Limsup random fractals

Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, ${\rm F}(\lambda):=\{ t\in[0,1]: \limsup_{h\to 0}{ | X(t+h)-X(t)| / \sqrt{ 2h|\log h|}} \ge \lambda\}$. They proved that for $\lambda \in (0,1]$, the Hausdorff dimension of ${\rm F}(\lambda)$ is $1-\lambda^2$ a.s. We prove that for any analytic set $E \subset [0,1]$, the supremum of the $\lambda$ such that $E$ intersects ${\rm F}(\lambda)$ a.s. equals $\sqrt{\text{dim}_p E }$, where $\text{dim}_p E$ is the packing dimension of $E$. We derive this from a general result that applies to many other random fractals defined by limsup operations. This result also yields extensions of certain ``fractal functional limit laws'' due to Deheuvels and Mason (1994). In particular, we prove that for any absolutely continuous function $f$ such that $f(0)=0$ and the energy $\int_0^1 |f'|^2 \, dt $ is lower than the packing dimension of $E$, there a.s. exists some $t \in E$ so that $f$ can be uniformly approximated in $[0,1]$ by normalized Brownian increments $s \mapsto [X(t+sh)-X(t)] / \sqrt{ 2h|\log h|}$; such uniform approximation is a.s. impossible if the energy of $f$ is higher than the packing dimension of $E$.


Introduction
The connections between hitting probabilities of stochastic processes, capacity and Hausdorff dimension are key tools in the analysis of sample paths; see Taylor (1986).
In this paper we show that for certain random fractals that are naturally defined by limsup operations (Brownian fast points being the canonical example), hitting probabilities are determined by the packing dimension dim P (E) of the target set E, rather than its Hausdorff dimension dim H (E). See Mattila (1995) for definitions of these dimension indices.
(1.2) Kaufman (1975) subsequently showed that any analytic set E with dim H (E) > λ 2 , a.s.intersects F(λ).The next theorem shows that packing dimension is the right index for deciding which sets intersect F(λ).It is a special case of a general result that we state in §2.Equivalently, Moreover, if dim P (E) > λ 2 then dim P (F (λ) ∩ E) = dim P (E), a.s.
Remark 1.2 For compact sets E, we can sharpen (1.3) to a necessary and sufficient criterion for E to contain λ-fast points: P(F(λ) ∩ E = ?)= 1 if and only if E is not a union of countably many Borel sets E n with dim P (E n ) < λ 2 .(In this case, we say that E / ∈ {dim P < λ 2 } σ .)See Theorem 2.1 and its application to fast points in §2.
Remark 1. 3 For traditional random fractals such as the range and the level sets of Brownian motion (as well as those of many other stable processes), it is well known that Hausdorff and packing dimensions a.s.coincide.An interesting feature of limsup random fractals is that generally their Hausdorff and packing dimensions differ.For instance, as regards to fast points, this follows from comparing the result (1.2) of Orey and Taylor with a result of Dembo, Peres, Rosen and Zeitouni (1998).In fact, Corollary 2.4 of the latter paper implies that dim P (F(λ)) = 1 a.s.for every λ ∈ [0, 1).This equality is a special case of the final assertion of Theorem 1.1 with E = [0, 1].Remark 1.4 Theorem 1.1 can be viewed as a probabilistic interpretation of the packing dimension of an analytic set E ⊂ R 1 + .Similarly, we can provide a probabilistic interpretation for the packing dimension of a multi-dimensional set E ⊂ R d + .Let W denote white noise on R d + , viewed as an L 2 (P)-valued random measure; see Walsh (1986)  Together with Theorem 1.1, equations (1.5) and (1.6) complete the bounds in Theorem 1.1 of Khoshnevisan and Shi (1998).
Remark 1. 6 It is interesting to note the following intersection property of the set of λ-fast points.Let F (λ) be an independent copy of F (λ).Then, This is different from the intersection properties of the images of Brownian motion or stable Lévy processes; see e.g.Hawkes (1971).
Another application of our method is to functional extensions of Theorem 1.1.Let C ac [0, 1] denote the collection of all absolutely continuous functions f : (The Sobolev norm squared is sometimes called the energy of f .)Let H denote the Hilbert space of all f ∈ C ac [0, 1] of finite energy.Strassen (1964) proved that for all f ∈ H with f H 6 1, and for all 0 6 t 6 1, lim inf The null set in question depends on the choice of t.Deheuvels and Mason (1994) analyzed the exceptional t's, and obtained functional extensions of the Orey-Taylor Theorem (1.2).To explain their result, consider the normalized increment process and note the single logarithm in the normalization, in contrast with the iterated logarithm in Strassen's theorem.For any f ∈ H , let Deheuvels and Mason (1994 H a.s., for any f ∈ H such that f H 6 1.Our next theorem extends their result as well as Theorem 1.1 above, by giving a hitting criterion for D(f).
In words, given an analytic set E ⊂ [0, 1] and a function f ∈ H , if f 2 H < dim P (E), then there a.s.exists a time t ∈ E such that the (normalized) Brownian increments process ∆ h [t] converges uniformly to f along a sequence of h values tending to 0, but if f 2 H > dim P (E) then a.s.there is no such time in E.
We conclude the Introduction with a final application to extensions of Chung's law of the iterated logarithm for Brownian motion.Let X d be standard Brownian motion in R d , and denote by the radius of the smallest ball centered at X d (t) that contains X d [t, t+h].Theorem 4 of Ciesielski and Taylor (1962) states that for all t ∈ [0, 1], where q d is the smallest positive root of the Bessel function J (d−2)/2 ; in particular, q 1 = π/2.The null set in question depends on t, and we show that any set in R + of positive packing dimension contains random times t at which the behavior of the Brownian path is markedly different: Remark 1.9 For all λ 1, define (1.12) One can think of S(λ) as the collection of all points of (sporadic) slow escape of order λ.Theorem 1.8 means that for any analytic set E ⊂ [0, 1] and every λ > 0, Furthermore, we will show in §2 that dim P (S(λ)) = 1 while dim H (S(λ)) = 1 − λ −2 for all λ 1.

