3B2v8:06a=w ðDec 5 2003Þ:51c
XML:ver:5:0:1
SPA : 1352
Prod:Type:FTPFTP
pp:1222ðcol:fig::NILÞ
ED:MangalaK:Mangala
PAGN:Dini SCAN:Shobha
ARTICLE IN PRESS
1
Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
www.elsevier.com/locate/spa
3
5
7
Level crossings of a two-parameter random walk
9
15
F
Department of Mathematics, University of Utah, 155 S, 1400 E JWB 233, Salt Lake City, UT 84112–0090,
USA
b
Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 810/107, A-1040 Vienna, Austria
c
Laboratoire de Probabilités UMR 7599, Université Paris VI, 4 place Jussieu, F-75252 Paris Cedex 05,
France
O
13
a
O
11
Davar Khoshnevisana,,1, Pál Révészb,2, Zhan Shic,3
17
PR
Received 27 September 2004; accepted 27 September 2004
19
Abstract
23
We prove that the number ZðNÞ of level crossings of a two-parameter simple random walk
in its first N N steps is almost surely N 3=2þoð1Þ as N ! 1: The main ingredient is a strong
approximation of ZðNÞ by the crossing local time of a Brownian sheet. Our result provides a
useful algorithm for simulating the level sets of the Brownian sheet.
r 2004 Published by Elsevier B.V.
29
TE
27
EC
25
D
21
MSC: 60J55; 60G50; 60F17
R
Keywords: Level crossing; Local time; Random walk; Brownian sheet
O
R
31
33
41
43
45
N
39
Corresponding author. Tel.: +1 801 581 3896; fax: +1 801 581 4148.
E-mail addresses: davar@math.utah.edu (D. Khoshnevisan), revesz@ci.tuwien.ac.at (P. Révész),
zhan@proba.jussieu.fr (Z. Shi).
1
Research supported by NSF and NATO grants.
2
Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T
029621.
3
Research supported by a grant from NATO.
U
37
C
35
0304-4149/$ - see front matter r 2004 Published by Elsevier B.V.
doi:10.1016/j.spa.2004.09.010
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
2
1
1. Introduction
3
Recall that a two-parameter real-valued Brownian sheet W ¼ fW ðs; tÞgs;tX0 is a
centered Gaussian process with covariance
17
19
21
23
25
27
29
3
2
F
cf. [1,12]. Here, dim refers to Hausdorff dimension. Other, more delicate, features of
the level sets can be found in [5–11,13].
One expects that the level sets of two-parameter random walks are uniform in
local time. Informally speaking, this and (1.2) together imply that for any reasonable
discrete approximation AN of W 1 f0g \ ½0; 12 ; one might expect that #AN N 3=2 ;
as N ! 1; here, # denotes cardinality, and ‘‘’’ stands for any reasonable notion of
asymptotic equivalence.
This paper is motivated, in part, by our desire to find a good algorithm for
simulating the zero-set of W inside a given box that we take to be ½0; 12 to be
concrete. A natural way to try and do this is by first performing a random-walk
approximation to W, and then approximating the zero-set of W by that of the walk.
With this in mind, let fX i;j gi;jX1 denote an array of i.i.d. random variables with
PfX i;j ¼ 1g ¼ PfX i;j ¼ 1g ¼ 12; and consider the two-parameter random walk
fS m;n ; m; nX0g defined as
Sm;n ¼
m X
n
X
X i;j
8m; nX1;
O
R
with the added stipulation that Sm;n ¼ 0 whenever mn ¼ 0: It is then possible to show
that as N ! 1;
33
fN 1 SbNsc;bNtc g0ps;tp1 ¼)fW ðs; tÞg0ps;tp1 ;
39
41
43
45
C
where ) denotes weak convergence in a suitable space. Here, convergence in
DDR ½0;1 ð½0; 1Þ will do, but we will not need this fact in the sequel; see [14, Theorem
4.1.1, Chapter 6] for a variant of this statement. Suffice it to say that the factor of
N 1 is the central limit scaling that comes from adding OðN 2 Þ i.i.d. variates.
A natural approximation of the zero-set W 1 f0g \ ½0; 12 would then be the
random set
N
37
(1.4)
U
35
(1.3)
R
i¼1 j¼1
31
(1.2)
8a 2 R;
O
15
dimðW 1 fagÞ ¼
O
13
It is known that the level sets of W have a rich and complicated structure. For
instance, if W 1 fag :¼ fs; tÞ 2 R2þ : W ðs; tÞ ¼ ag; then it follows that with probability
one,
PR
11
(1.1)
D
9
8s; t; s0 ; t0 X0:
TE
7
EfW ðs; tÞW ðs0 ; t0 Þg ¼ minðs; s0 Þ minðt; t0 Þ
EC
5
UN ¼ fði; jÞ 2 f0; . . . ; Ng2 : Si;j ¼ 0g;
(1.5)
where N is a large integer. While this algorithm is intuitively attractive, it does not
perform well. Indeed, by the local limit theorem ([19, Theorem 2.8]),
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
1
3
5
7
Ef#UN g ð2pÞ1=2
N X
N
X
ðijÞ1=2 4ð2pÞ1=2 N;
as N ! 1:
One can use this, in conjunction with a monotonicity argument, to show that with
probability one,
lim N 3=2 #UN ¼ 0:
(1.7)
19
21
23
25
27
F
O
O
Si;j S i;jþ1 o0:
Define
XN :¼ fði; jÞ 2 ½0; N2 \ Z2 : ði; jÞ is a crossingg
zi ðNÞ :¼ #fj 2 ½0; N \ Z2 : ði; jÞ is a crossingg
ZðNÞ :¼ z1 ðNÞ þ þ zN ðNÞ:
(1.8)
PR
17
D
15
ð1:9Þ
TE
13
In light of (1.2) and its proceeding discussion, (1.7) suggests that UN might be too
thin to properly simulate the zero-set W 1 f0g \ ½0; 12 of the Brownian sheet.
In this paper, we present an alternative algorithm for simulating the zero-set of W,
and show that our approximation has the correct size of N ð3=2Þþoð1Þ as N ! 1: Our
suggested approximation is a natural one that is based on the ‘‘crossing numbers’’ of
the approximating two-parameter random walk S.
A lattice point ði; jÞ is called a (vertical) crossing for the random walk S if
In words, ZðNÞ ¼ #XN is the total number of crossings of the random walk in the
first N N steps. We propose to show that XN is a good approximation to the zeroset of the Brownian sheet in ½0; 12 ; at least in the sense that ZðNÞ ¼ #XN is
sufficiently thick in the following asymptotically sense.
EC
11
(1.6)
i¼1 j¼1
N!1
9
3
Theorem 1.1. With probability one, ZðNÞ ¼ N ð3=2Þþoð1Þ as N ! 1:
31
Fig. 1 shows the simulation of the level set of a two-parameter simple walk,
together with the vertical crossings of the same random walk. The figure speaks for
itself, and the Matlab code is added as a brief appendix at the end of the paper.
