Mathematics Division, National Center for Theoretical Sciences at Taipei
NCTS/TPE-Math Technical Report 2006-004
Hausdorff and Packing Dimension Results for the Sample Paths
of Self-similar Random Fields
Yimin Xiao ∗
Michigan State University
Narn-Rueih Shieh
National Taiwan University
March 25, 2006
Abstract
Let X = {X(t), t ∈ RN } be a random field with values in Rd . We develop measure
theoretic methods for determining the Hausdorff and packing dimensions of the image
X(E) for any given closed set E ⊂ RN . We show that these results are applicable to
Gaussian random fields, self-similar stable random fields with stationary increments, the
(N, d)-stable sheets and the Rosenblatt process.
Running head: Hausdorff and Packing Dimension Results for Self-similar Random Fields.
2000 AMS Classification numbers: 60G15, 60G17, 60J35, 28A80.
Key words: Self-similar stable random fields, the stable sheets, Gaussian random fields,
the Rosenblatt process, Hausdorff dimension, packing dimension, packing dimension profiles,
images.
1
Introduction
Fractal dimensions such as Hausdorff dimension, box-counting dimension and packing dimension are very useful in characterizing roughness or irregularity of stochastic processes and
random fields. Many authors have studied the Hausdorff dimension and exact Hausdorff measure of the images of Markov processes and Gaussian random fields. We refer to Taylor (1986a)
and Xiao (2004) for extensive surveys on results and techniques for Markov processes, and to
Adler (1981) and Kahane (1985) for results on Gaussian random fields. It is well known [cf.
∗
Research partially supported by NSF grant DMS-0404729.
1
Kahane (1985, Chapter 18)] that if X = {X(t), t ∈ R N } is a fractional Brownian motion in
Rd of index H (0 < H < 1), then for every Borel set E ⊂ R N ,
n
o
1
dimH X(E) = min d,
dimH E
a.s.,
(1.1)
H
where dimH denotes the Hausdorff dimension. There have been various effort to extend (??)
to other non-Markovian processes and/or random fields, but only with partial successes. See,
for example, Kˆono (1986), Lin and Xiao (1994), Benassi, Cohen and Istas (2003). In order to
establish a result similar to (??) for a random field X, it is standard to determine upper and
lower bounds for dimH X(E) separately. While the capacity argument [based on Frostman’s
theorem] is very useful for determining the lower bound, the methods based on the classical
covering argument for obtaining an upper bound for dim H X(E) are quite restrictive and require
to impose strong conditions on X. As such, the aforementioned authors have to assume that
the random fields either have H¨older continuous sample functions or have first moments. Hence
their results are not applicable to stable random fields in general.
The first objective of this paper is to develop a measure theoretic method for studying
upper bound for the Hausdorff dimension of the image set X(E) of a random field X under
minimal conditions. Our method is new and is more powerful than those mentioned above.
The second objective of the present paper is to study the packing dimension of the image
set X(E) by using again the measure theoretic methods. Packing dimension and packing
measure were introduced by Tricot (1982) and Tricot and Taylor (1985) as a dual concept to
Hausdorff dimension and Hausdorff measure. It has been known that the packing dimension
of a set E characterizes different aspects of the geometry of E than the Hausdorff dimension
does; and more information on the fractal structure of E is available when both the Hausdorff
and packing dimensions of E are known. Furthermore, packing dimension may even be needed
for solving problems that only involve Hausdorff dimension, see Xiao (2004, Sections 7 and
12) for further information. Because of these reasons, packing dimension has become a very
useful tool in analyzing fractal sets and sample paths of stochastic processes. See Tricot (1982),
Tricot and Taylor (1985), Saint Raymond and Tricot (1988), Taylor (1986a, 1986b), Perkins
and Taylor (1987), Falconer and Howroyd (1997), Pruitt and Taylor (1996), Talagrand and
Xiao (1996), Xiao (1997, 2004) and the references therein for more information.
Let X = {X(t), t ∈ RN } be a fractional Brownian motion with values in R d of index
H ∈ (0, 1). Talagrand and Xiao (1996) proved that, if E ⊂ R N is a Borel set, then the packing
dimension analogue of (??) may fail and the packing dimension of E alone is not enough to
determine dimP X(E), where dimP denotes packing dimension. Later Xiao (1997) proved that
dimP X(E) =
1
DimHd E
H
a.s.,
(1.2)
where Dims E is the packing dimension profile of E defined by Falconer and Howroyd (1997)
[see §2 for its definition]. The arguments in Talagrand and Xiao (1996) and Xiao (1997) rely on
the modulus of continuity of fractional Brownnian motion, thus can not be applied to processes
with discontinuous sample paths.
Recently, Schilling and Xiao (2005) have extended (??) to a large class of Markov processes
with jumps including stable L´evy processes, certain Feller processes associated to pseudodifferential operators and stable-like processes on fractals. In this paper, we show that their
2
arguments can be modified to establish packing dimension results for non-Markov processes
and random fields. Our results are applicable to Gaussian random fields, self-similar stable
fields with stationary increments, the stable sheets and the Rosenblatt process.
Let X = {X(t), t ∈ RN } be a random field with values in Rd , which will simply be called
an (N, d)-random field. Recall that X = {X(t), t ∈ R N } is said to be H-self-similar if for every
constant c > 0, we have
d X(c t), t ∈ RN = cH X(t), t ∈ RN ;
(1.3)
and X is said to have stationary increments if for every h ∈ R N ,
d
d X(t + h) − X(h), t ∈ RN = X(t) − X(0), t ∈ RN .
(1.4)
In the above, X = Y denotes the two processes have the same finite dimensional distributions.
Fractional Brownian motion is a very important example of self-similar Gaussian random fields
with stationary increments. A systematic account on non-Gaussian self-similar stable processes
can be found in Samorodnitsky and Taqqu (1994).
The rest of this paper is organized as follows. In Section 2, we recall briefly the definitions
and some basic properties of Hausdorff dimension, packing dimension and packing dimension
profiles of sets and Borel measures. Let X = {X(t), t ∈ R N } be an (N, d)-random field and let
µ be a finite Borel measure on RN . In Section 3, we establish upper and lower bounds for the
Hausdorff and packing dimensions of the image measure µ X := µ ◦ X −1 under certain general
conditions. In Section 4, we derive analogous results for the image set X(E), where E ⊂ R N
is a closed set. Finally in Section 5, we show that the general theorems in Sections 3 and 4 are
applicable to several classes of random fields.
Throughout this paper we will use λn to denote the Lebesgue measure on Rn , no matter
the value of n. When n ≥ 2, we use hx, yi to denote the inner product and k · k to denote the
Euclidean norm in Rn . We will use K to denote an unspecified positive constant which may
differ from line to line. Some specific constants in Section i will be denoted by K i,1 , Ki,2 and
so on.
2
Hausdorff dimension, packing dimension and packing dimension profiles
In this section we recall briefly the definitions and some basic properties of Hausdorff dimension,
packing dimension and packing dimension profiles for Borel measures and sets. More detailed
information can be found in Falconer (1990), Mattila (1995) and Edgar (1998).
2.1
Hausdorff dimension of sets and measures
Let Φ be the class of functions φ : (0, δ) → (0, 1) which are right continuous, increasing with
φ(0+) = 0 such that there exists a finite constant K > 0 for which
φ(2s)
≤K
φ(s)
for 0 < s <
3
1
δ.
2
(2.1)
For φ ∈ Φ, the φ-Hausdorff measure of E ⊂ R N is defined by
φ-m(E) = lim inf
ε→0
X
φ(2ri ) : E ⊂
i
∞
[
B(xi , ri ), ri < ε ,
i=1
(2.2)
where B(x, r) denotes the open ball of radius r centered at x. The Hausdorff dimension of E
is defined by
dimH E = inf α > 0 : sα -m(E) = 0 .