Remark
Compare these to Equations (1.1) and (1.12), respectively.In Section 2, we will show that the aforementioned dimension results as well as the hitting results hold for F = (λ) and S = (λ) in place of F(λ) and S(λ), respectively.In particular, we will see that for G(λ) denoting either F = (λ) or while this probability is 0 if dim P (E) < λ 2 .Moreover, for all λ ∈ [0, 1], we have dim

The General Results
In this section, we state a general theorem for a class of multi-parameter, R 1 where N is a fixed positive integer.We then use this theorem to derive the results announced in the Introduction.
We impose three conditions on Y ; the first two are distributional, while the last is a condition on the sample functions.Throughout, R N + is endowed with the coordinatewise partial order: Each of the following conditions is sufficient for E / ∈ {dim P < λ} σ : • E has positive packing measure in dimension λ.
(More generally, it suffices that the packing measure of E is positive in some gauge ϕ that satisfies lim r→0+ r ε−λ ϕ(r) = 0 for some ε > 0.) Indeed, the sufficiency of the first condition is obvious; to see the sufficiency of the second condition, suppose that it holds and E = ∪ n E n with dim P (E n ) < λ.By regularization, we can represent E as a countable union E = ∪ n,j E n,j with dim M (E n,j ) < λ.Each E n,j is nowhere dense in E, so Baire's theorem yields the desired contradiction.
It follows from Joyce and Preiss (1995) that any analytic set E ⊂ R d with positive λ-dimensional packing measure, contains a compact subset E such that E / ∈ {dim P < λ} σ .
We are ready to state the main result of this section (see §4 for a proof).Denote

and 3. Then, for any analytic set
If E is compact, we can determine when the supremum in (2.1) is attained: Finally, suppose that y 0 < λ We record that the upper bound in Theorem 2.1 does not require any correlation bound.
Proposition 2.2 Suppose that the R 1 + -valued stochastic process Y satisfies Conditions 1 and 3.Then, for any analytic set Next, we determine the hitting properties for the level sets of the limsup considered in (2.1).

