GAUSSIAN ISOTROPIC RANDOM FIELDS
ANATOLIY MALYARENKO
Dedicated to my teacher Professor M. I. Yadrenko on the occasion of his seventieth birthday.
A. We consider a class of Gaussian isotropic random fields related to multi-parameter fractional Brownian motion. We calculate both the lo-cal and global moduli of continuity as well as the Hausdorff and packing dimensions of the exceptional random sets of fast points for that fields.
1. INTRODUCTION
1.1. A formulation of the problem. Let (Ω, F,P) be a probability space. Let ξ(x, ω) : X ×Ω 7→ R be a measurable and separable random function on a complete separable metric space (X, ρ). The local modulus of continuity measures the rate of escape of the function ξ(x) from a fixed point x0 ∈ X.
As a canonical example, consider the Gaussian process W(t) on [0,+∞) with zero mean and the correlation function
B(s, t) = Eξ(s)ξ(t)= 1
2(|s|+ |t| − |s − t|).
It is called Brownian motion or Wiener process. Then, according to the local law of the iterated logarithm(Khintchin, 1924), for each t ∈ [0,+∞), (1) lim sup
s→t
|W(s) − W(t)| p
2|s − t| log log(|s − t|−1) = 1 P-almost surely.
Thus, in this case the function f (r) = p2r log log(r−1) is the local modulus
of continuity of the Wiener process.
Let Nt denotes the P-null set outside which the rate of escape of the
Wiener process from the point t is defined by (1). The union ∪t≥0Nt is
Date: June 10, 2005.
2000 Mathematics Subject Classification. Primary 60D05, 60G15; Sec-ondary 60G17.
Key words and phrases. Modulus of continuity, fast point, isotropic ran-dom field.
not a P-null set. Indeed, according to L´evy modulus of continuity (L´evy, 1937): lim sup s→t sup t∈[0,1] |W(s) − W(t)| p
2|s − t| log(|s − t|−1) = 1 P-almost surely.
The function g(r) = p2r log(r−1) is the global modulus of continuity of
the Wiener process. Thus, the sample path of the Wiener processP-almost surely contains fast points. In that points the rate of escape is faster than in (1).
For some Gaussian random functions the local modulus of continuity coincides with the global one, see (Dudley, 1967). But this is not the case for the Wiener process.
It is very easy to prove that the set of all fast points of the Wiener process on the interval [0, 1] has zero Lebesgue measure P-almost surely. Indeed, letL denotes the Lebesgue measure. Let B([0, 1])) denotes the σ-algebra of Borel sets on the interval [0, 1]. Consider the set
A = (ω, t) ∈ Ω × [0, 1]: lim sup s→t |W(s, ω) − W(t, ω)| p 2|s − t| log log(|s − t|−1) , 1 . Clearly, the set A belongs to the σ-algebra F × B([0, 1]). Then we have
P×L (A ) = "
Ω×[0,1]
χA(ω, t) dP(ω) dt < ∞, where χA denotes the indicator of the setA . By Fubini theorem
P×L (A ) = Z 1 0 Z Ω χA(ω, t) dP(ω) ! dt = 0,
because χA(ω, t) = 0 forP-almost every ω. Once more by Fubini theorem we obtain Z Ω Z 1 0 χA(ω, t) dt ! dP(ω) = 0.
It follows that P-almost surely χA(ω, t) = 0 almost everywhere on the interval [0, 1]. The proof is complete.
For any set of zero Lebesgue measure, it is interesting to calculate its Hausdorff dimension dimH. All necessary definitions are assembled in
Subsection 1.3. The first step in this direction was made by (Orey and Taylor, 1974). They defined the set of λ-fast points as
F≥(λ) = t ∈ [0, 1] : lim sup s→t |W(s) − W(t)| p 2|s − t| log(|s − t|−1) ≥ λ
and proved that for 0 < λ ≤ 1
dimH(F≥(λ))= 1 − λ2 P-almost surely.
Different generalisations were obtained later in (Kaufman, 1975; Khosh-nevisan and Shi, 2000; KhoshKhosh-nevisan et al., 2000).
Our goal is to prove similar theorems for isotropic Gaussian random fields with homogeneous increments.