For a finite Borel measure in RN , its Hausdofff dimension (or lower Hausdorff dimension
as it is sometimes called) is defined by
dimH µ = inf dimH E : µ(E) > 0 and E ⊂ RN is a Borel set ,
(2.3)
and its upper Hausdorff dimension is defined by
dim∗H µ = inf dimH E : µ(RN \E) = 0 and E ⊂ RN is a Borel set .
(2.4)
We will make use of the following equivalent characterizations for dim H µ and dim∗H µ proved
by Hu and Taylor (1994, Corollary 4.2):
−β
N
dimH µ = sup β > 0 : lim sup r µ B(x, r) = 0 for µ-a.a. x ∈ R
(2.5)
r→0
and
∗
dimH µ = inf β > 0 : lim sup r
−β
µ B(x, r) > 0 for µ-a.a. x ∈
r→0
RN
.
(2.6)
The Hausdorff dimensions of an analytic set E ⊂ R N and finite Borel measures on E are
related by the following identity:
dimH E = sup dimH µ : µ ∈ M+
(2.7)
c (E) ,
where M+
c (E) denotes the family of finite Borel measures with compact support in E. This
can be verified by (??) and Frostman’s lemma.
2.2
Packing dimension and packing dimension profile
For φ ∈ Φ, Taylor and Tricot (1985) defined the set function φ-P (E) on R N by
X
φ(2ri ) : B(xi , ri ) are disjoint, xi ∈ E, ri < ε .
φ-P (E) = lim sup
ε→0
(2.8)
i
The set function φ-P is not an outer measure because it fails to be countably subadditive. But
it gives rise to a metric outer measure φ-p on R N as follows
X
[ φ-P (En ) : E ⊂
En .
(2.9)
φ-p (E) = inf
n
n
4
φ-p(E) is called the φ-packing measure of E. If φ(s) = s α , then sα -p(E) is called the αdimensional packing measure of E. The packing dimension of E is defined by
dimP E = inf α > 0 : sα -p (E) = 0 .
(2.10)
The packing dimension can also be defined through the upper box-counting dimension, which is
more convenient to use in many applications. For any ε > 0 and any bounded set E ⊂ R N , let
N (E, ε) be the smallest number of balls of radius ε needed to cover E. The upper box-counting
dimension of E are defined as
dimB E = lim sup
ε→0
log N (E, ε)
.
− log ε
Tricot (1982) proved that
dimP E = inf
sup dimB En : E ⊂
n
∞
[
n=1
En ,
(2.11)
see also Falconer (1990, p.45). It is well known that 0 ≤ dim H E ≤ dimP E ≤ N for every set
E ⊂ RN .
Similar to (??) and (??), the packing dimension of a Borel measure µ on R N (or lower
packing dimension as it is sometimes called) is defined by
dimP µ = inf{dimP E : µ(E) > 0 and E ⊂ RN is a Borel set},
(2.12)
and the upper packing dimension of µ is defined by
dim∗P µ = inf{dimP E : µ(RN \E) = 0 and E ⊂ RN is a Borel set}.
For a finite Borel measure µ on RN and for any s > 0, let
Z
µ
Fs (x, r) =
min{1, r s ky − xk−s } dµ(y)
(2.13)
(2.14)
RN
be the s-dimensional potential. Falconer and Howroyd (1997) defined the packing dimension
profile and the upper packing dimension profile of µ as
o
n
(2.15)
Dims µ = sup β ≥ 0 : lim inf r −β Fsµ (x, r) = 0 for µ-a.a. x ∈ RN
r→0
and
n
o
Dim∗s µ = inf β > 0 : lim inf r −β Fsµ (x, r) > 0 for µ-a.a. x ∈ RN ,
(2.16)
0 ≤ Dims µ ≤ Dim∗s µ ≤ s
(2.17)
r→0
respectively. From these definitions it is easy to show that
and if s ≥ N , then
Dim∗s µ = dim∗P µ,
Dims µ = dimP µ,
5
(2.18)
see Falconer and Howroyd (1997) for a proof. Note that the identities in (??) give equivalent
characterizations of dimP µ and dim∗P µ in terms of the potential FNµ (x, r).
Similar to (??), it can be easily proved that
dimP E = sup dimP µ : µ ∈ M+
(2.19)
c (E) ,
see Falconer and Howroyd (1997, p.277). Motivated by this, Falconer and Howroyd (1997)
defined the s-dimensional packing dimension profile of E ⊂ R N by
Dims E = sup Dims µ : µ ∈ M+
(2.20)
c (E) .
It follows from (??) and (??) that
0 ≤ Dims E ≤ s
and Dims E = dimP E
if s ≥ N.
(2.21)
By the above definition, one can also verify that for any Borel set E ⊂ R N with dimH E =
dimP E, we have
Dims E = min s, dimP E .
(2.22)
Packing dimension profiles for finite Borel measures and sets carry more information about
local and global geometric properties of the measures and sets than packing dimension does.
By using packing dimension profiles, Falconer and Howroyd (1997) determined the packing
dimension of projections. It turns out they are very useful in studying various random fractals
as well.
In Section 3 we will make use of the following lemma, which is a consequence of Proposition
18 in Falconer and Howroyd (1997).
Lemma 2.1 Let σ : R+ → [0, N ] be any one of Dims µ, Dim∗s µ, Dims E. Then σ(s) is nondecreasing and continuous.
3
Hausdorff and packing dimensions of the image measures
Let X = {X(t), t ∈ RN } be an (N, d)-random field defined on some probability space (Ω, F, P).
We assume throughout this paper that (t, ω) 7→ X(t, ω) is B(R N )×F-measurable, where B(RN )
is the family of Borel subsets of RN .
For a general random field X which is neither continuous nor Markovian, the classical
covering methods for determining the Hausdorff and packing dimensions of X(E) are not
applicable [It should be pointed out that dimH X(E) and dimP X(E) may not be determined
by the uniform H¨older exponent even when X is continuous; see Example ?? below]. Our
approach to these problems is measure theoretic. In this section, we derive upper and lower
bounds for the Hausdorff and packing dimensions of the image measures of X and, in Section
??, we prove analogous results for the image set X(E).
We will make use of the following two conditions:
(C1) There exist positive and finite constants H 1 λ, β, h0 and K3,1 such that λβ > 1 and for
all t ∈ RN and h ∈ (0, h0 ),
λ
H1
≤ K3,1 | log h|−λβ .
(3.1)
P
sup kX(s) − X(t)k ≥ h | log h|
ks−tk≤h
6
(C2) There exist positive constants H 2 and K3,2 such that for all s, t ∈ RN and r > 0,
P kX(s) − X(t)k ≤ ks − tkH2 r ≤ K3,2 min 1, r d .
(3.2)
Remark 3.1 Both Conditions (C1) and (C2) implicitly require the random field X to have
certain approximate scaling property (or self-similarity). When X is H 1 -self-similar, Condition
(C1) is satisfied whenever the tail probability of sup ks−tk≤h kX(s) − X(t)k decays no slower
than a polynomial rate. Hence it is satisfied by many random fields including Gaussian and
stable random fields. See Proposition ?? below and Section ??. On the other hand, Condition
(C2) is easily satisfied if the random vector X(s) − X(t) /ks − tkH2 has a bounded density
function, which is the case for all the random fields considered in this paper.
The following proposition gives a simple sufficient condition for a self-similar process X =
{X(t), t ∈ R} to satisfy Condition (C1). More precise information can be obtained if further
distributional properties of X are known; see Section 5.
Proposition 3.2 Let X = {X(t), t ∈ R} be a separable, H-self-similar process in R d with
stationary increments. If there exist positive constants β > 0 and K 3,3 such that Hβ > 1 and
n
o
P kX(1)k ≥ u ≤ K3,3 u−β ,
∀ u ≥ 1.