Throughout this paper, we write log x :¼ lnðx _ eÞ:
O
R
33
R
29
N
37
C
35
39
41
43
45
U
2. Brownian sheet and invariance
To prove Theorem 1.1, we need to analyze the crossings of the walk
simultaneously at all levels. With this in mind, for each x 2 R we say that ði; jÞ is a
(vertical) x-crossing for the random walk if
ðS i;j xÞðS i;jþ1 xÞo0:
Next, we can define
(2.1)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
4
1
3
5
7
9
11
O
F
13
O
15
PR
17
19
D
21
TE
23
25
EC
27
R
29
O
R
31
33
Fig. 1. The zeros vs. the vertical crossings.
41
43
45
N
39
zi ðx; nÞ :¼ #fj : 0pjon : ði; jÞ is an x-crossingg
m
X
zi ðx; nÞ:
Zðx; m; nÞ :¼
U
37
C
35
ð2:2Þ
i¼0
Thus, Zðx; m; nÞ denotes the number of x-crossings in the first m n steps. We
remark that ZðNÞ ¼ Zð0; N; NÞ:
When N is large, the entire process ðx; s; tÞ7!Zðx; bNsc; bNtcÞ is close to the
crossing local times of a Brownian sheet that we describe next.
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
3
5
For any fixed s40; fLs ðx; tÞgtX0 denotes the local time at x of the process
t7!W ðs; tÞ: This is the density of the occupation measure, as described by the
formula,
Z þ1
Z t
f ðxÞLs ðx; tÞ dx ¼
f ðW ðs; uÞÞ du:
(2.3)
0
1
11
13
15
17
The following strong approximation constitutes a central portion of this paper.
Theorem 2.1. Possibly in an enlarged probability space, there exists a coupling for the
two-parameter random walk S and the Brownian sheet W such that for any 40 the
following holds almost surely: As N ! 1;
max jS m;n W ðm; nÞj ¼ OðN 1=2 ðlog NÞ3=2 Þ;
max
max sup jZðx; m; nÞ Cðx; m; nÞj ¼ oðN ð4=3Þþ Þ;
1pmpN
1pnpN
25
27
29
D
where Zðx; m; nÞ is the x-crossing number of the random walk, and Cðx; s; tÞ is the
crossing local time of W.
Remark 2.2. Theorem 2.1 implies weak convergence as a corollary. Indeed, we can
note the following scaling relationship which comes only from the Brownian scaling
of W:
W ðNs; NtÞ CðNx; Ns; NtÞ
ðdÞ
;
¼ fðW ðs; tÞ; Cðx; s; tÞÞgs;tX0;x2R ; (2.6)
N
N 3=2
s;tX0; x2R
39
41
43
45
O
R
C
37
) fðW ðs; tÞ; Cðx; s; tÞÞgs;t2½0;1;x2R :
N
35
where ¼ denotes the equality of finite-dimensional distributions. Then the following
is a consequence of Theorem 2.1, (2.6), and a standard estimate of modulus of
continuity: As N ! 1;
S bNsc;bNtc ZðNx; bNsc; bNtcÞ
;
N
N 3=2
s;t2½0;1; x2R
ð2:7Þ
2
Here, ) denotes weak convergence in the Skorohod space DR2 ðR ½0; 1 Þ:
U
33
R
ðdÞ
31
ð2:5Þ
TE
23
x2R
EC
21
PR
1pmpN 1pnpN
19
(2.4)
0
F
9
Ref. [21] introduces the process Ls as the line local time of W.
We define the crossing local time of W at level x as
Z s
pffiffiffi
uLu ðx; tÞ du:
Cðx; s; tÞ ¼
O
7
O
1
5
Remark 2.3. We have not considered other walks than the simple ones described
here. However, we suspect that, after making small adjustments, our methods would
apply to other walks such as Gaussian ones. (The local time of a two-parameter
Gaussian walk needs to be appropriately defined, of course.) We believe that the endresult is the same: Counting vertical crossings performs better than counting
(approximate) zeros.
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
6
7
9
11
13
Zðx; m; nÞ ¼
i¼0
m
Z
15
25
27
29
31
33
0
ð2:9Þ
D
PR
Such a route is fraught with technical difficulties. For one, it is not clear why (2.8)
should hold jointly for all i if Li is the line local time of a single Brownian sheet.
Moreover, the rate of approximation in (2.8) depends in a subtle way on i, and it is
not at all clear what information can be gleaned from this about the rate of
approximation in (2.9).
Our method provides an approach that is based on our attempt at solving the
following loosely stated problem: ‘‘How close are the local times LðX 1 Þ and LðX 2 Þ of
two processes X 1 and X 2 if X 1 X 2 ; and if LðX 1 Þ and LðX 2 Þ are sufficiently
smooth?’’. Interestingly enough, our solution to the mentioned problem uses (2.8),
but only for a fixed value of i.
The remainder of the paper is organized as follows: The two parts of Theorem 2.1,
namely the two claims in (2.5), are proved in distinct sections. The first assertion of
(2.5) is straightforward; it is proved in Section 3. We prove the second assertion of
(2.5) in Section 4. We do this by means of two technical lemmas. The said lemmas
themselves are proved, respectively, in Sections 5 and 6. Finally, we derive Theorem
1.1 in Section 7. Our derivation involves an application of (2.5) and a recent estimate
of the authors on the explosion rate of the local time along lines of W [15].
TE
23
i¼0
pffiffi
sLs ðx; nÞ ds ¼ Cðx; m; nÞ:
EC
21
m pffi
X
iLi ðx; nÞ
R
19
zi ðx; nÞ O
R
17
m
X
F
5
O
3
us say a few words about the proof of (2.5). Recall that Zðx; m; nÞ ¼
PLet
m
z
ðx;
nÞ; where for each i, n7!zi ðx; nÞ is the number of x-level crossings of the
i
i¼0
one-parameter, simple random walk fS i;j gj¼0;1;2;... : There is a rich literature for level
crossings of such random walks. For example, we can appeal to [4, Theorem 1.2] to
see that for each fixed i,
pffi
zi ðx; nÞ iLi ðx; nÞ:
(2.8)
pffi
In words, this means that zi ðx; nÞ can be well approximated by i Li ðx; nÞ where
Li ðx; nÞ is the local time of a Brownian motion with infinitesimal variance i. One
might then conjecture that such local times can be embedded in the line local times Li
of a single Brownian sheet W; cf. (2.3). If this were so, then
O
1
C
35
3. Proof of Theorem 2.1 (2.5): first part
39
Our proof of the first part of (2.5) relies on Bernstein’s inequality that we now
recall; cf. [20, p. 855].