In Subsection 1.2 we define the above mentioned class of random fields and describe some its subset in terms of the incremental variance. Af-ter discussing different concepts of dimension for both deterministic and random sets (Subsection 1.3) we formulate our results in Subsection 1.4. Proofs of the results, remarks, and examples are presented in Section 2. 1.2. The object of investigations. Let ξ(x), x ∈ RN be a zero mean ran-dom field satisfying the condition E|ξ(x)|2 < ∞, with the correlation func-tion B(x, y). We denote by k · k and (·, ·) the usual norm and scalar product on the space RN, respectively. The random field ξ(x) has homogeneous increments, if the incremental variance
σ2(x, y)= E(ξ(x) − ξ(y))2 satisfies the condition
σ2(x+ z, y + z) = σ2(x, y), z ∈ RN.
According to (Yaglom, 1957), the correlation function of the random field with homogeneous increments has the form
(2) B(x, y) = Z
RN
(ei(p,x) − 1)(e−i(p,y) − 1) dµ(p)+ (Bx, y). Here µ denotes the measure on RN \ {0} which satisfies the condition (3)
Z
RN
kpk2dµ(p)
1+ kpk2 < ∞,
and B denotes a fixed positive semi-definite matrix.
Let O(N) denotes the group of all orthogonal matrices of the nth order. Definition 1. (Yadrenko, 1983) A random field ξ(x) is called isotropic if the expectation Eξ(x) depends only on the norm kxk, and for any g ∈ O(N) we have
Let ξ(x) be a random field with zero mean and the correlation function (4) B(x, y) =
Z
RN
(ei(p,x) − 1)(e−i(p,y)− 1)
kpkN+2αf(kpk) dp, 0 < α < 1,
where f (kpk) ≥ 0, p ∈ RN. It is easy to check that the random field ξ(x) is isotropic. Moreover, it has homogeneous increments, because (4) coincides with (2) for B = 0 and
(5) dµ(p) = dp
kpkN+2αf(kpk).
Substituting (5) to (3) and changing to spherical coordinates, we obtain that the function f should satisfy the condition
(6)
Z ∞
0
r1−2α
(1+ r2) f (r)dr < ∞.
In what follows ξ(x) denotes Gaussian zero mean isotropic random field with homogeneous increments and correlation function (4) satisfying con-dition (6). Calculating the incremental variance of the random field ξ(x), we obtain: σ2(x, y) = B(x − y, x − y) = 2 Z RN 1 − cos(p, x − y)) kpkN+2αf(kpk) dp.
Changing to the spherical coordinates, we have: σ2(x, y) = 4π (N−1)/2 Γ((N − 1)/2) Z ∞ 0 tN−1dt Z π 0
1 − cos(kx − ykt cos u) tN+2αf(t) sin
N−2u du.
Using formulae 2.5.3.1 and 2.5.55.7 from (Prudnikov et al., 1986) and the formula Jν(z) = z 2 ν 1 Γ(ν + 1)0F1(ν+ 1; −z2/4) we obtain σ2(x, y) = 4π N/2 Γ(N/2) Z ∞ 0 1 −0F1(N/2; −kx − yk2t2/4) t2α+1f(t) dt,
where0F1(N/2; −kx − yk2t24) is the generalised hypergeometric function: pFq(a1, . . . , ap; b1, . . . , bq; z) = ∞ X k=0 (a1)k. . . (ap)k (b1)k. . . (bq)k zk k!, p ≥ 0, q ≥ 0, Jν(z) denotes the Bessel function of the ν-th order and (a)0 = 1, (a)k =
the random field ξ(x) depends only on the distance r between the points x and y: (7) σ2(r) = 4π N/2 Γ(N/2) Z ∞ 0 1 −0F1(N/2; −r2t2/4) t2α+1f(t) dt.
Let f (t) = 1. Making the change of variable tr = s, we obtain σ2(r) = Cr2α, where (8) C = 4π N/2 Γ(N/2) Z ∞ 0 1 −0F1(N/2; −s2/4) s2α+1 ds.
The random field C−1/2ξ(x) is called the multi-parameter fractional Brown-ian motion. In particular, for N = 1 and α = 1/2 we obtain two indepen-dent Wiener processes. One of them corresponds to the half-axis t ≥ 0. The second one corresponds to the half-axis t ≤ 0.
1.3. Different concepts of dimension. We assembled here definitions con-cerning dimension for both deterministic and random sets. For more de-tails, see (Mattila, 1995) and (Khoshnevisan and Shi, 2000).