(3.3)
Then there exists a positive constant K 3,4 such that for all u ≥ 1,
P sup kX(t)k ≥ u ≤ K3,4 u−β .
(3.4)
t∈[0,1]
In particular, Condition (C1) is satisfied with H 1 = H and any λ satisfying λβ > 1.
Proof Without loss of generality, we can assume d = 1. Since the self-similarity index H > 0,
we have X(0) = 0 a.s.
Let T ∗ = {tn , n ≥ 0} be a separant for X = {X(t), t ∈ [0, 1]}. We assume 0 = t 0 < t1 <
t2 · · · < tn < · · · . For any n ≥ 2, consider the random variables Y k (1 ≤ k ≤ n) defined by
Yk = X(tk ) − X(tk−1 ).
For 1 ≤ i < j ≤ n, let Si,j =
u ≥ 1,
Pj
k=i Yk .
By the properties of X and (??), we have that for any
X
j
n
o
Yk ≥ u = P X(tj ) − X(ti−1 ) ≥ u
P k=i
o
u
(tj − ti−1 )H
Hβ
tj − ti−1
.
n
= P X(1) ≥
≤ K3,3 u−β
(3.5)
Thus, Condition (3.4) of Theorem 3.2 of Moricz, Serfling and Stout (1982) is satisfied with
g(i, j) = tj − ti−1 , α = Hβ and φ(t) = tβ . It is easy to see the function g(i, j) satisfies their
7
condition (1.2) with Q = 1. Therefore, it follows from Theorem 3.2 of Moricz, Serfling and
Stout (1982) that there exists a constant K 3,4 (independent of n) such that for all u ≥ 1,
P
j
X
Yk ≥ u ≤ K3,4 u−β .
max X(tj ) ≥ u = P max 1≤j≤n
1≤j≤n
(3.6)
k=1
Letting n → ∞ yields (??).
Remark 3.3
• Theorem 3.2 of Moricz, Serfling and Stout (1982) requires Hβ > 1 in our context. The
question whether their Theorem 3.2 holds under the condition Hβ = 1 is still open; see
Moricz, Serfling and Stout (1982, p.1039) for details. Even though Proposition ?? holds
for any strictly stable L´evy process X with index α ∈ (0, 2] [in this case we have H = 1/α
and β = α], it is not known whether Proposition ?? holds in general if Hβ = 1.
• If Hβ < 1, generally (??) does not imply (??) as shown by the linear fractional stable
processes with index α ∈ (0, 1). See Maejima (1983) or Samorodnitsky and Taqqu (1994).
For any Borel measure µ on RN , the image measure µX of µ under the mapping t 7→ X(t)
is defined by
µX (B) := µ t ∈ RN : X(t) ∈ B
for all Borel sets B ⊂ Rd .
3.1
Upper bounds
First we consider the upper bound for the Hausdorff dimension of the image measure µ X .
Proposition 3.4 Let X = {X(t), t ∈ RN } be a random field with values in Rd . If Condition
(C1) is satisfied, then for any finite Borel measures µ on R N ,
n 1
o
dimH µX ≤ min d,
dimH µ ,
a.s.
(3.7)
H1
and
o
n 1
dim∗H µ ,
dim∗H µX ≤ min d,
H1
a.s.
(3.8)
Proof First note that for any fixed s ∈ R N and the sequence hn = 2−n (n ≥ 1), it follows
from (??) that for every integer n ≥ log(1/h 0 ),
−H1 n
n λ
X(t) − X(s) ≥ 2
(log 2 )
P
sup
≤ K n−βλ .
(3.9)
kt−sk≤2−n
Since
P∞
n=1
n−βλ < ∞, the Borel-Cantelli lemma implies that almost surely,
sup X(t) − X(s) ≤ 2−H1 n nλ ,
∀n ≥ n0 ,
kt−sk≤2−n
8
(3.10)
where n0 = n0 (ω, s) depends on ω and s. By Fubini’s theorem, we derive that, for any finite
Borel measure µ on RN , almost surely (??) holds for µ-a.a. s ∈ R N .
Now we fix an ω ∈ Ω such that (??) is valid µ-a.a. s ∈ R N and prove that (??) holds.
Since dimH µX ≤ d holds trivially, we only need to prove dim H µX ≤ H11 dimH µ. Without loss
of generality, we assume dimH µX > 0 and take any γ ∈ (0, dimH µX ). Then by (??) we have
Z
−γ
lim sup r
1l{ky−xk≤r} dµX (y) = 0
(3.11)
r→0
Rd
for µX -a.a. x ∈ Rd . Here and in the sequel, 1lA denotes the indicator function (or random
variable) of the set (or event) A. Equivalent to (??), we have
Z
−γ
lim sup r
1l{kX(t)−X(s)k≤r} dµ(t) = 0
(3.12)
r→0
RN
for µ-a.a. s ∈ RN . Let us fix s ∈ RN such that both (??) and (??) hold. For any ε > 0, we
choose n1 ≥ n0 such that nλ ≤ 2εn for all n ≥ n1 . By (??) we can write
Z
1l{kX(t)−X(s)k≤r} dµ(t)
RN
≥
≥
∞ Z
X
n=n1 2
∞ Z
X
n=n1
=
Z
−n−1 ≤kt−sk<2−n
1l{kX(t)−X(s)k≤r} dµ(t)
(3.13)
2−n−1 ≤kt−sk<2−n
kt−sk<2−n1
1l{kt−sk≤r1/(H1 −ε) } dµ(t)
1l{kt−sk≤r1/(H1 −ε) } dµ(t).
Hence, we have
Z
RN
1l{kt−sk≤r1/(H1 −ε) } dµ(t) ≤
Z
+
Z
RN
1l{kX(t)−X(s)k≤r} dµ(t)
kt−sk≥2−n1
Note that
lim r −γ
r→0
Z
kt−sk≥2−n1
(3.14)
1l{kt−sk≤r1/(H1 −ε) } dµ(t).
1l{kt−sk≤r1/(H1 −ε) } dµ(t) = 0
(3.15)
because the indicator function takes value 0 when r > 0 is sufficiently small. It follows from
(??), (??) and (??) that with r = ρH1 −ε ,
Z
lim sup ρ−(H1 −ε)γ
1l{kt−sk≤ρ} dµ(t)
ρ→0
RN
Z
−γ
(3.16)
= lim sup r
1l{kt−sk≤r1/(H1 −ε) } dµ(t)
r→0
RN
Z
≤ lim sup r −γ
1l{kX(t)−X(s)k≤r} dµ(t) = 0.
r→0
RN
9
Thus we have proven (??) holds almost surely for µ-a.a. s ∈ R N . This implies dimH µ ≥
(H1 − ε)γ a.s. Since ε > 0 and γ < dimH µX are arbitrary, (??) follows.
In order to prove (??), it is sufficient to show dim ∗H µX ≤ H11 dim∗H µ a.s. Let ω ∈ Ω be fixed
as above. We take an arbitrary β > dim∗H µ. By (??) we have
Z
−β
1l{kt−sk≤ρ} dµ(t) > 0 for µ-a.a. s ∈ RN .
(3.17)
lim sup ρ
ρ→0
RN
By using (??), we derive that for x = X(s)
Z
Z
1l{ky−xk≤r} dµX (y) ≥
1l{kt−sk≤r1/(H1 −ε) } dµ(t)
Rd
RZN
−
1l{kt−sk≤r1/(H1 −ε) } dµ(t).
(3.18)
kt−sk≥2−n1
It follows from (??), (??) and (??) that
lim sup r −β/(H1 −ε)
Z
1l{ky−xk≤r} dµX (y)
Z
1l{kt−sk≤r1/(H1 −ε) } dµ(t) > 0
Rd
r→0
≥ lim sup r
−β/(H1 −ε)
r→0
(3.19)
RN
for all s ∈ RN satisfying (??). This implies dim∗H µX ≤ β/(H1 − ε) a.s. Letting ε ↓ 0 and
β ↓ dim∗H µ yields (??). This finishes the proof of Proposition ??.