Let fZk gkX1 be a sequence of independent mean-zero variables such that for some
c40; EfjZk jn gp12vk n! cn2 for all nX2: Then, for any x40 and nX1;
(
)
X
n
1
x2
P Zk Xx p2 exp Pn
:
(3.1)
k¼1 2 k¼1 vk þ cx
41
43
45
U
N
37
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
13
15
p
17
19
21
Z
F
11
0
c1 log m
nyn1 dy þ
Z
0
O
9
where c1 ; c2 and c3 are constants that do not depend on x; m: Since fX i;j gi;jX1 are
i.i.d., we can arrange things so that fW j gjX0 are independent processes. P
Now, let us fix mX1; and consider the process Z ¼ fZj gjX1 where Zj ¼ m
i¼1 X i;j W j ðmÞ: Clearly, Z is a sequence of independent (but not necessarily i.i.d.) mean-zero
variables. According to [16] for any nX2 and jX1;
Z 1
nyn1 PfjZj j4yg dy
EfjZj jn g ¼
1
nðc1 log m þ xÞn1 c2 ec3 x dx:
0
The first integral equals ðc1 log mÞn ; whereas the second is
Z c1 log m Z 1 þ
nðc1 log m þ xÞn1 c2 ec3 x dx
c1 log m
0
ð3:3Þ
Z
c1 log m
Z
1
dx þ nc2
ð2xÞn1 ec3 x dx
0
c1 log m
Z 1
Z 1
ec3 x dx þ nc2
ð2xÞn1 ec3 x dx
pnð2c1 log mÞn1 c2
pnð2c1 log mÞ
23
25
n1
O
7
PR
5
D
3
Fix jX1; and consider the sequence fX i;j gi;jX1 of i.i.d. random variables. According
to [16], possibly in an enlarged probability space, there exists a standard Brownian
motions W j such that for all x40 and mX1;
(
)
X
m
P X W j ðmÞ4c1 log m þ x pc2 ec3 x ;
(3.2)
i¼1 i;j
c2
e
0
27
c3 x
TE
1
7
0
EC
nð2c1 log mÞn1 c2 n!c2 2n1
þ
:
p
c3
ðc3 Þn
29
ð3:4Þ
31
39
41
43
45
O
R
8nX2:
(3.5)
C
37
c4 ðlog mÞ2
ðc4 log mÞn2 n!
2
By Bernstein’s inequality (cf. (3.1)), for any x40 and m; nX1;
(
)
X
n X
m
n
X
1
x2
P X W j ðmÞXx p2 exp :
j¼1 i¼1 i;j j¼1
2c4 nðlog mÞ2 þ x log m
N
35
U
33
EðjZj jn Þp
R
Therefore, there exists an absolute constant c4 such that
(3.6)
It is manifest that fW ðj þ 1; kÞ W ðj; kÞgj;kX0 has the same law
P as fW j ðkÞgj;kX0 :
Standard methods can then be used to prove that we can embed f ‘j¼1 W j ðkÞgj;kX1 in
a two-parameter Brownian sheet W, although we may possibly need to work in an
enlarged probability space. (In the paper, we often use several embedding schemes in
enlarged spaces. This can be justified by a coupling argument as in [3, p. 53].)
Therefore, for any x40 and NX1;
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
8
1
P max
max jSm;n W ðm; nÞj4x
1
x2
2
p2N exp :
2c4 Nðlog NÞ2 þ x log N
1pmpN 1pnpN
3
5
13
15
17
4. Proof of Theorem 2.1 (2.5): second part
F
11
Take x ¼ ð7c4 Þ1=2 N 1=2 ðlog NÞ3=2 ; so that the expression on the right-hand side is
summable in N. Applying the Borel–Cantelli lemma readily yields the first assertion
of (2.5). &
In this section, we prove the second assertion of (2.5) of Theorem 2.1. This is done
by virtue of two technical estimates—Propositions 4.1 and 4.2—as well as two
supporting lemmas.
Our first proposition controls the oscillations of the process C, viz.,
O
9
ð3:7Þ
O
7
23
max
jCðx; m; nÞ Cðy; m; nÞj ¼ oðN 1þmaxða=2;1=4Þþ Þ:
sup
1pmpN 1pnpN jxyjpN a
(4.1)
D
21
max
PR
Proposition 4.1. For any a; 40; the following holds almost surely: As N ! 1;
19
27
Proposition 4.2. Given a; 40; the following holds almost surely: As N ! 1;
max
31
37
O
R
max
sup
sup
41
43
8N large:
(4.3)
Proof. We can appeal to the proof of [18, Lemma 1.2] to deduce that for any
a; b; l40;
(
)
P
45
jW ðu; vÞ W ði; jÞjpN y
U
0pi;jpN ipupiþ1 jpvpjþ1
39
(4.2)
Lemma 4.3. For any fixed y4 12 ; the following is almost surely valid:
C
35
jZðx; m; nÞ Zðy; m; nÞj ¼ oðN 1þmaxða=2;1=4Þþ Þ:
We postpone proving these Propositions until Sections 5 and 6, respectively. In the
remainder of this section, we use the latter propositions to prove the second assertion
of (2.5) of Theorem 2.1.
N
33
sup
EC
max
0pmpN 1pnpN jxyjpN a
R
29
TE
25
Theorem 2.1 asserts that the processes Z and C are asymptotically close to one
another. The following analogue of Proposition 4.1 states that their asymptotic
moduli of continuity are also close, viewed on an appropriate scale.
sup
ðs;tÞ2½0;a½0;b
jW ðs; tÞj4l p4PfjW ða; bÞj4lg:
(4.4)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
1
3
Therefore, for any i; jpN;
(
P
sup
9
)
jW ðu; vÞ W ði; jÞj4N y
sup
ipupiþ1 jpvpjþ1
5
p4PfjW ð1; 1Þj4N y g þ 4PfjW ði; 1Þj4N y g þ 4PfjW ð1; jÞj4N y g
1
p12 exp N 2y1 :
2
11
Consequently,
(
X
P max
sup
sup
0pi;jpN ipupiþ1 jpvpjþ1
NX1
13
)
p12
15
X
NX1
jW ðu; vÞ W ði; jÞj4N
y
1 2y1
N exp N
;
2
F
9
2
ð4:6Þ
PR
21
&
Lemma 4.4. For any fixed 40; the following holds almost surely: As N ! 1;
sup Zðx; N; NÞ ¼ oðN ð3=2Þþ Þ:
x2R
(4.7)
D
19
Our final lemma is an almost-sure uniform bound for Z.
O
which is finite. The Borel–Cantelli lemma completes our verification.
17
ð4:5Þ
O
7
23
31
EC
PfSi;j S i;jþ1 o0gp
i¼0 j¼0
N N
1
X
X
i¼0 j¼0
c5
pffiffiffiffiffiffiffiffiffiffi pc6 N 3=2 ;
jþ1
(4.8)
where c5 and c6 are two unimportant constants; cf. [19, Theorem 2.8] for the local
limit theorem. To this, we apply Markov’s inequality to obtain
R
29
N N
1
X
X
PfZð0; N; NÞ4N ð3=2Þþ gpc6 N :
O
R
27
EfZð0; N; NÞg ¼
TE
Proof. Applying the local limit theorem, it is not hard to see that
25
(4.9)
2=
37
Zð0; N; NÞ ¼ OðN ð3=2Þþ Þ;
C
35
Let N k ¼ bk c and use the Borel–Cantelli lemma to see that, with probability one,
ð3=2Þþ
for all large k, Zð0; N k ; N k ÞpN k
: By the monotonicity of N7!Zð0; N; NÞ;
sup
41
43
45
jxjpN
U
39
a.s.
(4.10)
Proposition 4.2 then shows that
N
33
jZðx; N; NÞj ¼ oðN ð3=2Þþ2 Þ;
a.s.
(4.11)
1þ
It remains to replace supjxjpN 1þ by supx2R in the above.