For β > 0, define the β-dimensional Hausdorff measure of the subset A of the metric space (X , ρ) as follows (Hausdorff, 1918):
Hβ(A ) = lim ε↓0 inf X j (diam(Aj))β: A ⊂ ∪jAj, diam(Aj) < ε ,
where diam(Aj) = supx,y∈Ajρ(x, y).
The Hausdorff dimension of the set A is defined as dimH(A ) = inf{β: Hβ(A ) = 0}.
The packing dimension was introduced in (Tricot, 1980). A sequence of closed balls
B(xj, rj) = {x ∈ X : ρ(x, xj) ≤ rj}
is called an ε-packing of the set A ⊆ X , if xj ∈ A , rj < ε, and the balls
B(xj, rj) are pairwise disjoint. Denote
pβ(A ) = lim ε↓0 sup X j (2rj)β ,
where the supremum is taken over all possible ε-packings of the set A . Unfortunately, pβ in not a measure. But after the regularisation
Pβ(A ) = inf X n pβ(An) : A ⊆ ∪nAn
we obtain the β-dimensional packing measure Pβ. The packing dimension of the set A is defined as follows:
dimP(A ) = inf{β: Pβ(A ) = 0}.
Let B(RN) denotes the σ-algebra of all Borel subsets of the space RN. A mapping ω 7→ A (ω) from the space of elementary events Ω to B(RN) is called a random set (Khoshnevisan and Shi, 2000) if the function
χ(ω, x) = 1, x ∈ A (ω), 0, x < A (ω)
is measurable in the product space (Ω × RN, F × B(RN)). We can econom-ically cover the set A (ω) by closed balls with rational radii and rational coordinates of their centra. Thus both dimH(A (ω)) and dimP(A (ω)) are
random variables.
1.4. The formulation of results. We present some notation. Z+ denotes
the set of non-negative integers. For any s = (s1, . . . , sN) ∈ ZN+ we denote
|s| = s1 + · · · + sN. For any function g(x) ∈ L2(RN) we denote by ˆg(p) its
Fourier transform
ˆg(p) = Z
RN
ei(p,x)g(x) dx.
Now we formulate conditions onto the function f (t). The first condition was already mentioned earlier.
Condition 1. f (t) is a non-negative function satisfying (6).
Condition 2. The function f (kpk) is infinitely differentiable on RN. More-over, for any s ∈ Z+N there exists a constant C = Cs ≥ 0 such that for any
p ∈ RN ∂|s|f(kpk) ∂s1p 1. . . ∂sNpN ≤ Cs(1+ kpk)−|s|.
Condition 3. The function f−1(kpk) is locally integrable on RN. Condition 4. The pseudo-differential operator with symbol f−1(kpk)
(Ag)(x) = 1 (2π)N
Z
RN
is non-negatively defined.
Condition 5. There exist a limit limt→∞ f(t) satisfying
0 < lim
t→∞ f(t) < ∞.
Theorem 1. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and incremental variance(7). Let the func-tion f(t) satisfies Conditions 1–5. Then
• for any x ∈ RN the local modulus of continuity of the random field ξ(x) is given by the law of the iterated logarithm:
(9) P lim sup y→x |ξ(x) − ξ(y)| p
2σ2(kx − yk) log log(kx − yk−1) = 1
= 1;
• for any compact set D ⊂ RN with non-empty interior the D-global modulus of continuity of the random field ξ(x) is given by the L´evy modulus of continuity: (10) P lim sup x,y∈D y→x |ξ(x) − ξ(y)| p 2σ2(kx − yk) log(kx − yk−1) = √ N = 1.
Condition 6. The function r 7→ σ2(r) is twice continuously differentiable. Moreover, there exists positive constants c0 and r0 such that for any r ∈
(0, r0) (11) d2σ2(r) dr ≤ c0 σ2(r) r2 .
Recall that the setE ⊂ RN is called analytic if it can be represented as an image of the Borel set under some continuous mapping. In what follows D denotes the fixed compact subset of the space RN having non-empty
interior.
The next theorem shows that the maximal possible rate of escape of the random field ξ(x) from the analytic setE ⊆ D is determined by the packing dimension of the set E .