Now we turn to the packing dimensions of µ X . The following result is proved by Schilling
and Xiao (2005).
Proposition 3.5 Let X = {X(t), t ∈ RN } be a random field with values in Rd . If Condition
(C1) is satisfied, then for any finite Borel measures µ on R N ,
dimP µX ≤
1
DimH1 d µ,
H1
a.s.
(3.20)
dim∗P µX ≤
1
Dim∗H d µ,
1
H1
a.s.
(3.21)
and
3.2
Lower bounds
As we will see in next section, it is quite standard to determine the lower bound for dim H X(E)
using Condition (C2) and a capacity argument based on Frostman’s theorem. However, it is
more difficult to obtain a sharp lower bound for dim P X(E) directly. To overcome this difficulty,
we first consider the lower bounds for the packing dimensions of the image measures under an
(N, d)-random field.
Proposition 3.6 Let X = {X(t), t ∈ RN } be an (N, d)-random field satisfying condition (C2).
Then for every finite Borel measure µ on R N ,
dimP µX ≥
1
DimH2 d µ,
H2
10
a.s.
(3.22)
and
dim∗P µX ≥
1
Dim∗H d µ,
2
H2
a.s.
(3.23)
Proof We only prove (??); the proof of (??) is similar and is omitted.
We will assume DimH2 d µ > 0. Otherwise, there is nothing to prove. Note that, for any
fixed s ∈ RN , Fubini’s theorem imples
Z
µ
EFd X X(s), r = E
min 1, r d kv − X(s)k−d dµX (v)
d
Z R
(3.24)
d
−d
=
E min 1, r kX(t) − X(s)k
dµ(t).
RN
The last integrand in (??) can be written as
E min 1, r d kX(t) − X(s)k−d
= P kX(t) − X(s)k ≤ r + E r d kX(t) − X(s)k−d · 1l{kX(t)−X(s)k≥r} .
By Condition (C2), we obtain that for all s, t ∈ R N and r > 0,
rd
.
P kX(t) − X(s)k ≤ r ≤ K3,2 min 1,
kt − skH2 d
(3.25)
(3.26)
Denote the distribution of X(t) − X(s) by Γ s,t (·). Let ν be the image measure of Γs,t (·) under
the mapping T : z 7→ kzk from Rd to R+ . Then the second term in (??) can be written as
Z
Z ∞ d
rd
r
1
l
Γ
(dz)
=
ν(dρ)
d {kzk≥r} s,t
d
kzk
ρ
d
R
r
(3.27)
Z ∞ d
r
P kX(t) − X(s)k ≤ ρ dρ,
≤d
ρd+1
r
where the last inequality follows from an integration-by-parts formula. Hence, by (??) and
(??) we derive that the second term in (??) can be bounded by
Z ∞
d 1
ρ
d
Kr
min 1,
dρ
ρd+1
kt − skH2
r
(
(3.28)
1
if r ≥ kt − skH2 ,
d
H2
≤K
r
log kt−sk
if r < kt − skH2 .
r
kt−skH2
It follows from (??), (??), (??) and (??) that for any 0 < ε < 1 and s, t ∈ R N ,
r d−ε
E min{1, r d kX(t) − X(s)k−d } ≤ K3,5 min 1,
.
kt − skH2 (d−ε)
(3.29)
For any γ ∈ (0, DimH2 d µ), by Lemma ??, there exists ε > 0 such that γ < Dim H2 (d−ε) µ. It
follows from (??) that
Z
r d−ε
−γ/H2
min 1,
lim inf r
dµ(t) = 0 for µ-a.a. s ∈ RN .
(3.30)
r→0
|t − s|H2 (d−ε)
R+
11
By (??), (??), (??) and Fatou’s lemma we have that for µ-a.a. s ∈ R N
−γ/H2 µX
E lim inf r
Fd X(s), r
r→0
Z
r d−ε
−γ/H2
dµ(t) = 0.
≤ K3,5 lim inf r
min 1,
r→0
|t − s|H2 (d−ε)
RN
(3.31)
By using Fubini’s theorem again, we see that almost surely,
µ
lim inf r −γ/H2 Fd X X(s), r = 0 for µ-a.a. s ∈ RN .
r→0
Hence dimP µX ≥
γ
H2
a.s. Since γ can be arbitrarily close to Dim H2 d µ, we have
dimP µX ≥
1
DimH2 d µ,
H2
a.s.
(3.32)
This proves (??).
Combining Propositions ?? and ??, we have the following result on the packing dimensions
of the image measures.
Theorem 3.7 Let X = {X(t), t ∈ RN } be an (N, d)-random field and let H be a positive
constant. If for every ε > 0, X satisfies conditions (C1) and (C2) with H 1 = H − ε and
H2 = H + ε, respectively. Then for every finite Borel measure µ on R N ,
Proof
dimP µX =
1
DimHd µ,
H
a.s.
(3.33)
dim∗P µX =
1
Dim∗Hd µ,
H
a.s.
(3.34)
For any ε > 0, Propositions ?? and ?? imply that
1
1
Dim(H+ε)d µ ≤ dimP µX ≤
Dim(H−ε)d µ
H +ε
H −ε
a.s.
Letting ε → 0 along rational numbers and using Lemma ??, we derive (??). The proof of (??)
is similar.
4
Hausdorff and packing dimensions of the image sets
In order to determine the upper bounds for the Hausdorff and packing dimensions of the image
set X(E), we will need some lemmas. Lemma ?? is a consequence of Theorem 2.1.11 in Edgar
(1998, p.73), which is more general than Theorem 1.20 in Mattila (1995) that was used in Xiao
(1997).
Lemma 4.1 Let E ⊂ RN be a closed set and let f : RN → Rd be a Borel function. If ν is a
finite Borel measure on Rd with support in f (E), then there exists a Borel measure µ on R N
such that ν = µf and the support of µ is contained in E.
12
Lemma 4.2 Let E ⊂ RN be a closed set. Then for any Borel measurable function f : R N →
Rd ,
dimH f (E) = sup dimH µf : µ ∈ M+
(4.1)
c (E)
and
dimP f (E) = sup dimP µf : µ ∈ M+
c (E) .
(4.2)
Proof Denote the right hand side of (??) by γ E . It follows from (??) that dimH f (E) ≥ γE .
+
On the other hand, for any ν ∈ M+
c (f (E)), Lemma ?? implies that there exists an µ ∈ M c (E)
such that ν = µf . This and (??) imply dimH f (E) ≤ γE . Hence we have proved (??).
The proof of (??) is similar [when N = 1 it is also proved in Schilling and Xiao (2005)]. Remark 4.3 If we assume f is continuous, then (??) and (??) hold for all analytic sets
E ⊂ RN , as shown by Xiao (1997).
The following proposition gives upper bounds for the Hausdorff and packing dimension of
the image sets.
Proposition 4.4 Let X = {X(t), t ∈ RN } be an (N, d)-random field satisfying Condition
(C1). Then for all closed sets E ⊂ RN ,
1
dimH E ,
a.s.
(4.3)
dimH X(E) ≤ min d,
H1
and
dimP X(E) ≤
1
DimH1 d E,
H1
a.s.
(4.4)
Proof It is clear that (??) follows from (??) and Proposition ??. On the other hand, (??)
follows from (??) and (??).
Now we consider the lower bounds for the Hausdorff and packing dimensions of X(E).
Proposition 4.5 Let X = {X(t), t ∈ RN } be an (N, d)-random field satisfying condition (C2).
Then for every closed set E ⊂ RN ,
1
dimH X(E) ≥ min d,
dimH E ,
a.s.
(4.5)
H2
and
dimP X(E) ≥
1
DimH2 d E,
H2
a.s.