By the law of the iterated logarithm law for Brownian sheet,
jW ðs; tÞj
lim sup pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1;
s;t!1
4st log logðstÞ
a.s.;
(4.12)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
10
11
13
Therefore, with probability one, when N is sufficiently large, Zðx; N; NÞ ¼ 0 for all
jxjXN 1þ : Accordingly, (4.11) implies our lemma since 40 is arbitrary. &
We are in position for presenting our
Proof of Theorem 2.1 (2.5): second part. By our definition (cf. 2.4) of the crossing
local time Cðx; s; tÞ; for any Borel set A R;
Z
Z s Z t
pffiffiffi
Cða; s; tÞ da ¼
du dv u 1fW ðu;vÞ2Ag :
(4.14)
A
19
21
23
25
27
In particular, for any x 2 R and b412;
Z
Z
xþN b
x
Z
m
Cða; m; nÞ da ¼
n
du
0
dv
0
pffiffiffi
u 1fxpW ðu;vÞpxþN b g :
N b Cðx; m; nÞ þ oðN bþ1þmaxðb=2;1=4Þþ Þ
Z m
Z n
pffiffiffi
¼
du
dv u 1fxpW ðu;vÞpxþN b g :
0
0
31
i¼0 j¼0
Z m
p
43
45
dv
pffiffiffi
u 1fxpW ðu;vÞpxþN b g
0
m X
n pffi
X
p
i 1fxN y pW ði;jÞpxþN b þN y g þ OðN 3=2 Þ:
35
41
n
O
R
0
33
Z
du
R
m X
n pffi
X
i 1fxþN y pW ði;jÞpxþN b N y g þ OðN 3=2 Þ
ð4:17Þ
C
i¼0 j¼0
N
On the other hand, according to (2.5), for any aob; almost surely as N ! 1;
1faþN y pSi;j pbN y g p1fapW ði;jÞpbg p1faN y pSi;j pbþN y g ;
(4.18)
U
39
ð4:16Þ
Lemma 4.3 then shows us that almost surely, as N ! 1;
29
37
(4.15)
We apply Proposition 4.1 to this and deduce that with probability one the following
exists for all N large: For all m; npN;
TE
17
0
EC
15
0
F
9
(4.13)
O
7
a.s.
O
5
max jS m;n j ¼ oðN 1þ Þ;
max
1pmpN 1pnpN
PR
3
cf. [22]. This, together with (2.5), implies that
D
1
uniformly in i; jpN: Recall Zðx; m; nÞ from (2.2) and note that
Z
m X
n pffi
X
Zða; m; nÞ da ¼
i 1fSi;j 2Ag :
A
i¼0 j¼0
From this it follows that with probability one, as N ! 1;
(4.19)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
x2N y þN b
Zða; m; nÞ dapN b Cðx; m; nÞ þ oðN bþ1þmaxðb=2;1=4Þþ Þ
xþ2N y
p
5
9
11
13
Zða; m; nÞ da;
ðN b 4N y ÞZðx; m; nÞpN b Cðx; m; nÞ þ oðN bþ1þmaxðb=2;1=4Þþ Þ
pðN b þ 4N y ÞZðx; m; nÞ;
max sup jZðx; m; nÞ Cðx; m; nÞj
max
0pmpN 1pnpN x2R
ðbyÞ
p4N
sup Zðx; N; NÞ þ oðN 1þmaxðb=2;1=4Þþ Þ
x2R
¼ oðN ðbyð3=2ÞÞ Þ þ oðN 1þmaxðb=2;1=4Þþ Þ:
PR
D
5. Proof of Proposition 4.1
23
33
R
hZiOrl pc7 sup
kZkm
:
m
(5.2)
C
Our proof of Proposition 4.1 is based on the following convenient formulation of
Dudley’s metric entropy theorem.
N
37
(5.1)
They stand for the L ðPÞ-norm and the Orlicz (pseudo-)norm (associated with the
convex function f ðxÞ ¼ ex 1) of Z; respectively. (To be precise, hiOrl is not a norm,
but it is equivalent to one.) The two norms are related to each other as follows: There
exists an absolute constant c7 such that
mX1
35
hZiOrl ¼ inffa40 : E½expða1 jZjÞp2g:
O
R
31
and
EC
kZkp ¼ fE½jZjp g1=p ;
TE
Let us start by fixing the basic notation in this section. For any real-valued
random variable Z; we write
p
29
ð4:22Þ
We could take b ¼ 23 þ and y ¼ 12 þ : Since 40 can be chosen arbitrarily small,
this yields (2.5). &
21
27
ð4:21Þ
uniformly in m; npN and in x 2 R: Consequently, by Lemma 4.4, for any 40;
17
25
ð4:20Þ
x2N y
uniformly in m; npN: Combining Proposition 4.2 with the above, we arrive at
15
19
xþ2N y þN b
F
7
Z
O
3
Z
O
1
11
45
If mD o1; then there exists a universal constant c8 ; such that for all l40;
39
41
U
43
Lemma 5.1 (Lacey [17, Theorem 3.1]). Let fX ðtÞ; t 2 Tg be a real-valued stochastic
process, and let d X ðs; tÞ ¼ hX s X t iOrl be the natural pseudo-metric on T induced by X.
Let NðrÞ ¼ Nðr; T; d X Þ be the minimum number of d X -balls of radius r needed to cover
T, and let
Z D
D :¼ sup d X ðs; tÞ and mD :¼
log NðrÞ dr:
(5.3)
ðs;tÞ2T 2
0
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
12
1
3
(
P
)
sup jX s X t j4c8 ðl þ D þ mD Þ pc8 el=D :
(5.4)
ðs;tÞ2T 2
5
11
13
15
F
9
I 1 ðpÞ ¼ kLs ðx; nÞ Ls ðy; nÞkp ; and
O
7
Fix 1pi; npN; and choose k 2 f0; 1; 2; kmax g where kmax :¼ bN 2 =n1=2 c:
We
pffiffiffiintend to
pffiffiffi apply Fact 5.1 to the process fLs ðx; nÞ; ðs; xÞ 2 T ¼ ½i; i þ 1 ½k n; ðk þ 1Þ ng; where Ls ðx; nÞ is the line local time of the Brownian sheet W.
This was defined in (2.3).
We begin by estimating the entropy number NðrÞ induced by d X : This will be
done in successive steps.
Choose some ðs; xÞ 2 T and ðt; yÞ 2 T: In order to bound d X ððs; xÞ; ðt; yÞÞ; we start
by estimating kLs ðx; nÞ Lt ðy; nÞkp for pX1: This can be reduced to estimating
17
I 1 ðpÞpc9 p
PR
21
Lemma 5.2. For every n 2 ð0; 12Þ; there exists a constant c9 ¼ c9 ðnÞ 2 ð0; 1Þ such that
for all x; y 2 R; for all integers i; nX1; s 2 ½i; i þ 1; and pX1;
n1=2ð1nÞ
jx yjn :
i1=2ð1þnÞ
TE
23
ðdÞ
31
33
EC
29
Since L1 ðx; tÞ is a standard Brownian local time, we can use the following inequality
of [2]: For any n 2 ½0; 12Þ and pX1;
1
jL1 ðx; tÞ L1 ðy; tÞj
o1:
(5.8)
c9 ðnÞ ¼ sup sup sup
jx yjn
pX1 p 0ptp1 xay
p
R
27
Proof. Writing ¼ for equality of distributions, and use scaling to deduce that
rffiffiffi n
x
y
ðdÞ
Ls ðx; nÞ Ls ðy; nÞ ¼
L1 pffiffiffiffiffi ; 1 L1 pffiffiffiffiffi ; 1 :
(5.7)
s
sn
sn
O
R
25
(5.6)
D
19
ð5:5Þ
O
I 2 ðpÞ ¼ kLs ðy; nÞ Lt ðy; nÞkp :
Consequently, and writing c9 ¼ c9 ðnÞ for brevity, we have
n1=2ð1nÞ
jx yjn :
s1=2ð1þnÞ
Since s 2 ½i; i þ 1; this has the desired result.
