Random fractals generated by oscillations of the uniform empirical process

Statistics & Probability Letters 50 (2000) 285 – 291

Random fractals generated by oscillations of the uniform empirical process Zacharie Dindar ∗ L.S.T.A., UniversitÃe Paris VI, Paris, France Received December 1998; received in revised form April 2000

Abstract The Hausdor dimension of the set generated by exceptional oscillations of the uniform empirical process is studied. We c 2000 Elsevier Science correct a former result obtained by Deheuvels and Mason (1995, Ann. Probab. 23, 355 –387). B.V. All rights reserved MSC: primary 62G30; secondary 28A80 Keywords: Empirical process; Oscillations; Fractals

1. Introduction and statement of main result Pn Let {Un : n¿1} be i.i.d. uniform (0; 1) random variables. For each n¿1 set Fn (t) = n−1 i=1 1I{Ui 6t} and denote by n (t) = n1=2 (Fn (t) − t) the uniform empirical process. For any h ¿ 0 and t ∈ [0; 1] consider the increment functions of s ∈ [0; 1] deÿned by n (h; t; s) = n (t + sh) − n (t). Let {hn : n¿1} denote a sequence of constants such that 0 ¡ hn ¡ 1, hn ↓ 0; nhn ↑ ∞; nhn =log n → ∞ and (log(1=hn ))=(log log n) → ∞, and set bn = (2hn log(1=hn ))1=2 . Deheuvels and Mason (1992) have established a functional limit law showing the a.s. convergence in an appropriate set metric of {bn n (hn ; t; :): t ∈ [0; 1)} to the Strassen set S = {f: |f|H } (cf. Strassen, 1964). Here, for each function f on [0; 1], we set |f|H = R1 2 { 0 f˙ (s) ds}1=2 when f is absolutely continuous with Lebesgue derivative f˙ and f(0) = 0. We set |f|H = ∞ otherwise. Introduce the subsets of [0; 1) deÿned by L(f) = {t ∈ [0; 1): lim inf kb−1 n n (hn ; t; :) − fk = 0}; n→∞

L =

[

L(f):

f: |f|H ¿ ∗

Correspondence address: 45, rue Vineuse, 75016 Paris, France. E-mail address: [email protected] (Z. Dindar).

c 2000 Elsevier Science B.V. All rights reserved 0167-7152/00/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 0 ) 0 0 1 0 5 - X

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Deheuvels and Mason (1995) showed that L(f) and L constitute random fractals and evaluated their Hausdor dimension (see e.g. Falconer, 1990), denoted hereafter by dim(:). The aim of this paper is to reÿne their results by proving the following limit law. Theorem 1.1. We have; with probability 1; for all f∈ S and 0661; dim L(f) = 1 − |f|2H

and

dim L = 1 − 2 :

(1.1)