Theorem 2.3
Suppose that the R 1 + -valued stochastic process Y satisfies Conditions 1,2 and 3.For each λ > 0, define the random set Then, for any compact set E ⊂ [0, 1] N and for all λ ∈ (y 0 , y 1 ], Our next corollary gives the Hausdorff dimension and packing dimension of the intersection of Γ(λ) (or Γ = (λ)) with an independent copy of itself.Its proof follows easily from Step Three of our proof of Theorem 2.1 and Lemma 3.4.

Corollary 2.4 Suppose that
We conclude this section by using Theorems 2.1 and 2.3 to prove the assertions of the Introduction.

Proof of Theorem 1.1
Recall that X is linear Brownian motion.We let N = 1 and define the process Y as follows: It follows from the stationarity of increments and the scaling property of Brownian motion that Y satisfies Condition 1 with y 0 = 0 and any y 1 .Condition 2 also holds for ψ(h) := h due to the independence of increments of Brownian motion.
instead of that of Lévy.
Equations (1.5) and (1.6) of Remark 1.5 are proved by direct and elementary means.We verify (1.5) to illustrate the basic idea.In light of Theorem 1.1 of Khoshnevisan and Shi (1998), it suffices to demonstrate that with probability one, lim sup We can assume dim M (E) > 0 and fix a 0 < δ < dim M (E).Then, there exist at least where {n k } is some sequence which tends to infinity.We note that for any constant η > 0 we can choose {n k } such that Elementary properties of Brownian motion can be used to show that for any 0 < β < δ and for all k 1, inf where lim k→∞ ε k = 0. Let By the Chebyshev's inequality, we have for any 1/2 < ε < 1,

Suppose we have chosen the sequence {n
This verifies (2.5) and completes our proof of (1.5) of Remark 1.5.

Proof of Theorem 1.7
As usual, extend the Sobolev norm to the space C[0, 1] of all continuous functions, by setting g H := ∞ for every continuous function g : [0, 1] → R 1 which is not in H .We will use the following form of Schilder's theorem (See Chapter 5 of Dembo and Zeitouni 1998): Let X be linear Brownian motion.For any Borel set denote by F o and F the interior and closure of F , respectively.Then Fix f ∈ H with f H > 0 and define R(γ) := min g 2 H , where the minimum is taken over the Then, Condition 1 with y 0 = 0 and y 1 = f 2 H follows from Schilder's theorem.Condition 2 follows with ψ(h) := h, due to the independence of the increments of X.Finally, Condition 3 follows from Lévy's uniform modulus of continuity.Note that for λ 0, For all η 0, define In particular, we can take η = 0, and note that R(0 This establishes Theorem 1.7.

Proof of Theorem 1.8
Let X d denote Brownian motion in R d , and for all t 0 and h > 0, define In order to verify Condition 1 we use the 'small-ball' estimate of Lévy (1953) (see also (3.2) in Ciesielski and Taylor (1962)), where ξ d,k are positive constants and q d,k are the positive zeros of the Bessel function J (d−2)/2 with q d,1 = q d .Note that as a → 0, the above probability is equivalent to we see that Condition 1 is satisfied with y 0 = 0 and any y 1 .As in our proofs of Theorems 1.1 and 1.6, Conditions 2 and 3 follow from the independence of increments and Lévy's modulus of continuity.Therefore the conclusion of Theorem 1.8 follows from (2.1), and Remarks 1.9 and 1.11 follow from Theorems 2.1 and 2.3, respectively.