C
35
39
41
43
45
(5.9)
&
U
37
N
I 1 ðpÞpc9 p
Our estimate for I 2 ðpÞ of (5.5) is derived by similar methods.
Lemma 5.3. For every l 2 0; 14 ; there exists a constant c10 ¼ c10 ðlÞ 2 ð0; 1Þ such that
for all pX1; integers n; iX1; and s; t 2 ½i; i þ 1;
I 2 ðpÞpc10 p n1=2 ilð1=2Þ ðt sÞl :
(5.10)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
1
3
13
Proof. We recall the following result of [17, Proposition 4.2]: For any pX1; 0oho1
and 0olo14;
sup kL1þh ðx; 1Þ L1 ðx; 1Þkp pc10 hl p:
(5.11)
x2R
5
7
9
Assuming spt without loss of generality, and using the scaling property, we have:
For all ipsptpi þ 1 and y 2 R;
rffiffiffi n
y
y
1=2 lð1=2Þ
p
ffiffiffiffi
ffi
p
ffiffiffiffi
ffi
I 2 ðpÞ ¼
ðt sÞl :
L1
; 1 Lt=s
;1 pc10 p n s
s
sn
sn
p
(5.12)
11
19
21
23
41
43
45
EC
R
O
R
sup
sup jLs ðx; nÞ Lt ðx; nÞjpc16
C
ðs;tÞ2½i;iþ12 x2R
N 1=2 log N
:
i1=2ð1þnÞ
(5.15)
N
Proof. We prove this by applying Lemma 5.1 to the process ðx; sÞ7!X x;s ¼ Ls ðx; nÞ;
where nX1 is an arbitrary integer.
Recall D and mD from (5.3), and note that thanks to Lemma 5.4,
pffiffiffi
n1=2ð1nÞ ð nÞn þ n1=2
N 1=2
Dpc12
p2c
:
(5.16)
12
i1=2ð1þnÞ
i1=2ð1þnÞ
Next, we estimate mD : Owing to Lemma 5.4, the minimum number NðrÞ of d X -balls
of radius r needed to cover T satisfies
U
39
ð5:14Þ
We now use this to estimate the asymptotic modulus of continuity of the line local
times.
Lemma 5.5. Given a fixed n 2 0; 12 ; there exists a constant c16 ¼ c16 ðnÞ 2 ð0; 1Þ such
that almost surely for all large N, and 1pi; npN;
33
37
O
F
n1=2ð1nÞ jx yjn þ n1=2 ðt sÞ1=2n
;
i1=2ð1þnÞ
:¼ c9 þ c10 : This immediately yields our lemma. &
pc11 p
where c11
35
O
kLs ðx; nÞ Lt ðy; nÞkp pI 1 ðpÞ þ I 2 ðpÞ
27
31
(5.13)
Proof. We choose l ¼ 2n; and appeal to Lemmas 5.2 and 5.3, as well as Minkowski’s
inequality to deduce that
25
29
n1=2ð1nÞ jx yjn þ n1=2 jt sj1=2n
:
i1=2ð1þnÞ
PR
17
hLs ðx; nÞ Lt ðy; nÞiOrl pc12
D
15
TE
13
Apply this to s; t 2 ½i; i þ 1 to finish. &
Lemma 5.4. For every n 2 0; 12 ; there exists a constant c12 ¼ c12 ðnÞ 2 ð0; 1Þ such that
for all integers i; nX1; x; y 2 R; and ðs; tÞ 2 ½i; i þ 12 ;
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
14
1
PfAikn gpc8 N 6 ;
(
Aikn :¼
19
21
23
25
27
37
39
41
43
45
F
(5.19)
N 1=2 log N
;
(5.20)
i1=2ð1þnÞ
with c16 ¼ 2c12 c15 : On the other hand, it follows from (4.12) that with probability
one, when N is sufficiently large, sup0pspN sup0ptpN jW ðs; tÞjoN 2 ; so that Ls ðx; nÞ ¼
0 for s; npN and jxj4N 2 : Our lemma follows as a consequence. &
jLs ðx; nÞ Lt ðx; nÞjpc15 D log Npc16
We present one final supporting lemma.
Lemma 5.6. If a; 40 and n 2 0; 12 are held fixed, then with probability one for all
large N and all i; npN;
jLi ðx; nÞ Li ðy; nÞjpi1=2ð1þnÞ N anþ1=2ð1nÞþ :
(5.21)
R
jxyjpN a
P
O
R
Proof. Let a40 and use scaling to see that for each i and n, supjxyjpN a jLi ðx; nÞ Li ðy; nÞj is distributed as ðn=iÞ1=2 supjxyjpN a =pffiffiffi
in jL1 ðx; 1Þ L1 ðy; 1Þj: Hence, for any
b40 and pX1;
(
)
sup
jxyjpN a
jLi ðx; nÞ Li ðy; nÞj4b
C
35
jLs ðx; nÞ Lt ðy; nÞj4c15 D log N ; ð5:18Þ
p
np=2 pbp
sup pffiffiffi jL1 ðx; 1Þ L1 ðy; 1Þj
jxyjpN a = in
i
p
p
np=2 N a np jL
ðx;
1Þ
L
ðy;
1Þj
1
1
pffiffiffiffi pbp
sup
i
jx yjn
in xay
p
np=2 N a np
pffiffiffiffi :
pðpc9 ðnÞÞp bp
i
in
N
33
sup
pffiffi
pffiffi
ðs;tÞ2½i;iþ12 ðx;yÞ2½k n;ðkþ1Þ n2
By the Borel–Cantelli lemma, almost surely for all large N, 1pi; npN; ðs; tÞ 2
½i; i þ 12 ; and jxjpN 2 ; we have
U
31
sup
)
and c15 :¼ c8 ð7 þ c14 Þ: Therefore,
(
)
N [
N
[
[
c8
3c8
P
Aikn pN 2 ð2N 2 þ 1Þ 6 p 2 :
N
N
i¼1 n¼1 jkjpkmax
sup
29
where
O
17
(5.17)
D
15
1=2ð3nÞ 1=n
N
:
r3
TE
13
pc13 N
1=2
RD
Thus, mD p 0 logðNðrÞÞ drpc14 D log N; where c14 depends only on n: Applying
Lemma 5.1 with l :¼ 6D log N yields the following:
9
11
1=n
O
7
1=2nð3nÞ
r3 i3=2ð1þnÞ
PR
5
NðrÞpc13 n
EC
3
1=2
The last inequality follows from (5.8).
ð5:22Þ
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
9
11
13
15
17
19
21
Since this is summable in N, we can use the Borel–Cantelli lemma to see that with
probability one, for all large N and all i; npN;
sup
jxyjpN a
jLi ðx; nÞ Li ðy; nÞjpn1=2ð1nÞ i1=2ð1þnÞ N anþ ;
thus leading to our lemma.
