εTi .
ρ = 0 implies that aT ≤ εT for any ε > 0 and large enough T , hence we have from (6.7) for large i
sup (Y (λTi + s) − Y (λTi )) = Y (λTi + aTi ) − Y (λTi ) =
0≤s≤aTi
Z
λTi +aTi
λ Ti
ds
W (s)
a Ti
´√
.
p
1 − ε − ε(1 + ε)
2Ti log log Ti
√
Since ε > 0 is arbitrary, (6.3) follows with C3 = 1/ 2. This completes the proof of Theorem 1.2(i).
≤ ³√
u
t
7. Proof of Theorem 1.2(ii)
If ρ = 1, then (1.15) is equivalent to (1.3) of Theorem A. So we may assume that 0 < ρ < 1. It
suffices to show (1.15) when aT = ρT .
First we prove the upper bound
(7.1)
lim sup
T →∞
inf 0≤t≤T −ρT sup0≤s≤ρT |Y (t + s) − Y (t)|
√
≤ ρ,
8T log log T
a.s.
Let k be the largest integer for which kρ < 1 and put xi = iρ, i = 0, 1, . . . , k, xk+1 = 1. It suffices
to show that if f ∈ S defined by (1.5), then
min
1≤i≤k+1
|f (xi ) − f (xi−1 )| ≤ ρ.
Assume on the contrary that
|f (xi ) − f (xi−1 )| > ρ,
Then
k+1
X
i=1
k
∀i = 1, 2, . . . , k + 1.
X ρ2
ρ2
ρ2
(f (xi ) − f (xi−1 ))2
>
+
= kρ +
≥ 1,
xi − xi−1
ρ
1 − kρ
1 − kρ
i=1
contradicting (2.12) of Fact 2.5. This proves (7.1).
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The lower bound
(7.2)
lim sup
T →∞
inf 0≤t≤T −ρT sup0≤s≤ρT |Y (t + s) − Y (t)|
√
≥ ρ,
8T log log T
a.s.
follows from the fact that by Theorem B the function f (x) = x, 0 ≤ x ≤ 1 is a limit point of
√
Y (xt)
8T log log T
and for this function
min
0≤x≤1−ρ
|f (x + ρ) − f (x)| = ρ.
This completes the proof of Theorem 1.2(iia).
u
t
Now assume that
aT (log log T )2
= 0.
T →∞
T
(7.3)
lim
Define λT as in (6.4). Then according to Chung’s LIL (cf. Fact 2.6)
π
|W (λT )| ≥ √ (1 − ε)
8
(7.4)
s
T
log log T
for ε > 0 and all T sufficiently large. But according to Fact 2.4,
sup |W (λT + s) − W (λT )|
0≤s≤aT
p
≤ (2 + ε)aT (log(T /aT ) + log log T ) ≤
s
(2 + ε)εT
.
log log T
Assuming W (λT ) > 0, on choosing suitable ε > 0 we get for some c26 > 0
W (λT + s) ≥ W (λT ) −
s
(2 + ε)εT
≥ c26
log log T
s
T
.
log log T
Hence
inf
sup |Y (t + s) − Y (t)| ≤ Y (λT + aT ) − Y (λT )
0≤t≤T −aT 0≤s≤aT
=
Z
aT
0
ds
aT
≤
W (λT + s)
c26
r
log log T
T
for all large T .
The case when W (λT ) < 0 is similar. This shows the upper bound in (1.16).
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For the lower bound we use Fact 2.7: with probability one
s
T
π
T
(7.5)
gT ≤
,
max |W (u)| ≤ √
,
2
0≤u≤T
(log log T )
2 log log T
i.o.
According to Theorem 1.2(i) we have for any ε > 0 and all large T
inf
(7.6)
sup |Y (t + s) − Y (t)|
0≤t≤T (log log T )−2 0≤s≤aT
√
(K4 − ε)aT
(K4 − ε)aT log log T
p
≥ r³
≥
.
´
(1 + ε)T
T
(log log T )2 + aT log log T
On the other hand, if T (log log T )−2 ≤ t ≤ T − aT , and (7.5) is satisfied, then
√
Z t+aT
aT 2 log log T
ds
√
≥
, i.o.
(7.7)
|Y (t + aT ) − Y (t)| =
|W (s)|
π T
t
Combining (7.6) and (7.7) for ε > 0 with probability one
Ã
√ ! √
2 aT log log T
K4 − ε
inf
sup |Y (t + s) − Y (t)| ≥ min √
,
,
0≤t≤T −aT 0≤s≤aT
T
1+ε π
i.o.
This shows the lower bound in (1.16). The proof of Theorem 1.2(iib) is complete by applying
the 0-1 law for Brownian motion.
u
t
Acknowledgements
The authors are indebted to Marc Yor for helpful remarks. We thank also the referee for
useful suggestions. Cooperation between the authors was supported by the joint French–Hungarian
Intergovernmental Grant ”Balaton” (grant no. F-39/00).
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