Theorem 2. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and incremental variance(7). Let the func-tion f(t) satisfies Conditions 1–6. Then for any analytic set E ⊆ D
P sup x∈E lim sup y→x |ξ(x) − ξ(y)| p 2σ2(kx − yk) log(kx − yk−1) = p dimP(E ) = 1.
Following (Orey and Taylor, 1974) and (Khoshnevisan et al., 2000), de-fine the set of λ-fast points for the random field ξ(x) as
F≥(λ) = x ∈ D : lim sup y→x |ξ(x) − ξ(y)| p 2σ2(kx − yk) log(kx − yk−1) ≥ λ . and the set of λ-level fast points as
F=(λ) = x ∈ D : lim sup y→x |ξ(x) − ξ(y)| p 2σ2(kx − yk) log(kx − yk−1) = λ . For compact sets E Theorem 2 can be strengthened to a necessary and sufficient conditions on E to contain λ-fast or λ-level fast points.
Theorem 3. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and incremental variance(7). Let the func-tion f(t) satisfies Conditions 1–6. Then for any compact set E ⊆ D and any λ > 0 the next three conditions are equivalent:
a: P{F≥(λ) ∩E , ∅} = 1.
b: P{F=(λ) ∩E , ∅} = 1.
c: E is not a union of countably many Borel sets En with dimP(En) <
λ2.
Moreover, if any of the above conditions fails, then
P{F≥(λ) ∩E , ∅} = P{F=(λ) ∩E , ∅} = 0.
Next, we find the packing dimension and inequalities for the Hausdorff dimension of both the setF≥(λ) ∩E of λ-fast points and the set F=(λ) ∩E
of λ-level fast points.
Theorem 4. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and incremental variance(7). Let the func-tion f(t) satisfies Conditions 1–6. Let E ⊆ D be an analytic set and λ ∈ (0, pdimP(E )]. Let one of the next conditions be satisfied:
• λ < pdimP(E ).
• E is a compact set which is not a union of countably many Borel sets En with dimP(En) < λ2.
Then we have
P{dimP(F≥(λ) ∩E ) = dimP(E )} = 1
and
In particular, ifE = D and λ ∈ (0, √N], then (12) P{dimP(F≥(λ)) = N} = 1
and
(13) P{dimH(F≥(λ)) = N − λ2} = 1.
Theorem 5. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and incremental variance(7). Let the func-tion f(t) satisfies Conditions 1–6. Let E ⊆ D be a compact set which is not a union of countably many Borel sets En with dimP(En) < λ2 for some
λ > 0. Then we have
P{dimP(F=(λ) ∩E ) = dimP(E )} = 1
and
P{dimH(E ) − λ2 ≤ dimH(F=(λ) ∩E ) ≤ dimP(E ) − λ2} = 1.
In particular, ifE = D and λ ∈ (0, √N], then (14) P{dimP(F=(λ)) = N} = 1
and
(15) P{dimH(F=(λ)) = N − λ2} = 1.
2. PROOFS AND REMARKS
2.1. Proof of Theorem 1. First consider one particular case. Let ξ(x) be the multi-parameter fractional Brownian motion. Then f (t) = C, where C is calculated in (8), and
(16) σ2(r) = r2α.
The remark after the proof of Lemma 1.1 from (Benassi et al., 1997) shows that the random field ξ(x) satisfies the conditions of Theorem 1.3 of the above cited paper. According to that theorem we have
(17) P lim sup y→x |ξ(x) − ξ(y)| p
2kx − yk2αlog log(kx − yk−1) = lim supx→y
σ(x, y) kx − ykα = 1 for any x ∈ RN and
(18) P lim sup x,y∈D y→x |ξ(x) − ξ(y)| p 2kx − yk2αlog(kx − yk−1) = √ N sup y∈D lim sup x→y σ(x, y) kx − ykα = 1.
It is clear from (16) that the right hand side of the equality in figure brack-ets in (17) is equal to 1, and the right hand side of the equality in figure brackets in (18) is equal to √N.
For any x0 ∈ RN consider the next function:
(19) c2x0(u) = lim ε↓0 E " (ξ(x0 + εu) − ξ(x0))2 ε2α # .
It is clear that c2x0(u) = kukα, which is the α-homogeneous non-trivial func-tion.