Discrete Limsup Random Fractals
In this section, we establish a general result pertaining to "discrete limsup random fractals".This will be used in the proof of the results of Section 2.
Throughout, let us fix an integer N 1.For every integer n 1, let D n denote the collection of all hyper-cubes of the form [k for some γ > 0, where log 2 is the logarithm in base 2.
We refer to γ as the index of the limsup random fractal A.
Condition 5: a bound on correlation length.For each ε > 0, define Suppose that δ > 0 satisfies If Condition 5 holds for all δ > 0, then we say that Condition 5 * holds.
We are ready to state and prove the main result of this section.
Proof.First, we show that dim P (E) < γ implies that A ∩ E = ?, a.s.By regularization (see Mattila 1995), it suffices to show that whenever dim M (E) < γ, then A ∩ E = ?, a.s.Fix an arbitrary but small η > 0 such that dim M (E) < γ − η.By the definition of upper Minkowski dimension, we can find θ ∈]0, γ − η[, such that for all n 1, On the other hand, by Condition 4, for all n large enough, It follows from (3.1) and (3.2) that for each n 1 Since θ < γ − η, the Borel-Cantelli lemma implies that there exists a random variable n 0 , such that E ∩ A(n) = ?a.s.for all n n 0 .This shows that A ∩ E = ?, a.s.
We claim that for all n 1, the open set B(n) ∩ E is a.s.dense in (the complete metric space) E .If so, Baire's category theorem (see Munkres 1975) implies that E ∩ ∞ n=1 B(n) is dense in E and in particular, nonempty.Since A = ∩ ∞ n=1 B(n), the result follows.Fix an open set V such that V ∩ E = ?.It suffices to show that A(n) ∩ V ∩ E = ?for infinitely many n, a.s.Indeed, this will imply that B(n) ∩ V ∩ E = ?for all n a.s.; by letting V run over a countable base for the open sets, we will conclude that B(n) ∩ E is a.s.dense in E .
Thus fix an open set V such that V ∩ E = ?. Let N n denote the total number of hyper-cubes that N n 2 nγ 1 for infinitely many integers n.In other words, #(N) = ∞, where , where the sum is taken over all I ∈ D n such that I ∩ E ∩ V = ?.In words, S n is the total number of hyper-cubes I ∈ D n such that I ∩ V ∩ E ∩ A(n) = ?.We need only show that S n > 0 for infinitely many n.We want to estimate Fix ε > 0 and for each I ∈ D n , let G n (I) denote the collection of all J ∈ D n such that (i) J ∩ V ∩ E = ?, and (ii) Cov(Z n (I), Z n (J)) 6 εp 2 n .
If J ∈ D n satisfies (i) but not (ii), then we say that J is in B n (I).(The notation is meant to indicate 'good' and 'bad' choices of J.) Thus, Cov(Z n (I), Z n (J)).
For the remaining covariance, use the fact that all Z n (I)'s are either 0 or 1.In particular, Recalling the notation of Condition 5, we can deduce that Combining this with the Chebyshev's inequality, we obtain: By Conditions 4 and 5, there exists sequences a n and b n such that lim n a n = lim n b n = 0 and p n = 2 −nγ(1+an) and f (n, ε) 6 2 n(δ+bn) .Thus, by (3.3) and the inequality Since ε > 0 is arbitrary, we see that P(S n = 0) → 0 as n → ∞ in N.
By independence, there exists one null set outside which for all open hyper-cubes V of 'rational end-points' with V ∩ E = ?, we have In particular, dim H (A) = N − γ, a.s.
Proof.The right-hand inequality in (3.1) does not require condition 5 * , and can be verified by a direct first-moment calculation: By regularization, it suffices to prove that dim H (A ∩ E) 6 dim M (E) − γ a.s.
Let N n denote the total number of hyper-cubes I ∈ D n such that I ∩ E = ?. Define S n := Z n (I), where the sum is taken over all I ∈ D n such that I ∩ E = ?.Then, where ξ = dim M (E) and ε n → 0. Thus E n S n 2 −nθ < ∞ for any θ > ξ − γ.Finally, for any n 0 , the intersection A ∩ E has a cover consisting of S n intervals in D n for each n n 0 .By picking n 0 large, we see that the θ-dimensional Hausdorff measure of A ∩ E vanishes, whence (3.2) follows.
The left-hand inequality in (3.1) is not as easy to prove from scratch, but it follows from Theorem 3.1 by a co-dimension argument; similar arguments can be found, e.g., in Peres (1996b) or Khoshnevisan and Shi (1998).For later use, we state it in greater generality.
Proof.For α < dim H (E) − γ, let Υ α be a random closed set (independent of A) in the cube [0, 1] N , that has Hausdorff dimension N − α a.s., and satisfies P( set can be obtained, e.g., as the closed range of an N − α stable process if α > N − 2, and as a fractal-percolation limit set in general; see Hawkes (1971), Peres (1996a) and the references therein.With the latter choice, the intersection of Υ α with an independent copy of Υ β has the same distribution as Υ α+β for any α, β 0. Consequently, where the L ∞ norm is taken in the underlying probability space; see Peres (1996b).Let Υ α be a union of countably many i.i.d.copies of Υ α .Then P( s. in the product space; here, we used the fact that any analytic set of Hausdorff dimension > γ contains a compact set of Hausdorff dimension > γ.Therefore, dim H (A ∩ E) α a.s.Taking α → dim H (E) − γ completes our proof of the lemma and the corollary.