We are ready to present our
Proof of Proposition 4.1. According to [17, Theorem 2.2], the following holds with
probability one:
Z 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
u sup Lu ðx; NÞ du ¼ Oð Nloglog N Þ
x2R
0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
sup sup uLu ðx; NÞ ¼ Oð Nloglog N Þ:
ð5:25Þ
D
TE
27
Let Cðx; s; tÞ be the crossing local time of W, as in (2.4). For 1pm; npN;
Z m
pffiffiffi
u Lu ðx; nÞ du
Cðx; m; nÞ ¼
0
Z m
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffi
uLu ðx; nÞ du
¼ Oð N log log N Þ þ
EC
25
(5.24)
&
1pupN x2R
23
ð5:23Þ
F
7
O
5
O
3
pffiffiffiffi
Let 2 ð0; 1Þ; p :¼ 41 ; and b :¼ ðn=iÞ1=2 ðN a = inÞn N to obtain,
(
)!
N [
N
n1=2 N a n
[
pffiffiffiffi N
sup jLi ðx; nÞ Li ðy; nÞj4
P
a
i
in
i¼1 n¼1 jxyjpN
4=
4c9 ðnÞ
pN 2
N 4 :
PR
1
15
1
i¼1
31
35
C
jxyjpN
(5.27)
a
1
3
uniformly
a 1 in 1pm; npN: Since max 2ð3 nÞ; an þ 2 n þ can be as close to 1 þ
max 2; 4 as possible, Proposition 4.1 follows. &
N
39
jCðx; m; nÞ Cðy; m; nÞj ¼ OðN 1=2ð3nÞ log N þ N anþð3=2Þnþ Þ;
sup
U
37
ð5:26Þ
The last inequality following from Lemma 5.5 and (5.25), and Oð Þ is uniform
in
1pm; npN and x 2 R: From this and Lemma 5.6, we can see that for any n 2 0; 12 ;
almost surely,
O
R
33
R
m pffi
X
¼
i Li ðx; nÞ þ OðN 1=2ð3nÞ log NÞ:
29
41
6. Proof of Proposition 4.2
43
45
section is devoted to estimating the increments of x7!Zðx; m; nÞ ¼
PThis
m
i¼0 zi ðx; nÞ: By the definition of zi ðx; nÞ in (2.2),
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
16
1
15
17
19
21
23
25
(6.1)
PfjSi;j W ðijÞj4c1 logðijÞ þ zgpc2 ec3 z :
(6.3)
Strictly speaking, we really should write W i instead of W (our Wiener process W
depends on i). The same remark applies to the forthcoming event A and Wiener
process B.
Fix 2 0; 12 and d 2 0; 2 ; and consider the event
N
\
E :¼
fjS i;j W ðijÞjpN d g
(6.4)
8ipN:
EC
By (6.3), for large N, say NXN 0 ;
PðE c Þp
N
X
j¼1
31
c17 N 5 ¼ c17 N 4 :
(6.5)
R
29
S i;jþ1 4xg :
The two sums on the right-hand side represent the numbers of down- and upcrossings, respectively. Thus,
n1
X
(6.2)
1fSi;j 4x; Si;jþ1 oxg p1:
zi ðx; nÞ 2
j¼0
P
It remains to study the increments of x7! n1
j¼0 1fS i;j 4x; S i;jþ1 oxg : Our first step is to
estimate S i;j by a Wiener process. An obvious candidate would be the Brownian
sheet in (3.6) that was used in Section 3 to prove a strong approximation of Si;j :
Unfortunately, the error term in this approximation scheme is too large for our
needs. So, we proceed very carefully in replacing S i;j by another Brownian motion.
Fix ipN: Since Si;j is the sum of ðijÞ i.i.d. symmetric Bernoulli random variables,
by the KMT theorem (3.2) there exists a standard Wiener process fW ðtÞ; tX0g such
that for any z40 and some absolute constants c1 ; c2 and c3 ;
j¼1
27
1fSi;j ox;
j¼0
F
13
j¼0
n1
X
O
11
þ
O
9
S i;jþ1 oxg
PR
7
1fSi;j 4x;
D
5
n1
X
TE
3
zi ðx; nÞ ¼
37
1fSi;j 4x;
j¼0
C
35
n1
X
p
n1
X
N
33
O
R
On the event E, we have for ipN; npN and x 2 R;
S i;jþ1 oxg p
1fW ðijÞ4x;
n1
X
W ðiðjþ1ÞÞoxg
41
43
45
W ðiðjþ1ÞÞoxþN d g
þ 2 sup
n
X
a2R j¼0
U
j¼0
39
1fW ðijÞ4xN d ;
j¼0
1fapW ðijÞpaþN d g :
ð6:6Þ
Similarly, on E,
n1
X
1fSi;j 4x;
j¼0
S i;jþ1 oxg X
n1
X
j¼0
1fW ðijÞ4x;
W ðiðjþ1ÞÞoxg
2 sup
n
X
a2R j¼0
1fapW ðijÞpaþN d g :
(6.7)
If we write
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
1
Di;n
3
9
17
19
X
n1
Di;n 1E p sup 1fW ðijÞ4x;
jxyjpN a j¼0
þ 4 sup
33
35
37
39
41
43
45
1fapW ðijÞpaþN d g :
ð6:9Þ
F
X
n1
P sup
1fW ðjÞ4x;
x2R j¼0
(
)
pffiffiffi
x
1=4
log n
W ðjþ1Þoxg c20 nL pffiffiffi ; 1 Xc18 n
n
ð6:10Þ
D
pc19 nl ;
EC
TE
where L is the local time of B, and c20 :¼ EðbBþ ð1ÞcÞ: We also mention that in [4, Eq.
(1.13)], the preceding inequality was stated only for some l41; which sufficed for the
intended applications there. However, it is clear from its proof [4, pp. 272–273] that
by altering c19 ¼ c19 ðlÞ suitably, l can be made to be arbitrarily large.