(4.6)
Proof First we prove (??). For any 0 < γ < min d, H12 dimH E , we have γH2 < dimH E.
Hence there exists a Borel probability measure on E such that
Z Z
1
µ(ds)µ(dt) < ∞.
(4.7)
H2 γ
E E ks − tk
13
Let µX be the image measure of µ under X. This is a Borel probability measure on X(E). In
order to prove dimH X(E) ≥ γ a.s., by Frostman’s theorem [cf. Kahane (1985)], it is sufficient
to show
Z Z
1
µ (dx)µX (dy) < ∞
a.s.
(4.8)
γ X
Rd Rd kx − yk
By Fubini’s theorem, we have
Z Z Z Z
1
1
µ(ds)µ(dt).
µ (dx)µX (dy) =
E
E
γ X
kX(s) − X(t)kγ
E E
Rd Rd kx − yk
It follows from Condition (C2) that
Z ∞ n
o
1
−1/γ
P
kX(s)
−
X(t)k
≤
ρ
E
=
dρ
kX(s) − X(t)kγ
0
Z ∞
1
dρ
≤ K3,2
min 1, 1/γ
(ρ kt − skH2 )d
0
(4.9)
(4.10)
≤ K4,1 ks − tk−H2 γ ,
where in deriving the last inequality, we have used the fact that γ < d. Putting (??) into
(??), we see that (??) follows from (??). Hence we have proven dim H X(E) ≥ γ almost surely,
which, in turn, implies (??).
Next we prove (??). Note that for any γ < H12 DimH2 d E, by (??) there exists a Borel
measure µ ∈ M+
c (E) such that H2 γ < DimH2 d µ. It follows from (??) that dim P µX > γ a.s.
Hence by Lemma ?? we have dimP X(E) > γ a.s. Since γ < H12 DimH2 d E is arbitrary, we
obtain dimP X(E) ≥ H12 DimH2 d E a.s. The proof is completed.
The following is the main result of this section.
Theorem 4.6 Let X = {X(t), t ∈ RN } be an (N, d)-random field and let H be a positive
constant. If for every ε > 0, X satisfies conditions (C1) and (C2) with H 1 = H − ε and
H2 = H + ε, respectively. Then for every closed set E ⊂ R N ,
1
a.s.
(4.11)
dimH X(E) = min d, dimH E ,
H
and
dimP X(E) =
Proof
1
DimHd E,
H
a.s.
Clearly (??) and (??) follow from Propositions ??, ?? and Lemma ??.
(4.12)
Remark 4.7 By using Remark ??, one can prove the following somewhat stronger result: if
an (N, d)-random field X satisfies the assumptions of Theorem ?? and has continuous sample
paths, then for every analytic set E ⊂ R N , the formulaes (??) and (??) hold almost surely.
It is often desirable to compute dimP X(E) in terms of dimP E. The following is the packing
dimension analogue of (??). Note that when N > Hd, then (??) does not hold in general; see
Talagrand and Xiao (1996). In this sense, it is the best possible result of this kind.
14
Corollary 4.8 Let X = {X(t), t ∈ RN } be an (N, d)-random field satisfying the conditions of
Theorem ?? and let E ⊂ RN be a closed set. If either N ≤ Hd or E satisfies dimH E = dimP E,
then
1
dimP E ,
a.s.
(4.13)
dimP X(E) = min d,
H
Proof If N ≤ Hd, then (??) implies that for every Borel set E ⊂ R N , DimHd E = dimP E.
Hence Theorem ?? yields dimP X(E) = H1 dimP E a.s. On the other hand, if a Borel set E ⊂ R N
satisfies dimH E = dimP E, then (??) implies that DimHd E = min{Hd, dimP E}. Hence (??)
follows again from (??).
Remark 4.9 Note that in both (??) and (??) the exceptional null probability events depend
on E. It would be interesting to investigate whether or when there is a single exceptional
null probability event Ω0 such that, for all ω ∈
/ Ω0 , (??) and (??) hold simutaneously for all
closed (or Borel) sets E ⊂ RN . Such a “uniform dimension result” is applicable even when
E is random, hence it is useful in studying intersections and collisions of stochastic processes.
Uniform dimension results have been established for Brownian motion, stable L´evy processes
and fractional Brownian motion; see the survey papers of Pruitt (1975), Taylor (1986a) and
Xiao (2004) for further information.
5
Applications
The general results in Sections 3 and 4 can be applied to various classes of random fields. We
discuss some of the applications in this section.
5.1
Gaussian random fields
First we consider a Gaussian random field X = {X(t), t ∈ R N } in Rd defined by
X(t) = X1 (t), . . . , Xd (t) , ∀t ∈ RN ,
(5.1)
where X1 , . . . , Xd are independent copies of a real-valued, centered Gaussian random field
Y = {Y (t), t ∈ RN }, which satisfies the following condition: There exist positive constants
δ0 , K5,1 , K5,2 and a non-decreasing, right continuous function φ : [0, δ 0 ) → [0, ∞) such that for
all t ∈ RN and h ∈ RN with |h| ≤ δ0 ,
2 K5,1 φ(|h|) ≤ E Y (t + h) − Y (t)
≤ K5,2 φ(|h|).
(5.2)
Define the upper index of φ at 0 by
o
n
φ(r)
α∗ = inf β ≥ 0 : lim 2β = ∞
r→0 r
(5.3)
o
n
φ(r)
α∗ = sup β ≥ 0 : lim 2β = 0 .
r→0 r
(5.4)
with the convention inf ∅ = ∞. Analogously, we define the lower index of φ at 0 by
15
Clearly, 0 ≤ α∗ ≤ α∗ ≤ ∞. Xiao (2005) showed that it is possible to have α ∗ = ∞. However,
if Y is Gaussian random fields with stationary increments and its spectral measure has an
absolutely continuous part, then α ∗ ≤ 1; see Lemma 3.6 in Xiao (2005). A typical example
of such Gaussian random fields is the (N, d)-fractional Brownian motion of index H. In this
case, α∗ = α∗ = H. Many more examples can be found in Xiao (2005).
While it is well known that for every analytic set E ⊂ R N ,
1
1
dimH E
a.s.
(5.5)
min d, ∗ dimH E ≤ dimH X(E) ≤ min d,
α
α∗
the following packing dimension result is new, which extends Theorem 4.1 of Xiao (1997).
Theorem 5.1 Let X = {X(t), t ∈ RN } be an (N, d)-Gaussian random field defined by (??)
with Y satisfying (??). We assume that 0 < α ∗ ≤ α∗ ≤ 1. Then for every analytic set E ⊂ RN ,
1
1
Dimα∗ d E ≤ dimP X(E) ≤
Dimα∗ d E
∗
α
α∗
a.s.
(5.6)
In particular, if α∗ = α∗ = H ∈ (0, 1), then for every analytic set E ⊂ R N ,
dimP X(E) =
1
DimHd E
H
a.s.
(5.7)
Proof To verify that X satisfies Condition (C1), we make use of the following result, which
can be proved by using standard Gaussian principles [see, e.g., Xiao (2005, Lemma 3.10)]: If
(??) holds, then X has continuous sample paths and there exist positive constants K 5,3 and
K5,4 such that for all t ∈ RN , r > 0 small enough and u ≥ K5,3 φ1/2 (r), we have
P
sup
kX(s) − X(t)k ≥ u
ks−tk≤r
u2
.
≤ exp −
K5,4 φ(r)
(5.8)
It follows from (??) and (??) that, for any ε > 0, X satisfies Conditions (C1) with H 1 = α∗ −ε.
On the other hand, for every ε > 0, condition (C2) holds with H 2 = α∗ + ε. Therefore, (??)
follows from Propositions ??, ?? and Remark ??.