Remark 3.5
The sets Υ α used in the previous proof may be constructed as follows.Consider the natural tiling of the unit cube [0, 1] N by 2 N closed cubes of side 1/2.Let Ξ 1 be a random subcollection of these cubes, where each cube has probability 2 −α of belonging to Ξ 1 , and these events are mutually independent.At the k th stage, if Ξ k is nonempty, tile each cube Q ∈ Ξ k by 2 N closed subcubes of side 2 −k−1 (with disjoint interiors) and include each of these subcubes in Ξ k+1 with probability 2 −α (independently).Finally, define We record the next lemma for use in the next section.
Lemma 3.6 Let E ⊂ [0, 1] N be an analytic set.Then, for α 6 dim P (E), the sets Υ α defined above satisfy (3.4) for α 6 dim M (E).Let N n (E) denote the total number of hyper-cubes I ∈ D n that intersect E.
and therefore Thus the sum inside the expectation is finite a.s., whence dim M (E ∩ Υ α ) 6 β − α a.s., and ( As a consequence of Theorem 3.2, Corollary 3.3 and Lemma 3.4, the following result gives the Hausdorff dimension and packing dimension of the intersection of two independent limsup random fractals.

Corollary 3.7
Let A and A be two independent discrete limsup random fractals which satisfy Conditions 4 and 5 * with indices γ and γ respectively.Then, for any analytic set

Proofs of general theorems from Section 2 4.1 Proof of Theorem 2.1
The proof is divided into three steps.