Our proofs are based on an adaptation of the arguments of Deheuvels and Lifshits (1997) in the setting of empirical processes as considered by Deheuvels and Mason (1994,1995). We note that Theorem 1.1 improves upon Theorem 1.1 and Theorem 2:1 of Deheuvels and Mason (1995) in the sense that it shows the existence of an event of probability 1 such that (1.1) holds for all f ∈ S and 0661, whereas the above-mentioned authors prove this result for events depending upon f and . The proof of the theorem is given in the next section, together with additional details and references. 2. Proofs and complements Proof of (1.1) reduces to show with probability 1 that dim L 61 − 2 for any  ∈ [0; 1] and dim L(f)¿ R1 2 1 − 0 f˙ (s) ds for any f ∈ S. Since the proof of the upper bound can be derived from Deheuvels and Mason (1995), it suces to prove the lowerbound. For this sake, following notations and lemmas are needed. R1 2 Consider f ∈ S such that 0 ¡ 2 := 0 f˙ (s) ds ¡ 1 and deÿne a constant Á ∈ (0; 2−1 min(2 ; 1 − 2 )): Introduce the sequence {fj : j¿1}, elements of S such that limj → ∞ kf − fj k = 0 and deÿne the sequence {Fj : j¿1} of S such that for any j¿1, g ∈ Fj , fj ∈ Fj , we have j := #Fj ¡ ∞, Á ¡ |g|2H and |fj |H 6|f|H [here and elsewhere, #A denotes the cardinality of A]. Next, introduce the sequence {j : j¿1} of positive constants satisfying j exp(−(log j )2 ) ¡ 2−j and deÿne mj = #{i: i¿0; [tj (i); tj (i + 1)] ⊆ [0; 1]} where tj (i) = ihj for any j¿1 [here and elsewhere, buc6u ¡ buc + 1 denotes the integer part of u]. For any g bounded on [0; 1],  ¿ 0 and j¿1, deÿne Nj (g; ) = #Wj (g; ) where Wj (g; ) := {tj (i): 16i6mj , −1 kb−1 j j (hj ; tj (i); :) − gk ¡ }. Next, we set Nj (g; ; I ) := #{i: 16i6mj ; tj (i) ∈ I; kbj j (hj ; tj (i); :) − gk ¡ } for any I ⊆ [0; 1]. Lemma 2.1. ∀ ∈ (0; 1) and Â=Â() ¡ 1; there exists with probability 1 a j0 (; Â) ¡ ∞ such that; ∀j¿j0 (; Â) 1=2 and 16i6mj ; for tj (i) ∈ Wj (fj ; ) and t ∈ [tj (i); tj (i)+Âhj ]; kb−1 ()+kf−fj k: j j (hj ; t; :)−fk6(3=2)+2Â Proof. This last result can simply be derived from Stute (1982) and his classic oscillation moduli result of n (see also Shorack and Wellner, 1986, p. 542). For any h ∈ (0; 1), s ∈ [0; 1], t¿0 and constants n¿1, deÿne the quantity Ln (h; t; s) = n−1=2 {n (t + sh) − n (t) − nsh} where {n (t): t¿0} is a sequence of right continuous Poisson process with E(n (t)) = nt. We next deÿne the quantity Nj0 (g; ; I ) = #{i = 1; : : : ; mj : tj (i) ∈ I; Xi (g) = 1} where Xi (g) := 1{kb−1 are i.i.d. and follow a Bernoulli law with parameter pj (g; ) := P(X0 (g) = 1).  L (h ;tj (i);:)−gk¡} j

j

j

Lemma 2.2. ∀ ¿ 0; there exists with probability 1; a j1 () such that ∀j¿j1 (); uniformly over g ∈ S with |g|2

0 ¡ |g|2H ¡ 1; we have pj (g; )¿4−1 hj H : Proof. Set  ¿ 0 and consider g ∈ S with 0 ¡ |g|2H ¡ 1. Let {(t): t¿0} be a standard Poisson process and make use of the strong approximation results of KomlÃos et al. (1975a,b, 1976) (see e.g. Deheuvels and Mason,

Z. Dindar / Statistics & Probability Letters 50 (2000) 285 – 291

287

1994, p. 85) which allow the construction of a standard Wiener process {W (t): t¿0} on the same probability space. This combined with a large deviation result of Deheuvels and Lifshits (1993) (see also Lifshits, 1995, p. 251) imply for all large j, pj (g; ) ¿ P(kW k62−1=2 (log(1=hj ))1=2 ) exp(−log(1=hj )|g|2H ) n  o (2j hj log(1=hj ))1=2 − C3 log(j hj ) ; −C1 exp −C2 2 where C1 ; C2 ; C3 are absolute constants. A few calculations easily entail Lemma 2.2. Next, deÿne mj (I )=#{i: 06i6mj ; [tj (i); tj (i +1)] ⊆ I } and observe that, for any disjoint union E ⊆ [0; 1] of −1 −1 intervals with lengths greater than L¿3hj , we have for all j¿1, |E|(3hj )−1 6(h−1 j −2L )|E|6mj (E)6|E|hj : Lemma 2.3. ∀ ¿ 0; ∀ ∈ (0; 1 − 2 ); ∀ ∈ (0; 1); there exists a.s. a j2 (; ; ) ¡ ∞ such that; ∀j¿j2 (; ; ); uniformly over g ∈ Fj ; we have |Nj (g; ; E) − mj (E)pj (g; )|6mj (E)pj (g; );

∀E ⊆ [0; 1];

where E is a disjoint union of intervals with length L¿h1− j

2

−

(2.1)

.