5.2
Self-similar stable random fields
If X = {X(t), t ∈ R+ } is a strictly stable L´evy process in R d , the Hausdorff dimensions of
its image sets have been well studied. See Taylor (1986a) and Xiao (2004). The packing
dimension results similar to those in Sections 3 and 4 have also been obtained by Schilling and
Xiao (2005). Hence in this subsection, we will concentrate on non-Markov stable processes and
stable random fields.
Let X0 = {X0 (t), t ∈ RN } be an α-stable random field in R with the representation
Z
f (t, x) M (dx),
(5.9)
X0 (t) =
F
16
where M is an α-stable random measure on a measurable space (F, F) with control measure
m and skewness intensity β, f (t, ·) : F → R (t ∈ R N ) is a family of functions on F satisfying
Z
|f (t, x)|α m(dx) < ∞,
∀ t ∈ RN
(5.10)
F
and also the condition
Z f (t, x)β(x) ln |f (t, x)| m(dx) < ∞,
∀ t ∈ RN
(5.11)
F
if α = 1. For any integer n ≥ 1 and t1 , . . . , tn ∈ RN , the characteristic function of the joint
distribution of X0 (t1 ), . . . , X0 (tn ) is given by
X
X
n
n
α ξj f (tj , ·)
,
ξj X0 (tj ) = exp − E exp i
j=1
j=1
α,m
where ξj ∈ R (1 ≤ j ≤ n) and k · kα,m is the Lα (F, F, m) norm.
The class of α-stable random fields with representation (??) is very broad. In particular,
if a random field X0 = {X0 (t), t ∈ RN } is α-stable with α 6= 1 or symmetric α-stable, and
is separable in probability [that is, there is a countable subset T 0 ⊂ RN such that for every
t ∈ RN , there exists a sequence {tk } ⊂ T0 such that X0 (tk ) → X0 (t) in probability], then
X0 has a representation (??); see Theorems 13.2.1 and 13.2.2. in Samorodnitsky and Taqqu
(1994) for details.
For a separable α-stable random field in R given
by (??), Rosinski and Samorodnitsky
(1993) investigated the asymptotic behavior of P supt∈[0,1]N |X0 (t)| ≥ u as u → ∞ [see also
Theorem 10.5.1 in Samorodnitsky and Taqqu (1994)]. The following tail probability estimate
is a consequence of their result.
Lemma 5.2 Let X0 = {X0 (t), t ∈ RN } be a separable α-stable random field in R given in the
form (??). Assume that X0 has a.s. bounded sample paths. Then there exists a positive and
finite constant K5,5 , depending on α, f and m only, such that for all u > 0,
lim uα P
sup |X0 (t)| ≥ u = K5,5 .
(5.12)
u→∞
t∈[0,1]N
Remark 5.3 In the above lemma, it is crucial to assume X 0 has bounded sample paths almost
surely. Otherwise (??) may not hold as shown by the linear fractional stable process X 0 with
0 < α < 1 [see Example ?? below]. For an α-stable random field X 0 with representation (??),
some necessary conditions and sufficient conditions for X 0 to have a.s. bounded sample paths
can be found in Nolan (1989), Samorodnitsky and Taqqu (1994, Chapters 10 and 12), Marcus
and Rosinski (2005).
We define an α-stable random field X = {X(t), t ∈ R N } in Rd by
X(t) = X1 (t), . . . , Xd (t) ,
(5.13)
17
where X1 , . . . , Xd are independent copies of X0 . Of course, this definition is more restrictive
than that of a stable process in Rd . This is not essential and we have chosen this setting for
the sake of simpler presentation of the results.
The following theorem gives the Hausdorff and packing dimensions of the images of selfsimilar stable random fields.
Theorem 5.4 Let X = {X(t), t ∈ RN } be a separable α-stable random field in R d defined by
(??), where X0 is given in the form (??). Suppose that X 0 is H-self-similar, has stationary
increments, and its sample path is a.s. bounded. Then for every closed set E ⊂ R N ,
1
dimH X(E) = min d, dimH E ,
a.s.
(5.14)
H
and
dimP X(E) =
1
DimHd E,
H
a.s.
(5.15)
Proof It follows from the self-similarity and Lemma ?? that Condition (C1) with H 1 = H
is satisfied. On the other hand, Condition (C2) with H 2 = H is satisfied because X is H-selfsimilar, has stationary increments and the α-stable variable X(1) has a bounded continuous
density function. Therefore, both (??) and (??) follow from Theorem ??.
Remark 5.5 Note that, similar to the Gaussian case, it is the self-similarity index H that
plays an explicit rˆole in Theorem ??, not the stability index α. Of course, for any H-selfsimilar, α-stable processes with stationary increments, the two indices α and H are intimately
related; see Corollary 7.1.11 in Samorodnitsky and Taqqu (1994).
In the following we show that Theorem ?? can be easily applied to several important types
self-similar stable processes.
Example 5.6 [linear fractional stable processes] Let 0 < α < 2 and H ∈ (0, 1) be given
constants. we define an α-stable process X 0 = {X0 (t), t ∈ R+ } with values in R by
Z
X0 (t) =
hH (t, s) Mα (ds),
(5.16)
R
where Mα is a strictly α-stable random measure on R with Lebesgue measure as its control
measure and where
n
o
H−1/α
H−1/α
hH (t, s) = a (t − s)+
− (−s)+
n
o
(5.17)
H−1/α
H−1/α
− (−s)−
+ b (t − s)−
.
In the above, a, b ∈ R are constants with |a| + |b| 6= 0, t + = max{t, 0} and t− = max{−t, 0}.
Then the α-stable process X0 is H-self-similar with stationary increments, which is called an
(α, H)-linear fractional stable process. If H = α1 , then the integral in (??) is understood as
aM ([0, t]) for t ≥ 0 and as bM ([t, 0]) if t < 0. Hence X 0 is a strictly α-stable L´evy process.
Maejima (1983) proved that if αH < 1, then X 0 is a.s. unbounded on any interval of
positive length. On the other hand, if αH > 1 [i.e., 1 < α < 2 and 1/α < H < 1], then
18
the Kolmogorov’s continuity theorem implies that X 0 is a.s. continuous. See also Example
12.2.3 in Samorodnitsky and Taqqu (1994) for a proof using the metric entropy. In the latter
case, Takashima (1989) further studied the local and uniform H¨older continuity of the linear
fractional stable processes. His Theorems 3.1 and 3.4 showed that the local H¨older exponent
of X equals H. However, the exponent of the uniform H¨older continuity can not be bigger
than H − α1 [cf. Takashima (1989, Theorem 3.4)].
Now let X = {X(t), t ∈ R+ } be the (α, H)-linear fractional stable process with values in
d
R defined by (??). When αH > 1, the Hausdorff dimension of X(E) has been determined
by Lin and Xiao (1994) using different methods. It follows from Theorem ?? that, if αH > 1,
then for every analytic set E ⊂ R+ ,
dimP X(E) =
1
DimHd E,
H
a.s.
(5.18)
Note that, similar to the Hausdorff dimension results of Lin and Xiao (1994), the packing
dimension of X(E) does not depend on the uniform H¨older exponent of X.
There are several ways to define linear fractional α-stable fields, see Kokoszka and Taqqu
(1993). For example, for H ∈ (0, 1) and α ∈ (0, 2), define
Z N
N
∀ t ∈ RN ,
(5.19)
Z H (t) =
kt − skH− α − kskH− α Mα (ds),
RN
where Mα is an SαS random measure on RN with the N -dimensional Lebesgue measure as its
control measure. This is the stable analogue of the N -parameter fractional Brownian motion.
However, it follows from Theorem 10.2.3 in Samorodnitsky and Taqqu (1994) that, whenever
N ≥ 2, the sample paths of Z H is a.s. unbounded on any intervals in R N . Thus the results
of this paper do not apply to Z H when N ≥ 2. In general, little has been known about the
sample path properties of Z H .