Step One: the upper bound
Our strategy is to show that for all compact sets From this, we will deduce that By regularization, for all analytic sets and this will constitute the first step of our proof; see Mattila (1995) for details on regularization.
We now prove (4.1).Fix ρ > 2 and for all n 1 and all k ∈ Z N + , define Without loss of generality, we will prove the validity of (4.1) when the collection of all points of the form kρ −n such that (i) k ∈ Z N + satisfies k i 6 ρ n + 1 for all 1 6 i 6 N ; and (ii) I k,n ∩ E = ?.For brevity, write ξ := dim M (E).By definitions, there exists a sequence c n such that lim n c n = 0 and #G n 6 ρ nξ(1+cn) .Hence, for any γ ∈ (y 0 , y 1 ], P max t∈Gn max {m: where lim n d n = 0. We used Condition 1 in obtaining the last inequality.If we choose γ satisfying λ > γ > ξ = dim M (E), then we can pick ρ > 2 sufficiently close to 2 such that the sum (4.2) is finite.By the Borel-Cantelli lemma, lim sup Now we use Condition 3 to show (4.1).For any 0 < ε < λ − γ, h > 0 small and s ∈ E, there exist a positive integer n and t ∈ G n such that Furthermore, we can find a positive integer m such that such that 2 −n−1 6mρ −n 6 2 −n and |h − mρ −n | 6 ρ −n .Combining the triangle inequality, (4.3) and Condition 3, we see that for h > 0 small enough, or equivalently, for n large enough This proves (4.1).
Step Two: the lower bound By regularization, it suffices to show that for compact E ⊂ [0, 1] N with dim P (E) ∈ (y 0 , y 1 ], Since ψ is regularly varying of order 1, it follows that Condition 5 holds, with the same δ. Thus, if dim P (E) > γ > y 0 , then (4.4) holds.
Step Three: dimension estimates We can now complete our proof of Theorem 2.1.The results of steps one and two are equivalent to the following: This implies that if E = ∪ ∞ n=1 E n with dim P (E n ) < λ for all n 1, then P(Γ(λ) ∩ E = ?)= 0. Now suppose that the compact set E ⊂ [0, 1] N is not in {dim P < λ} σ .Let {U i } be a countable basis for the open sets in [0, 1] N .Define Then, E is compact, and every open set V that intersects E , satisfies dim P (E ∩ V ) λ.
(Otherwise, for some set U j from the basis, we would have ∅ = E ∩ U j ∈ {dim P < λ} σ whence E ∩ U j ∈ {dim P < λ} σ , a contradiction.)For n, m 1, denote By continuity of Y (t, h), the set U (n, m) is relatively open in E ; by the argument in step two above, U (n, m) is also dense in E .Since E is compact, Baire's theorem implies that the set is dense in E .This completes the characterization of hitting probabilities for Γ(λ).The remainder of our proof of Theorem 2.1 follows that of Corollary 3.3: let Y be an independent copy of Y .In analogy with the definition of Γ(λ), we can define for all λ > 0, Then, by Theorem 3.2 and the arguments of step two, P(Γ(λ) ∩ Γ (λ ) ∩ E = ?)= 1.Using the presented proof of Theorem 3.2 and applying our characterization of hitting probabilities, we can deduce now that dim P (Γ(λ) ∩ E) = dim P (E), a.s.

Proof of Theorem 2.3
If E is a countable union of Borel sets E n with dim P (E n ) < λ, it follows from Theorem 2.1 that Γ = (λ) ∩ E = ?a.s.Suppose that E is compact and not in {dim P < λ} σ .By Joyce and Preiss (1995), there exists a compact set E 0 ⊂ E that is not in {dim P < λ} σ and satisfies dim P (E 0 ) = λ.Using the argument in step three of §4.1, we can find a compact set E ⊂ E 0 , such that dim P (E ∩ V ) = λ for any open set V that intersects E .By Theorem 2.1, Γ(λ) ∩ E is dense in E a.s.On the other hand, Γ(λ + n −1 ) ∩ E = ?a.s.for any n 1.Thus, As this is Γ = (λ) ∩ E , the first part of the theorem is proved.Finally, the equality involving packing dimension follows as in Theorem 2.1.The asserted Hausdorff dimension estimate from below follows from Lemma 3.4, while the upper estimate follows from Theorem 2.1.

Fast Points for Gaussian Processes
In this section, we apply the general Theorems 2.1 and 2.3 to obtain information on the fast points of a large class of Gaussian processes with stationary increments.Throughout, suppose {G(t); t 0} is a mean zero Gaussian process with stationary increments such that for all s, t 0, We shall impose the following condition on the function σ.
Condition 6: There exist α ∈]0, 1[, a function L which is slowly varying at 0 and positive constants c 1 and c 2 such that c 1 s 2α L(s) 6 σ 2 (s) 6 c 2 s 2α L(s) for s 0 small enough.
(5.1) Furthermore σ 2 (s) is twice continuously differentiable and there exist positive constants c 3 and c 4 such that for s 0 small enough, dσ 2 (s) where ρ + := max{ρ, 0}, a := 2γ| log h| and Clearly, it suffices to show the existence of a function h → ψ(h) which is regularly varying at zero of order 1, such that uniformly for all t, s ∈ [0, 1] which satisfy |t − s| ψ(h), we have a 2 ρ → 0, as h → 0 + .
Let K(h) := σ 2 (h) (h 0) and use Taylor expansion about |t − s| to see that where 0 6 θ 1 , θ 2 61.By (5.3) it is easy to see that there exists a constant δ > 0 such that the function s 2α−2 L(s) is decreasing on [0, 2δ].Hence it follows from (5.1) and (5.2) that there exists a positive and finite constant c 5 such that for all t, s ∈ [0, 1] satisfying |t − s| 6 δ and all for some finite constant c 6 depending on θ only.On the other hand, by the continuity of K (s) there exists a positive and finite constant c 7 (which may depend on δ) such that for all t, s ∈ [0, 1] satisfying |t − s| δ and all h ∈ [0, δ/2] we have |ρ| 6 c 7 h 2−2α L(h) .
The following theorem shows that the rate of convergence of these approximations is completely determined by the gap dim P (E) − f 2 H .This theorem extends the results of Deheuvels and Mason (1998) (who considered E = [0, 1]); it also refines the one-dimensional case of Theorem 1.7.