Proof. Set g ∈ S with 0 ¡ |g|H ¡ 1,  ¿ 0,  ∈ (0; 1 − 2 ) and  ∈ (0; 1). For any j¿1, Nj (g; ; :) and mj (:) 2 − will be sucient are additive set functions, hence the proof of (2.1) for an interval with length |I |¿h1− j 0 0 0 0 for our needs. Fix  and  such that 0 ¡  ¡  and 0 ¡  ¡  and let J denote an interval of the form 2 0 −0 c. Set K(j) := bhj − c and suppose that (2.1) holds [tj (i); tj (i +k(j))) with 06i6(3=2)mj and k(j) := bh− j for E = J ,  = 0 . Then, deÿne K := K(I; j) = b|I |=h1− j

2

−0

c and observe that K(I; j)¿K(j). For any j large SK−2 SK+2 enough, choose K +2 disjoint intervals J1 ; : : : ; JK+2 of the form [tj (i); tj (i+k(j))) with ‘=1 J‘ ⊆ I ⊆ ‘=1 J‘ : 2 −0 for ‘ = 1; : : : ; K + 2. Therefore, ultimately in j, we get Moreover, |J‘ | ∼ h1− j (K+2 ) K+2 X X Nj (g; ; J‘ )6(1 + 0 ) mj (J‘ ) pj (g; ) Nj (g; ; I ) 6 ‘=1

‘=1

6 (1 + 0 )(K + 2)(K − 2)−1 {h−1 j |I |}pj (g; ) 6 (1 + )mj (I )pj (g; ): Likewise, a similar argument will prove with probability 1 that, for all j large enough and uniformly over g ∈ Fj , Nj (g; ; I )¿(1 − )mj (I )pj (g; ). Next, it suces to prove (2.1) when E is replaced by J of the form [tj (i); tj (i + k(j))) with 06i6(3=2)mj . To this purpose, combine Lemma 2.2, Lemma 3:1 of Deheuvels and Mason (1992), with a strong approximation result for binomial distribution (see e.g. Deheuvels and Mason, 1995), to prove that 0 2 h−1 j P(Nj (g; ; J ) ¿ (1 +  )mj (J )pj (g; ))6C1 exp(−(log j ) ): P∞ −1 This entails j=1 j hj Qj ¡ ∞ and the Borel–Cantelli lemma implies with probability 1, that for all j suciently large we have uniformly over g ∈ Fj ,

Nj (g; ; J )6(1 + 0 )mj (J )pj (g; ): Likewise, we prove with probability 1 that for all j suciently large and uniform over g ∈ Fj , Nj (g; ; J )¿(1− 0 )mj (J )pj (g; ): This concludes the proof of (2.1) when E = I and hence E is a disjoint union of intervals 2 − . with length L¿h1− j

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Towards the aim of proving with probability 1 that dim L(f)¿1−|f|2H for any f ∈ S, we use the following key result from Deheuvels and Mason (1995, p. 369) (see also Orey and Taylor, 1974). T∞ Lemma 2.4. Let K ⊆ [0; 1] be such that K := m=1 Em ; where E1 ⊇ · · · ⊇ Em ⊇ · · · for m = 1; 2; : : : ; and S Mm Em = k=1 Im; k ; with {Im; k : 16k6Mm } being; for each m¿1; a collection of disjoint closed subintervals of [0; 1] such that max16k6Mm |Im; k | → 0 and Mm → ∞ as m → ∞. Assume that there exist two constants
¿ 0 and d ¿ 0 such that the following property holds. For every interval I ⊆ [0; 1=2] with |I |6
; there exists a constant m(I ) such that for all m¿m(I ); Mm (I ) := #{Im; k ⊆ I : 16k6Mm }6d|I |c Mm :