Now let the function f (t) satisfies Conditions 1–5 and ξ(x) denotes the mean zero Gaussian random field with the incremental variance (7). The-orem 1.3 from (Benassi et al., 1997) is applicable, and we obtain (17) and (18). According to Proposition 1.3 from (Benassi et al., 1997) the random field ξ(x) is asymptotically self-similar of the order (α, 0), which means that for every x ∈ RN the random field
ξ(x+ ρu) − ξ(x)
ρα , u ∈ R
N
converges in law to a non-trivial limit for ρ ↓ 0. Then according to Theorem 1.4 from (Benassi et al., 1997) the limit (19) exists as the α-homogeneous non-trivial function. In the case of the isotropic random field this limit can be rewritten as
c2x0(u) = lim
ε↓0
σ2(εkuk) ε2α .
In particular, for any u with kuk = 1 we have
(20) c2x0(u) = lim
ε↓0
σ2(ε) ε2α , 0.
This means that the lim sup in the right hand sides of the equalities in figure brackets in (17) and (18) is actually a non-zero limit. Moreover, this limit does not depend on x by homogeneity of increments of the random field ξ(x). Consequently, the sup in the right hand side of the equality in figure brackets in (18) can be omitted. Now we divide both hand sides of the above mentioned equalities by that non-zero limit and obtain (9) and (10). 2.2. Proof of Theorems 2–5. To prove Theorems 2–5, consider the aux-iliary random field
η(x, r) = (ξ(x+ ru) − ξ(x))
2
2σ2(r) log(r−1) , x ∈ R N,
r ∈ (0, 1), where u ∈ RN is an arbitrary, but fixed vector with kuk = 1.
We denote by RN+ the set ∩Nj=1{ x ∈ RN: xj ≥ 0 }. For any y ∈ RN+ we
denote
[0, y] = ∩Nj=1{ x ∈ R+N: xj ≤ yj}.
Recall that the function ψ(t) defined in some right-side neighbourhood of the point t = 0 is called regularly varying at 0 of the order β, if for any s > 0 we have lim t↓0 ψ(st) ψ(t) = s β.
Properties of regularly varying functions can be found in (Bingham et al., 1987).
Lemma 1. The random field η(x, r) has the following properties:
A: For each x ∈ RN and r ∈ (0, 1) the random variables η(x, r) and η(0, r) have the same distribution. Moreover, for all x ∈ RN and r ∈ (0, 1) we have η(x, r) ≥ 0 and for each x ∈ RN and γ > 0
(21) lim
r↓0
logP{η(x, r) > γ} log r = γ.
B: For all ε > 0 and M > 0, there exists a function ψ= ψε,M, regularly
varying of the order1 at 0 and a number r0 = r0(ε, M) ∈ (0, 1) such
that for all r ∈(0, r0), for all x, y ∈ [0, M]N and for all γ ∈ [0, N],
(22) P{η(x, r) > γ/η(y, r) > γ} < (1+ ε)P{η(x, r) > γ}. C: For all ε > 0 and for all y ∈ RN+
P lim sup r1↓0 sup x∈[0,y] sup r2∈[r1−r11+ε,r1+r11+ε] |η(x, r1) − η(x, r2)| = 0 = 1 and P lim sup r↓0 sup x1,x2∈[0,y] kx1−x2k≤r1+ε |η(x1, r) − η(x2, r)| = 0 = 1.
Proof. The homogeneity of increments of the random field ξ(x) improves the first part of property A. To prove the second part, we write
P{η(x, r) > γ}= 2P( ξ(x+ ru) − ξ(x) σ(r) > q 2γ log(r−1) ) = 2Φ(a), (23)
whereΦ(x) denotes the tail of the standard normal distribution: Φ(x) = Z
∞
x
exp(−t2/2) dt,
and a = p2γ log(r−1). Now (21) is the direct consequence of (23) and the
following standard estimate (Lifshits, 1995, Chapter 1, (13)):
(24) 1 − x −2 √ 2πx exp(−x 2/2) ≤ Φ(x) ≤ 1 √ 2πx exp(−x 2/2).
To prove property B, we introduce the notation: ζ1 =
ξ(x+ ru) − ξ(x)
σ(r) , ζ2 =
ξ(y+ ru) − ξ(y)
σ(r) .