Example 5.7 [harmonizable stable processes] For any given constants 0 < α < 2 and H ∈
eH = {ZeH (t), t ∈ RN } with values in R is
(0, 1), the harmonizable fractional stable field Z
defined by
Z
eiht,λi − 1 f
H
e
Mα (dλ),
(5.20)
Z (t) = Re
N
RN |λ|H+ α
fα is a complex-valued SαS random measure on R N with the N -dimensional Lebesgue
where M
eH is H-selfmeasure as its control measure. It is easy to see that the α-stable random field Z
similar with stationary increments.
It follows from Theorems 10.4.2 in Samorodnitsky and Taqqu (1994) [which covers the case
eH has continuous
0 < α < 1] and Theorem 3 of Nolan (1989) [which is for 1 ≤ α < 2] that Z
sample paths almost surely. When N = 1, Kˆono and Maejima (1991) proved the following
eH : for any ε > 0,
H¨older continuity of Z
lim
h→0
sup
s, t ∈ [0, 1]
|s − t| ≤ h
|ZeH (t) − ZeH (s)|
=0
1 1
|t − s|H log |t − s| 2 + α +ε
a.s.
(5.21)
eH is different from that of the linear fractional stable
Note that the H¨older continuity of Z
processes and other self-similar stable processes with stationary increments [see Theorem 2 of
19
Kˆono and Maejima (1991)]. These results suggest that the harmonizable stable processes ZeH
behaves much nicer than the linear fractional stable processes.
Applying Theorem ?? to the harmonizable stable process in R d defined as in (??), still
denoted by ZeH , we have that for all 0 < α < 2, H ∈ (0, 1) and for every analytic set E ⊂ R N ,
we have
1
H
e
dimH Z (E) = min d, dimH E ,
a.s.
(5.22)
H
and
1
DimHd E,
a.s.
(5.23)
dimP ZeH (E) =
H
Example 5.8 [The Telecom process] Let α ∈ (1, 2) and κ ∈ (0, 1 − α1 ). The Telecom process
Zα = {Zα (t), t ∈ R+ } is an α-stable process with the following representation:
Z Z 1
−(1−κ)− α
Zα (t) =
(t ∧ v − u)+ − (0 ∧ v − u)+ (v − u)+
Mα (du, dv)
R R
(5.24)
Z Z Z t
1
−(1−κ)− α
1l{u < s < v}ds (v − u)+
=
Mα (du, dv),
R
R
0
where Mα is a symmetric α-stable random measure on R 2 with the Lebesgue measure as its
control measure. Pipiras and Taqqu (2000) have shown that this process arises naturally in
modelling Ethernet traffic. They have also shown that Z α is H-self-similar with stationary
increments, where H = α1 + κ, and it is different from the linear fractional stable motion.
For further information on Telecom processes, see Levy and Taqqu (2000), Pipiras and Taqqu
(2002).
Since αH > 1, we see that the sample paths of Z α are a.s. continuous. Hence Theorem ??
applies to the d-dimensional Telecom process, still denoted by Z α , defined by (??).
Example 5.9 [Self-similar fields of L´evy–Chentsov type] For N ≥ 2, we write R N = SN × R+
in polar coordinates, and let M be a symmetric α-stable random measure on R N with the
Lebesgue measure dx = dsdr, (s, r) ∈ S N × R+ as its control measure. For t ∈ RN let
V (t) = {(s, r) : 0 < r < hs, ti}. The α-stable field Y = {Y (t), t ∈ R N } of L´evy–Chentsov type
is defined by
Y (t) = M (V (t)), t ∈ RN .
See Samorodnitsky and Taqqu (1994, §8.3) for more information. In the case α = 2, Y is L´evy’s
multiparameter Brownian motion. It can be shown that Y is self-similar with index H = 1/α
and has stationary increments “in the strong sense”; see Theorem 8.3.2 in Samorodnitsky and
Taqqu (1994).
Shieh (1996) proved that Y has a version, still denoted by Y , such that the sample paths
Y (·) are locally bounded and that each projection process r → Y ((e, r)), e ∈ S N , is a oneparameter SαS L´evy process.
We now define an (N, d)-stable field X as in (??), where the components X 1 , . . . , Xd are
independent copies of Y . Then it is readily verified that the conditions (C1) and (C2) hold
with H1 = H2 = 1/α. Therefore, for every closed set E ⊂ R N ,
dimH X(E) = min d, α dimH E ,
a.s.
(5.25)
20
and
dimP X(E) = α Dimd/α E,
a.s.
(5.26)
Example 5.10
[The stable sheet] Recall from Ehm (1981) that an (N, d)-random field X =
X(t), t ∈ RN
+ is called a stable sheet with index α if X has the following properties:
(a) X has independent increments in the sense that, for every integer n ≥ 2 and disjoint
intervals Qj , j = 1, · · · , n in RN
+ , the increments X(Qj ), j = 1, 2, . . . , n are independent;
d
(b) For every interval Q ⊂ RN
+ and ξ ∈ R ,
E exp ihξ, X(Q)i = exp − λN (Q)ϕ(ξ) ,
(5.27)
where the characteristic exponent ϕ(ξ) satisfies ϕ(0) = 0 and
Z
α
gα hξ, yikξk −1 µ(dy).
ϕ(ξ) = −σkξk
Sd
In the above, σ > 0 is a constant,
πα gα (x) = |x|α 1 − isgn(x) tan
,
2
2i
g1 (x) = |x| + x log |x|.
π
if α 6= 1,
(5.28)
and µ is a probability measure on the unit sphere S d = {y ∈ Rd : kyk = 1} with
µ{y ∈ Sd : hθ, yi = 0} < 1 for all θ ∈ Sd .
The stable sheet X is called isotropic (in space) if its characteristic exponent is of the form
ϕ(ξ) = K kξkα , where K > 0 is a constant. The case α = 2 is just the Brownian sheet and
this is the only case that X has continuous sample paths.
It follows from (b) that the stable sheet X has the following self-similarity: for any
(c1 , · · · , cN ) ∈ (0, ∞)N ,
X(c1 t1 , . . . , cN tN ) : t ∈ RN
+
d
=
Y
N
j=1
1/α
cj
X(t) : t ∈ RN
+ .
Hence X is a special operator-self-similar stable field. Note that, unlike the other stable
processes considered in this section, the stable sheet does not have stationary increment in
the sense of (??). However, by using Properties (a) and (b) above one can show that the
increments of X over intervals are stationary; see Ehm (1981) for details.
Even though Theorem ?? can not be applied directly to the stable sheets, Theorems ??
and ?? [more precisely, their proofs] can. Thus we have the following corollary which extends
the result of Ehm (1981) on dimH X([0, 1]N ) as well as the packing dimension result for the
Brownian sheet in Xiao (1997).
21
Corollary 5.11 Let X = {X(t), t ∈ RN
+ } be an (N, d)-stable sheet with index α. Then for
every closed set E ⊂ (0, ∞)N ,
dimH X(E) = min d, α dimH E
a.s.
(5.29)
and
dimP X(E) = α Dimd/α E
Proof
a.s.
(5.30)
By Lemma 2.2 of Ehm (1981), we have for any interval [a, b] ⊂ R N
+ and u > 0,
P sup kX(t) − X(a)k ≥ u ≤ K5,6 u−α ,
(5.31)
t∈[a,b]
where K5,6 is a positive and finite constant depending on N and α only. Hence the stable sheet
X satisfies Condition (C1) with H1 = 1/α. On the other hand, for any constants 0 < ε < T
and s, t ∈ [ε, T ]N , the random variable X(s) − X(t) is α-stable with characteristic function
E eihξ,X(t)−X(s)i = exp − λN [0, s]∆[0, t] ϕ(ξ) ,
where [0, s]∆[0, t] denotes the symmetric difference of the intervals [0, s] and [0, t]. Since there
exist positive and finite constants K 5,7 and K5,8 , depending on ε, T and N only, such that
K5,7 ks − tk ≤ λN ([0, s]∆[0, t]) ≤ K5,8 ks − tk.