(2.2)

Then we have; dim K¿c. The following arguments are devoted to this particular construction. such that for any m¿1, we have Let {m : m¿1} and {m : m¿0}Qdenote two sequences of constants P∞ ∞ 0 ¡ m ¡ 2−1 , 0 = 0, m ¿0 with m=1 (1 + m )2 (1 − m )−2 62 and m=1 m 6Á=3. Next, we choose two decreasing sequences of constants {m : m¿1} and {Âm : m¿1} verifying 1 ¡ 2−1 , m ↓ 0, and the following technical conditions. (1i) Âm 6min(Â(m ); m2 =16); where Â(:) was introduced in Lemma 2.1, 2 2 (1ii) 2Âm1− −Á 6min(m ; m+1 ); (Âm =(1 − Âm ))2(1− )=3 6(1=6)m+1 ; 2 (1iii) (1 − Âm ) −1 6(1 + (5=6)m+1 )=(1 + (1=2)m+1 ): For any m¿1, we choose j0 (m ; m ; m ) such that for all j¿j0 (m ; m ; m ) Lemmas 2.1–2.3 hold together. The sets Em are constructed by an induction argument. Set j0 = 1, L0 = 2−1 and for any m¿1, set Lm = Âm hjm and L∗m−1 = Lm−1 − Lm . Next, for any m¿1, we choose jm with jm ¿ max{jm−1 ; j0 (m ; m ; m )} and such that the following hold. 2 1−2 −m 6L∗m−1 6Lm−1 ; (2i) 3Lm ¡ 3hjm 6h1− jm 6hjm ∗ (2ii) Lm−1 =Lm−1 61 + (1=2)m ; (2iii) 1 − m 61 − 2hjm =L∗m−1 ¡ 1; 2 +

(2iv) 2hjmm =hjm−1m−1 6Âm for m¿2; (2v) 6hjmm 6min(Âm ; Âm Mm−1 L∗m−1 ). ∗ Deÿne M0 = 1 and set for any m¿1, Mm := Njm (fjm ; m ; Em−1 ) with ! Mm Mm [ [ ∗ ∗ Im; k Im; k ; resp: Em = Em = k=1

(2.3)

k=1

∗ ∗ where Im; k (resp. Im; k ) is an interval of the form [tjm (k); tjm (k) + Lm ] (resp. [tjm (k); tjm (k) + Lm ]). Simply use ∗ ∗ ). (2i) to verify that the intervals Im; k (resp. Im; k ) are disjoints and for any m¿1, Em ⊆ Em−1 (resp. Em∗ ⊆ Em−1 We next prove that the induction process carries from m − 1 to stage m by showing that Mm ¿1 for any m¿1. |f |2H +m

Lemma 2.5. We have M0 L∗0 ¿3−1 : Moreover; if Mm−1 ¿1 for some m¿1; then Mm L∗m ¿4−1 hjmjm

:

Proof. First inequality is obvious from (2i). Suppose that {Mk ; jk ; Ek ; Ek∗ } are deÿned for any k ∈ {0; : : : ; m−1} ∗ , L = L∗m−1 Mm−1 and and that Mm−1 ¿1. Combining (2iii), Lemma 2.3 with  = m ,  = m ,  = m , E = Em−1 ∗ ∗ Lemma 2.2 with E = Em−1 , L = Lm−1 , we get ∗ −1 (1 − m )2 h−1 jm pjm (fjm ; m )6Mm (Mm−1 Lm−1 ) :

(2.4)

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Next, we simply combine (2ii), (2.4), Lemma 2.1 with  = m and j = jm to obtain |f |2

6−1 Âm Mm−1 L∗m−1 hjmjm H 64Mm L∗m : This, in combination with (2iv) entails Lemma 2.5. 2