Both ζ1 and ζ2 are standard normal random variables. Let ρ denotes the
correlation coefficient between these variables. From (Khoshnevisan and Shi, 2000, proof of Lemma 4.1) we have for ρ < 1/4
P{ζ1 > a, ζ2 > a} ≤ p 1 − ρ2 1 − 4ρ+ Φ a s 1 − 4ρ+ 1 − ρ2 2 , where ρ+ = max{ρ, 0}. Using (24), we obtain
P{ζ1 > a, ζ2 > a} P{ζ1 > a}P{ζ2 > a} ≤ (1 − ρ2)3/2 (1 − 4ρ+)2(1 − a−2)2 exp a 24ρ+− ρ 2 1 − ρ2 ! .
Next we follow (Khoshnevisan et al., 2000, Section 5). From the last inequality we conclude: to prove (22) it is enough to show that for any M > 0 there exists a function ψ(r), regularly varying of the order 1 at 0 such that uniformly for all x, y ∈ [0, M]N satisfying ky − xk ≥ ψ(r) and for all γ ∈ [0, N], we have lim r↓0 a 2 ρ = 0. Put u = y − x ky − xk. Calculating ρ, we have ρ = σ
2(ky − xk+ r) + σ2(ky − xk − r) − 2σ2(ky − xk)
2σ2(r) .
Use Taylor expansion around ky − xk. We obtain ρ = r 2 4σ2(r) " d2σ2(ky − xk+ θ 1r) dr2 + d2σ2(ky − xk − θ2r) dr2 # ,
where 0 ≤ θ1, θ2 ≤ 1. In view of (11) and (20) we see that there exist
positive constants c1 and r0 such that for any r ∈ (0, r0)
|ρ| ≤ c1 r ky − xk − r !2−2α . Put ψ(r) = r(log(r−1))1/(1−α). We conclude that for ky − xk ≥ ψ(r)
a2ρ ≤ c3γ
log(r−1) → 0, r ↓ 0.
Property B is proved.
Both parts of property C are trivial consequences of the triangle
inequal-ity and (10).
Recall that D denotes the fixed compact subset of the space RN hav-ing non-empty interior. The Gaussian random field ξ(x) is isotropic and has homogeneous increments. So after the shift and the rotation we may suppose that D ⊂ R+N.
Theorem 2 is the consequence of Lemma 1 and (Khoshnevisan et al., 2000, Theorem 2.1).
As in (Khoshnevisan et al., 2000), we abbreviate condition c) of Theo-rem 3 asE < {dimP < λ}σ and the negation of condition c) asE ∈ {dimP <
λ}σ. From Lemma 1 and (Khoshnevisan et al., 2000, Theorem 2.1) we
conclude that (25) P{F≥(λ) ∩E , ∅} = 1, E < {dimP < λ}σ 0, E ∈ {dimP < λ}σ.
which proves the equivalence of conditions c and a of Theorem 3. The equivalence of conditions c and b follows in the same manner from Lemma 1 and the following assertion of (Khoshnevisan et al., 2000, Theorem 2.3): (26) P{F=(λ) ∩E , ∅} = 1, E < {dimP < λ}σ 0, E ∈ {dimP < λ}σ.
The second part of Theorem 3 easily follows from (25) and (26).
Theorem 4 is the consequence of Lemma 1 and the last assertion of (Khoshnevisan et al., 2000, Theorem 2.1).
In the same manner, Theorem 5 is the consequence of Lemma 1 and the last assertion of (Khoshnevisan et al., 2000, Theorem 2.3).
It follows from our definition of the set D that dimP(D) = dimH(D) =
λ2 ∈ (0, N], so D < {dimP < λ}σ. This proves equations (12), (13), (14)
and (15).
2.3. Remarks.
Remark 1. Let the next condition be satisfied:
Condition 7. There exists a function L which is slowly varying at 0 and positive constants c4, c5, c6 and r1 such that for any r ∈ (0, r1)
c4r2αL(r) ≤ σ2(r) ≤ c5r2αL(r) and dσ2(r) dr ≤ c6 σ2(r) r .
Under Conditions 1, 6, and 7 Theorem 2 remains true. It can be proved exactly as in (Khoshnevisan et al., 2000, Section 5).
Remark 2. Let ξ(x) be the zero mean isotropic Gaussian random field with homogeneous increments and correlation function
B(x, y) = Z
RN
expl(i(p, x)) expl(−i(p, y)) kpkN+2l+2αg(kpk) dp, where l ∈ N, 0 < α < 1 and expl(z)= exp(z) − l X j=0 zj j!.