Proof. Let
Then, for t ∈ R 1 + , h > 0 and γ > f 2 Theorem 3.3 of de Acosta (1983) states that: For every f H < 1 and r > 0, By taking λ = 2| log h| in de Acosta's theorem, we see that Condition 1 is satisfied with y 0 = f 2 H and any y 1 .Conditions 2 and 3 are also satisfied due to the independence of increments and Lévy's modulus of continuity.Since S(f,c) = Γ(λ(f, c)), the assertions of the theorem follow from Theorem 2.1.

Condition 4 : 1 n
where k ∈ Z N + is any N -dimensional positive integer.In words, D n denotes the totality of all N -dimensional dyadic hyper-cubes.Suppose for each integer n 1, {Z n (I); I ∈ D n } denotes a collection of random variables, each taking values in {0, 1}.By a discrete limsup random fractal, we mean a random set of the form A := lim sup n A(n), where,A(n) := I∈Dn:Zn(I)=1 I o ,where I o denotes the interior of I. Discrete limsup random fractals and some of their dimension properties can be found inOrey and Taylor (1974), in Deheuvels and Mason (1998) and in Dembo, Peres, Rosen and Zeitouni (1998).The goal of this section is the determination of hitting probabilities for a discrete limsup random fractal A, under some conditions on the random variables {Z n (I); I ∈ D n }. the index assumption.Suppose that for each n 1, the mean p n := E[Z n (I)] is the same for all I ∈ D n and that lim n→∞ log 2 p n = −γ,

Theorem 3 . 1
Suppose A = lim sup n A(n) is a discrete limsup random fractal which satisfies Condition 4 with index γ, and Condition 5 for some δ > 0.Then, for any analytic set E ⊂ R N + , Finally, P(S n > 0 i.o.) lim sup n→∞ P(S n > 0) = 1 .Next, we derive an extension of Theorem 3.1.Theorem 3.2 Suppose A is a discrete limsup random fractal which satisfies Conditions 4 and 5 * .On the same probability space, consider A , an independent discrete limsup random fractal which satisfies Conditions 4 and 5 * and has exponent γ .Then for any analytic set E ⊂ R N + satisfying dim P (E) > γ ∨ γ , we have P(A ∩ A ∩ E = ?)= 1.In particular, if dim P (E) > γ, then dim P (A ∩ E) = dim P (E), a.s.Proof.Consider the closed set E and the open sets B(n) of the described proof of Theorem 3.1, and let {B (n)} be the sequence of open sets corresponding to the limsup random fractal A .Then, Theorem 3.1 shows that for any open set V such that V ∩ E = ?, P(B(n