A straightforward consequence of (6.9) and (6.10), with l ¼ 4d in (6.10), is as
follows: For any b40 and k40;
R
PfDi;n 1E 4b þ k þ 2c18 n1=4 log ng
(
)
pffiffiffi
pffiffiffi
x
y
pP
sup c20 nL pffiffiffiffi ; 1 c20 nL pffiffiffiffi ; 1 4b
in
in
jxyjpN a
(
)
n
X
þ P 4 sup
1fapW ðijÞpaþN d g 4k þ c19 n4=d
O
R
31
1fW ðijÞ4y;
j¼0
P
Let us consider the process x7! n1
j¼0 1fW ðijÞ4x; W ðiðjþ1ÞÞoxg : For each i, it is
Pn1
distributed as x7! j¼0 1fW ðjÞ4x=pffii; W ðjþ1Þox=pffiig : Therefore, by [4, Eq. (1.13)], there
exists a coupling of W and a standard Wiener process B such that for any l41; there
exist constants c18 and c19 such that for all nX1;
C
29
W ðiðjþ1ÞÞoyg a2R j¼0
N
27
W ðiðjþ1ÞÞoxg
n1
X
¼ P1 þ P2 þ c19 n4=d :
ð6:11Þ
U
25
n
X
a2R j¼0
21
23
(6.8)
O
15
1fSi;j 4y;
j¼0
then
11
13
S i;jþ1 oyg ;
O
7
S i;jþ1 oxg
n1
X
PR
5
X
n1
¼ sup 1fSi;j 4x;
jxyjpN a j¼0
17
Plugging this into (6.5) yields
PfDi;n 4b þ k þ 2c18 n1=4 log ngpP1 þ P2 þ c19 n4=d þ c17 N 4 :
To bound P1 ; we use (5.8) to see that for any n 2 0; 12 and pX1;
(6.12)
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
18
19
21
23
25
27
29
31
33
35
37
39
41
43
45
F
O
O
PR
17
where c21 ¼ ð8c20 c9 ðnÞ=dÞ4=d depends only on ðn; dÞ:
We now estimate P2 : The standard Gaussian density is bounded above by
ð2pÞ1=2 p12: Therefore, for any a 2 R; we have PfapW ðijÞpa þ 2N d gpN d ðijÞ1=2 :
We can apply this together with the Markov property to deduce that for any integer
pX1;
p
!p
d pffiffiffip
X
n
n
X
Nd
N n
pffi
pffiffiffi pc22
1fapW ðijÞpaþ2N d g pp!
;
(6.15)
j¼1
ij
i
j¼1
p
D
15
(6.14)
for some constant c22 ¼ c22 ðpÞ that depends only on p. By Markov’s inequality, for
any k40;
(
)
pffiffiffip
n
X
c22 N d n
pffi
sup P
1fapW ðijÞpaþ2N d g 4k p p
:
(6.16)
k
i
a2R
j¼1
TE
13
P1 pc21 N 4 ;
EC
11
pffiffiffiffi
Now let a‘ ¼ 2N d ‘; for ‘ ¼ 0; 1; 2; . . . ; ‘max ; where ‘max ¼ bN d inc: By the
usual estimate for Gaussian tails,
pffiffiffiffi ð2N d bN d incÞ2
P max jW ðijÞj4a‘max p2 exp (6.17)
p2 expðN 4d Þ:
1pjpn
2in
Pn
On the other hand, if max1pjpn jW ðijÞj4a‘max ; and if
j¼1 1fapW ðijÞpaþN d g 4k for
P
some a 2 R; then there exists ‘; with j‘jp‘max such that nj¼1 1fa‘ pW ðijÞpa‘ þ2N d g 4k:
By (6.16), this yields,
(
)
n
X
P sup
1fapW ðijÞpaþN d g 4k
R
9
ð6:13Þ
O
R
7
C
5
)
b
pffiffiffi
P1 ¼ P
sup pffiffiffi jLðx; 1Þ Lðy; 1Þj4
2c20 n
jxyjpN a = in
jLðx; 1Þ Lðy; 1Þj
b
p
ffiffiffiffi
4
pP sup
pffiffiffi
jx yjn
2c20 nðN a = inÞn
xay
p
ffiffiffiffi
p
pffiffiffi
2c20 nðN a = inÞn
:
pðpc9 ðnÞÞp
b
pffiffiffiffi
pffiffiffi
Choosing b ¼ nðN a = inÞn N d and p ¼ 4d gives
N
3
(
U
1
a2R j¼1
pð2‘max þ 1Þ
pffiffiffip
c22 N d n
pffi
þ 2 expðN 4d Þ:
kp
i
We replace k by 14 k to deduce that for all i; npN;
ð6:18Þ
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
1
P2 pc23
3
pc23
5
15
17
19
(6.20)
Plugging this and (6.14) into (6.12) implies that for i; npN;
rffiffiffi
pffiffiffi N a n
n
P Di;n 4 n pffiffiffiffi N d þ N 2d
þ 2c18 n1=4 log n
i
in
4
4d
pc24 N þ 2 expðN Þ þ c19 n4=d ;
n¼bN c
31
D
TE
which is summable for N. By the Borel–Cantelli lemma, almost surely for all large N,
ipN; and every N d pnpN;
rffiffiffi
pffiffiffi N a n d
n
2d
p
ffiffiffiffi
Di;n p n
N þN
(6.23)
þ 2c18 n1=4 log n:
i
in
EC
29
Thus, whenever mpN and N d pnpN;
a n
m
X
pffiffiffi
pffiffiffi
N
Di;n p nN 1ðn=2Þ pffiffiffi N d þ N 2dþð1=2Þ n þ N ð5=4Þþd
n
i¼1
33
39
41
43
45
eventually (in N):
ð6:24Þ
C
This last inequality uses the facts that npN and that do14:
Pm
1þd
inequality
Di;n p2n
If noN d ; we can use the
1trivial
1
to see that i¼1 Di;n p2N :
Therefore, whenever n 2 0; 2 ; 2 0; 4 and d 2 0; 2 ; with probability one, for all
large N and m; npN;
N
37
pN ð3=2Þnþanþd þ 2N ð5=4Þþd ;
U
35
ð6:22Þ
R
27
pc24 N 2 þ 2N 2 expðN 4d Þ þ c19 N 2 ;
O
R
25
ð6:21Þ
where c24 :¼ c21 þ c23 þ c17 : As a consequence,
0
1
rffiffiffi
N
N [
[
pffiffiffi N a n d
n
P@
Di;n 4 n pffiffiffiffi N þ N 2d
þ 2c18 n1=4 log n A
i
in
d
i¼1
21
23
F
13
P2 pc23 N 4 þ 2 expðN 4d Þ:
O
11
Taking k ¼ N 2d ðn=iÞ1=2 and p ¼ 1d ð5 þ dÞ yields
O
9
ð6:19Þ
PR
7
pffiffiffiffi pffiffiffip
N d in N d n
pffi
þ 2 expðN 4d Þ
kp
i
pffiffiffiffi pffiffiffip
N d in N d n
pffi
þ 2 expðN 4d Þ:
kp
i
19
m
X
Di;n pN ð3=2Þnþanþd þ 3N ð5=4Þþd :
(6.25)
i¼1
Since n 2 0; 12 is arbitrary, it can be chosen such that 32 n þ an þ do1 þ a2 þ 2d: In
view of the definition of Di;n in (6.8), and relation (6.2), we have proved that, almost
surely, for any 2 ð0; 12Þ; when N ! 1;
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
20
1
3
X
m
m
X
max max
sup zi ðx; nÞ zi ðy; nÞ
0pmpN 1pnpN jxyjpN a i¼0
i¼0
¼ oðN 1þða=2Þþ þ N ð5=4Þþ Þ:
5
ð6:26Þ
This completes our proof of Proposition 4.2.
&
7
25
27
29
31
33
35
37
39
41
F
O
O
PR
We now use the following recent estimate of the authors [15, Theorem 3.3]: For any
n 2 0; 12 ; there exists a small constant h0 ¼ h0 ðn; aÞ such that for all N40 and
h 2 ð0; h0 Þ;
(
)
N
nj log hj
P
inf Lu 0;
ph p exp :
(7.3)
1
2a
log j log hj
2 NpupN
D
23
aNpupN
aN
TE
21
EC
19
a.s.