By using the Fourier inversion formula, we derive
n
o
P kX(s) − X(t)k ≤ ks − tk1/α r ≤ K5,9 min 1, r d ,
(5.32)
(5.33)
where the constant K5,9 depends on K5,7 , α and d only. Hence X satisfies Condition (C2)
with H2 = 1/α for all s, t ∈ [ε, T ]N . It follows from Theorem ?? that for every closed set
E ⊂ [ε, T ]N , (??) and (??) hold almost surely. Since ε < T are arbitrary, the σ-stability of
dimH and dimP yields the desired conclusion.
Example 5.12 [The additive stable processes] An N -parameter, d-dimensional random field
X = {X(t), t ∈ RN
evy process if X has the following pathwise
+ } is called an additive L´
decomposition:
X(t) = X1 (t1 ) + · · · + XN (tN ),
∀t ∈ RN
+,
where X1 , · · · , XN are independent, classical L´evy processes on R d . Additive L´evy processes
arise naturally in the studies of the L´evy sheets and intersections of classical L´evy processes. We
refer to Khoshnevisan and Xiao (2004) for historical information, potential theoretic results for
additive L´evy processes and their applications to fractal geometry of ordinary L´evy processes.
When X1 , · · · , XN are strictly α-stable processes in R d , we call X an (N, d)-additive αstable process. It is an H-self-similar stable random field with H = 1/α.
Similar to the stable sheet case, we can verify that the (N, d)-additive α-stable process X
satisfies Conditions (C1) and (C2) with H 1 = H2 = 1/α, respectively. Therefore for every
closed set E ⊂ (0, ∞)N , both (??) and (??) hold almost surely.
22
5.3
The Rosenblatt process
Given an integer m ≥ 2 and a constant κ ∈ (1/2 − 1/(2m), 1/2), the Hermite process Y m,κ =
{Y m,κ (t), t ∈ R+ } of order m is defined by
Y
m,κ
(t) = K5,10
Z
0
Rm
Z t Y
m
(s −
0 j=1
κ−1
u j )+
ds
dB(u1 ) · · · dB(um ),
(5.34)
R0
where K5,10 > 0 is a normalizing constant depending on m and κ only and the integral Rm is
the m-tuple Wiener-Itˆo integral with respective to the standard Brownian motion excluding
the diagonals {ui = uj }, i 6= j. Of course the integral (??) is also well defined if m = 1. In
this case, the process is a fractional Brownian motion for which the problem considered in this
paper has been solved. So we will exclude it from this section.
The Hermite process Y m,κ is an H-self-similar process with stationary increment and H =
1 + mκ − m
2 ∈ (0, 1). It is a non-Gaussian process and often appears in non-central limit
theorems for certain processes defined as integrals or partial sums of nonlinear functionals
of some stationary Gaussian sequences with long range dependence; see Taqqu (1975, 1979),
Dobrushin and Major (1979), Major (1981) for further information.
It follows from Theorem 6.3 of Taqqu (1979) that the Hermite process Y m,κ has the following equivalent representation:
Y
m,κ
(t) = K5,11
Z
0
Rm
m
eit(u1 +···+um ) − 1 Y
|uj |κ−1 ZG (du1 ) · · · ZG (dum ),
i(u1 + · · · + um )
(5.35)
j=1
where K5,11 > 0 is a normalizing constant and ZG is a centered complex Gaussian random
measure on R with Lebesgue measure as its control measure; see Major (1981) for systematic
account on multiple Wiener-Itˆo integrals.
Mori and Oodaira (1986) have studied the functional laws of the iterated logarithms for the
Hermite process Y m,κ . The following lemma is a corollary of Lemma 6.3 in Mori and Oodaira
(1986). It can also be proven by using Theorem 6.6 in Major (1981) and a standard chaining
argument.
Lemma 5.13 There exist positive constants K 5,12 and K5,13 , depending on m only, such that
for all u ≥ K5,12 ,
m,κ
P max |Y
(t)| ≥ u ≤ exp − K5,13 u2/m .
(5.36)
t∈[0,1]
The case m = 2 has received considerable attention recently. The process Y 2,κ is called
the Rosenblatt process by Taqqu (1975) [or the fractional Rosenblatt motion as called by
Pipiras (2004)] in R. Its self-similarity index is given by H = 2κ. Besides its connections to
non-central limit theorems, the Rosenblatt process also appears in limit theorems for some
quadratic forms of random variables with long range dependence. Albin (1998a, 1998b) has
discussed distributional properties and the extreme value theory of Y 2,κ . In particular, Albin
(1998b, Section 16) has obtained bounds on the tail probability of max t∈[0,1] Y 2,κ (t) which
are more precise than (??). Pipiras (2004) has established a wavelet-type expansion for the
Rosenblatt process.
23
Now we consider the Rosenblatt process X 2,κ in Rd by letting its component processes
to be independent copies of Y 2,κ . This non-Gaussian process resembles in many ways the
d-dimensional fractional Brownian motion. However, since the tails of its distributions are
fatter than those of fractional Brownian motion, we also expect to see interesting differences.
It would be interesting to study further sample path properties of X 2,κ systematically. One of
the motivations of such studies is to see to what extent the Gaussian methods can be pushed
forward. A similar approach has been helpful in studying the symmetric stable processes; see
Samorodnitsky and Taqqu (1994).
We content ourselves with the following result on the Hausdorff and packing dimensions of
the images of X 2,κ .
Corollary 5.14 Let X 2,κ = {X 2,κ (t), t ∈ R} be a Rosenblatt process in R d defined above.
Then for every analytic set E ⊂ R, we have
1
2,κ
dimH E ,
a.s.
(5.37)
dimH X (E) = min d,
2κ
and
dimP X 2,κ (E) =
1
Dim2κd E,
2κ
a.s.
(5.38)
Proof Since X 2,κ is 2κ-self-similar with stationary increments, Lemma ?? implies immediately that X 2,κ satisfies Condition (C1) with H1 = 2κ. On the other hand, Albin (1998a) has
proven that the random variable Y 2,κ (1) has a bounded and continuous density. Thus X 2,κ
also satisfies Condition (C2) with H 2 = 2κ. Therefore (??) and (??) follow from Theorem ??
and Remark ??.
Remark 5.15 It is easily seen that if one could show that for m ≥ 3 the Hermite process
Y m,κ has the property that
P |Y m,κ (1)| ≤ x ≤ K5,14 x for all x > 0,
(5.39)
then Y m,κ would satisfy Condition (C2) and Theorem ?? would hold for the d-dimensional
Hermite process X m,κ as well, where H = 1 + mκ − m
2 . It would be interesting to prove (??)
and to further investigate sample path properties of the Hermite processes.
Remark 5.16 By using the multiple Wiener-Itˆo integral with respect to a Gaussian random
measure ZG in RN [cf. Major (1981)], one can define the N -parameter Hermite random fields in
a way similar to (??) and/or (??). Little has been known about their distributional properties
or sample path properties even in the special case of m = 2.
Acknowledgements. Part of this work was done during the second author’s visit to
Taiwan University and NCTS/Taipei. He would like to thank the staff for their support and
the good working conditions. The authors thank Professor Patrik Albin for discussions on the
Rosenblatt process.
24
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Narn-Rueih Shieh. Department of Mathematics, National Taiwan University, Taipei,
10617, Taiwan.
E-mail: [email protected]
Yimin Xiao. Department of Statistics and Probability, A-413 Wells Hall, Michigan State
University, East Lansing, MI 48824, U.S.A.
E-mail: [email protected]
URL: http://www.stt.msu.edu/~xiaoyimi
27