. 4 )
Choose δ > 0 such that γ + δ < dim P (E).Recall the notation of Section 3 and for all I ∈ D n , define π n (I) to be the element in I which is coordinatewise smaller than all other elements of I.For I ∈ D n , let Z n (I) denote the indicator function of the event (Y (π n (I), 2 (δ−1)n ) > γ).By Condition 1, the distribution of Z n (I) does not depend on the choice of I ∈ D n .Moreover, if we letp n := E[Z n (I)] for I ∈ D n , then lim n n −1 log 2 p n = −γ(1 − δ).In other words, we have verified Condition 4 with index γ(1 − δ), for the discrete limsup random fractal obtained from the Z n (I).Similarly, by Condition 2, for each ε > 0, there exists a regularly varying function ψ of order 1 at 0, such that whenever I, J ∈ D n satisfy |π n
for details.For any t ∈ R d + and h 0, let [t, t + h] denote the Cartesian product d i=1 [t i , t i + h].That is, [t, t + h] designates the hyper-cube of side h with 'left endpoint' t.Then, for any analytic set E ⊂ R d By reversing the order of sup and lim sup in Theorem 1.1, we obtain the following probabilistic interpretations of the upper and lower Minkowski dimensions of E, denoted dim M (E) and dim M (E), respectively; see Mattila (1995) for definitions.

Condition 1: stationarity and tail power law. For
and only if s i 6 t i for all 1 6 i 6 N .each t ∈ R N Given a set E ⊂ R d and λ > 0, we say that E / ∈ {dim P < λ} σ if E is not a union of countably many Borel sets E n with dim P Finally, by Lévy's modulus of continuity (cf.Orey and Taylor (1974)), we can deduce Condition 3.According to (2.1) and Theorem 2.1, Theorem 1.1 and Remark 1.2 both follow.This proves Theorem 1.1.
We conclude this subsection by sketching some proofs for the Remarks following Theorem 1.1.First, we prove(1.4)ofRemark 1.4.Take Y (t, h) to be |W [t, t + h]| 2 /(2h d | log h|) and use the modulus of continuity of Orey and Pruitt (1973, Theorem 2.1): It remains to show that if dim P (E) > γ + δ, then A ∩ E = ?, a.s.Indeed, suppose dim P (E) > γ + δ.By Joyce and Preiss (1995), we can find a closed E ⊂ E, such that for all open sets V , whenever E ∩ V = ?, then dim M (E ∩ V ) > γ + δ.It suffices to show that with probability one, A ∩ E = ?. Define the open sets B(n Since dim P (E) > γ + δ, Theorem 3.1 implies that there almost surely exists t ∈ E, such that Y (2 −n [t2 n ], 2 (δ−1)n ) γ for infinitely many n.In particular, Khoshnevisan and Shi (1998) work ofMarcus (1968).Standard estimates on the tails of Gaussian distributions reveal that Condition 1 holds with y 0 = 0 and any y 1 .To check Condition 2, we use the same argument as that in Section 4 ofKhoshnevisan and Shi (1998)to see that Bingham et al (1987)llows from Theorem 1.8.2 in Bingham et al (1987) that, without loss of generality, we may and will assume that L(s) varies smoothly near the origin with index 0.where L (n) (s) is the n-th derivative of L(s).It is known (cf.Bingham et al (1987)) that the slowly varying function L(s) can be represented byL(s) = exp θ(s) + h→0+ |G(t + h) − G(t)| 2σ 2 (h)| log h|= ( dim P (E)) 1/2 , a.s.

6 Rate of convergence in the functional LIL Theorem
Adler 1990) conclude that a 2 ρ → 0, as h → 0+ uniformly for all s, t ∈ [0, 1] with |t − s| ψ(h).This implies Condition 2. Finally, by Theorem 2.1, it suffices to check Condition 3. By standard entropy methods (seeAdler 1990), there exists a constant c 8 such that This fact, the triangle inequality and the first inequality in (5.2) together imply that Y (t, h) satisfies Condition 3. The theorem now follows from Theorem 2.1.As a canonical example, consider fractional Brownian motion β of index α, where α ∈]0, 1[ is fixed.In other words, (β(t); t 0) is a mean zero Gaussian process with β(0) = 0 andE[(β(t + h) − β(t)) 2 ] = h 2α .Clearly, this is a Gaussian process which satisfies the conditions of the above Theorem.In fact, similar considerations yield the following which completes Theorem 1.1 of Khoshnevisan and Shi (1998).