(7.1)
1 We start byR choosing
a 2 2 ; 1 and b41 such that 2a4b: We also recall that
N pffiffiffi
Cð0; N; NÞ ¼ 0 u Lu ð0; NÞ du: Thus,
Z N pffiffiffiffiffiffiffi
pffiffiffi
aN Lu ð0; NÞ duX að1 aÞN 3=2 inf Lu ð0; NÞ:
(7.2)
Cð0; N; NÞX
In [15] we obtained the above for N ¼ 1: This formulation is a consequence of the
=2
yields
case N ¼ 1 and scaling. Taking N k ¼ bbk c and h ¼ N k
(
)
1
X
Nk
(7.4)
P
inf
Lu 0;
pN k o þ 1:
1
2a
2 N k pupN k
k¼1
R
17
Cð0; N; NÞXN ð3=2Þ ;
O
R
15
By the Borel–Cantelli lemma, almost surely for all large k,
Nk
=2
inf
Lu 0;
4N k :
1
2a
2 N k pupN k
C
13
In view of (4.10), we need to check only the lower bound; namely that for any
40; ZðNÞXN ð3=2Þ almost surely for all large N. By means of (2.5), we have to
prove that for large N,
N
11
7. Proof of Theorem 1.1
Consequently, whenever N k1 pNpN k ;
U
9
inf
aNpupN
Lu ð0; NÞX
inf
aN k1 pupN k
=2
43
4N k
Lu ð0; N k1 ÞX
Xð2aNÞ=2 :
1
45
inf
1
2N k pupN k
(7.5)
Nk
Lu 0;
2a
ð7:6Þ
We have used the fact that N k1 Xð2aÞ N k for all sufficiently large k; this, in turn,
follows from the inequality 2a4b: In summary, for all large N,
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
pffiffiffi
Cð0; N; NÞX a ð1 aÞN 3=2 ð2aNÞ=2 XN 3=2 ;
1
3
yielding (7.1), whence Theorem 1.1.
21
(7.7)
&
5
17
19
21
23
25
11.
F
O
EC
27
% run a 1-dimensional simple walk for n time-steps
load seed
% The same random walk is always used
rand(‘state’,seed)
figure;
x = rand(n);
X ¼ 2 roundðxÞ 1;
% Generate Rademacher variables
S=cumsum(X);
% Sum the columns separately
W=S;
W(:,1) = S(:,1);
for i=2:n
Wð:; iÞ ¼ Wð:; i 1Þ þ Sð:; iÞ;
% W is the walk
end
for k=1:n
for j=1:n
if W(k,j) == 0
plot(k,j)
end
end
end
print -deps file.ps
O
15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
PR
13
D
11
The Matlab code for Fig. 1(a) follows. It presupposes the existence of a seed for
the random number generator, and that the seed is stored in a Matlab file called
‘‘seed.dat.’’
TE
9
Appendix: The Matlab code
R
29
31
12.
O
R
7
33
39
C
N
37
In order to simulate the vertical crossings (for the same walk), the third line of 11
above (i.e., ‘‘if Wðk; jÞ ¼¼ 0’’) needs to be replaced by ‘‘if Wðk; jÞ Wðk; j þ 1Þo0:’’
U
35
Acknowledgements
41
43
45
A part of this work was done while one of us (P. R.) was visiting the Laboratoire
de Probabilités et Modèles Aléatoires UMR 7599, Université Paris VI. We wish to
thank the laboratoire for their hospitality. The final draft of this paper has benefitted
from many suggestions made by an anonymous referee and one of associate editors.
SPA : 1352
ARTICLE IN PRESS
D. Khoshnevisan et al. / Stochastic Processes and their Applications ] (]]]]) ]]]–]]]
22
References
3
[1] R.J. Adler, The Geometry of Random Fields, Wiley, Chichester, 1981.
[2] M.T. Barlow, M. Yor, Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma, and
applications to local times, J. Funct. Anal. 49 (1982) 198–229.
[3] I. Berkes, W. Philipp, Approximation theorems for independent and weakly dependent random
vectors, Ann. Probab. 7 (1979) 29–54.
[4] A.N. Borodin, On the character of convergence to Brownian local time, II, Probab. Theory Related
Fields 72 (1986) 251–277.
[5] R.C. Dalang, T.S. Mountford, Nondifferentiability of curves on the Brownian sheet, Ann. Probab. 24
(1) (1996) 182–195.
[6] R.C. Dalang, T.S. Mountford, Points of increase of the Brownian sheet, Probab. Theory Related
Fields 108 (1) (1997) 1–27.
[7] R.C. Dalang, T.S. Mountford, Level sets, bubbles and excursions of a Brownian sheet, in: InfiniteDimensional Stochastic Analysis, Amsterdam, 1999, pp. 117–128; Verh. Afd. Natuurkd. 1. Reeks. K.
Ned. Akad. Wet. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000.
[8] R.C. Dalang, T.S. Mountford, Jordan curves in the level sets of additive Brownian motion, Trans.
Amer. Math. Soc. 353 (9) (2001) 3531–3545.
[9] R.C. Dalang, J.B. Walsh, Geography of the level sets of the Brownian sheet, Probab. Theory Related
Fields 96 (2) (1993) 153–176.
[10] R.C. Dalang, J.B. Walsh, The structure of a Brownian bubble, Probab. Theorey Related Fields 96 (4)
(1993) 475–501.
[11] R.C. Dalang, J.B. Walsh, Local structure of level sets of the Brownian sheet, in: Stochastic Analysis:
Random Fields and Measure-Valued Processes, Ramat Gan, 1993/1995, pp. 57–64; Israel Math.
Conf. Proc. 10, Bar-Ilan University, Ramat Gan, 1996.
[12] W. Ehm, Sample function properties of multiparameter stable processes, Z. Wahrsch. verw. Geb. 56
(2) (1981) 195–228.
[13] W.S. Kendall, Contours of Brownian processes with several-dimensional times, Z. Wahrsch. verw.
Geb. 52 (3) (1980) 267–276.
[14] D. Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields, Springer, New
York, 2002.
[15] D. Khoshnevisan, P. Révész, Z. Shi, On the explosion of the local times of Brownian sheet along
lines, Ann. Inst. H. Poincaré Probab. Statist. 40 (1) (2004) 1–24.
[16] J. Komlós, P. Major, G. Tusnády, An approximation of partial sums of independent r.v.’s and the
sample df. I, Z. Wahrsch. verw. Geb. 32 (1975) 111–131.
[17] M.T. Lacey, Limit laws for local times of the Brownian sheet, Probab. Theory Related Fields 86
(1990) 63–85.
[18] W.E. Pruitt, S. Orey, Sample functions of the N-parameter Wiener process, Ann. Probab. 1 (1973)
138–163.
[19] P. Révész, Random Walk in Random and Nonrandom Environments, World Scientific Publishing
Co., Inc., Teaneck, NJ, 1990.
[20] G.R. Shorack, J.A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York,
1986.
[21] J.B. Walsh, The local time of the Brownian sheet, Astérisque 52–53 (Temps Locaux) (1978) 47–62.
[22] G.J. Zimmerman, Some sample function properties of the two-parameter Gaussian process, Ann.
Math. Statist. 43 (1972) 1235–1246.
23
25
27
29
31
33
35
37
39
O
O
PR
21
D
19
TE
17
EC
15
R
13
O
R
11
C
9
N
7
U
5
F
1