Let ∆ denotes the Laplace operator in RN. According to (Benassi et al., 1997) the trajectories of the random field ξ(x) are l times continuously differentiable. If the function g(p) satisfies Conditions 1–7, then Theo-rems 1–5 and Remark 1 are applicable to the random field (−∆)l/2ξ(x). Remark 3. Let α = 1. In this case the function f (t) = C fails to satisfy Condition 1. However, if the function f (t) still satisfies Conditions 1–5, then we can apply (Benassi et al., 1997, Theorem 1.3). The local modulus of continuity looks like
P lim sup y→x |ξ(x) − ξ(y)| p
2σ2(kx − yk) log(kx − yk−1) log log log(kx − yk−1) = 1
= 1, and the global modulus of continuity (10) remains unchanged.
Conditions A and C of Lemma 1 can be proved even in this case. How-ever, the problem of proving Condition B remains open.
2.4. An example. Let ξ(x), x ∈ RN be the multi-parameter fractional Brownian motion with σ2(r) = r2α, 0 < α < 1. The function f (t) is positive constant and Conditions 1–6 are trivially satisfied. Theorems 1–5 are applicable.
In the case of an ordinary Brownian motion, when N = 1 and α = 1/2, the set D can be chosen to be random. In particular, (Khoshnevisan and Shi, 2000, Theorem 1.3) states that with probability one,
(27) lim sup y↓0 sup x∈D |ξ(x + y)| p
ylog(y−1) = supx∈D lim supy↓0
|ξ(x+ y)| p
ylog(y−1) = 1,
where D = { x ∈ [0, 1]: ξ(x) = 0 }. Theorem 1.5 from (Khoshnevisan and Shi, 2000) states that with probability one,
(28) dimH(D ∩ F≥(λ)) =
1 2 − λ
2.
This coincides with Theorems 2 and 4, because dimP(D) = 1/2 almost
surely (Taylor, 1986). The Markov property of the Brownian motion is essentially used in the proof of (27) and (28). The question of the extend-ing (27) and (28) to the case of the (multi-parameter) fractional Brownian motion remains open.
REFERENCES
Benassi, A., S. Jaffard, and D. Roux. 1997. Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13, no. 1, 19–90.
Bingham, N. H., C. M. Goldie, and J. L. Teugels. 1987. Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge.
Dudley, R. M. 1967. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1, 290–330.
Hausdorff, F. 1918. Dimension und ¨ausseres Mass, Math. Ann. 79, 157– 179.
Kaufman, R. 1975. Large increments of Brownian motion, Nagoya Math. J. 56, 139–145.
Khintchin, A. 1924. Uber einen Satz der Wahrscheinlichkeitsrechnung, Fund. Math. 6, 9–12.
Khoshnevisan, D. and Z. Shi. 2000. Fast sets and points for fractional Brownian motion, S´eminaire de Probabilit´es, XXXIV, pp. 393–416. Khoshnevisan, D., Y. Peres, and Y. Xiao. 2000. Limsup random fractals,
L´evy, P. 1937. Th´eorie de l’Addition des Variables Al´eatoires, Gauthier-Villars, Paris.
Lifshits, M. A. 1995. Gaussian random functions, Mathematics and its ap-plications, vol. 322, Kluwer Academic Publishers, Dordrecht.
Mattila, P. 1995. Geometry of sets and measures in Euclidean spaces. Frac-tals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge.
Orey, S. and S. J. Taylor. 1974. How often in a Brownian path does the law of the iterated logarithm fails?, Proc. London Math. Soc. 28, 174–192. Prudnikov, A. P., Yu. A. Brychkov, and O. I. Marichev. 1986. Integrals and
series. Vol. 1. Elementary functions, Gordon & Breach Science Publish-ers, New York.
Taylor, S. J. 1986. The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc. 100, 383–406.
Tricot, C. 1980. Rarefaction indices, Mathematika 27, 46–57.
Yadrenko, M. I. 1983. Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publica-tions Division, New York.
Yaglom, A. M. 1957. Certain types of random fields in n-dimensional space similar to stationary stochastic processes, Teor. Veroyatnost. i Primenen. 2, 292–338.
M¨ U, B 883, SE–721 23 V¨, S E-mail address: amo@mdh.se