)=3 By Lemma 2.5 and deÿnition of m and Âm , we get Mm ¿(64=4)h−2(1− . Since M0 = 1, we obtain Mm ¿1 jm for all m¿0. Therefore, the existence of {Mm ; jm ; Em∗ ; Em } for any m¿0 is established. In the sequel, we prove the existence of c, d and
of Lemma 2.4 for any interval I ⊆ [0; 1=2] with |I |6
. Proof of (2.2) is achieved considering several cases. (a) Case 1: For m¿1, assume that I ⊆ Im−1; k0 for some k0 ∈ {1; : : : ; Mm−1 }. 2 −m . Applying Lemma 2.3 with  = m ,  = m ,  = m , E = I , L = |I | and Case 1a: Suppose that |I |¿h1− jm j = jm , combined with Lemma 2.5, (2.4) and deÿnition of Em entail for any m¿2 Mm (I ) 6(1 + m )(1 − m )−2 |I |(Mm−1 L∗m−1 )−1 (2.5) Mm 2 (2.6) 64(1 + m )(1 − m )−2 |I |1− −m−1 : 2

For m = 1, M0 L∗0 ¿3−1 and (2.5) entails M1 (I )=M1 63(1 + 1 )(1 − 1 )−2 |I |1− . 2 −m and choose an interval I ⊆ I 0 ⊆ [0; 1=2] such that |I 0 | = Case 1b: Suppose that 2−1 Âm hjm 6|I |6h1− jm

h1− jm

2

−m

. Apply (2.5) with I 0 replacing I , combined with Lemma 2.5 and (2iv) to obtain for any m¿2

2 2 Mm (I ) −2m m − −m−1 6 4(1 + m )(1 − m )−2 h1− hjm hjm−1 jm Mm

6 4(1 + m )(1 − m )−2 |I |1−

2

−2m

:

(2.7) −2

1−2 −21

. For m = 1 simply use (2v) to obtain M1 (I )=M1 6(1 + 1 )(1 − 1 ) |I | Case 1c: Finally suppose |I | ¡ (1=2)Âm hjm . Recall that for any k = 1; : : : ; Mm , |Im; k | = Lm = Âm hjm to obtain for any m¿1, Mm (I )=Mm = 0. (b) Cases 2 and 3: We now turn to the case where I is not necessarily a subset of Im−1; k , k = 1; : : : ; Mm−1 . For any m¿1, let H (m) denote the statement that for any J ⊆ [0; 1=2], (m  ) m Y 1 + i 2 X 2 Mm (J ) 64 i : (2.8) |J |1− −2
m where
m := Mm 1 − i i=1

i=1

From (2.5) and (2.6), observe that (2.8) is veriÿed for I ⊆ Im−1; k0 . As a sequel, we prove that H (m) is true for any m¿1 with the help of the following lemma. Lemma 2.6. Let m¿2. Suppose that H (m − 1) holds and assume that J ⊆ [0; 1=2] is an interval with endpoints belonging to the set {tjm−1 (i); 16i6mjm−1 }. Then; we have (m  2 ) 2 1 + (1=2)m Y 1 + i Mm (J ) 64 (2.9) |J |1− −2
m−1 : Mm 1 + m 1 − i i=1

Proof. Assume that J has endpoints belonging to {tjm−1 (i); 16i6mjm−1 }. Therefore, J ∩ Em−1 is a disjoint union of Mm−1 (J ) intervals Im−1; k such that Im−1; k ⊆ J for 16k6Mm−1 . Therefore, X Mm (Im−1; k ) Mm (Im−1; k ) Mm (J ) = 6Mm−1 (J ) max : 16k6Mm−1 Mm Mm Mm k: 16k6M ; m−1

Im−1; k ⊆ J

Now, combine (2.5), (2ii), H (m − 1) and the last inequality to obtain (2.9).

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(b1) Case 2: Suppose that I veriÿes I ∩ Im−1; k0 6= ∅ for some k0 ∈ {1; : : : ; Mm−1 }. Let I 0 denote the smallest closed interval which contains every Im−1; k such that I 0 ∩ Im−1; k 6= ∅ with k ∈ {1; : : : ; Mm−1 } and deÿne the set I 00 = I ∩ I 0 . Case 2a: Let m¿2 and suppose I 0 =Im−1; k0 i.e. I 0 contains at most one interval Im−1; k with k ∈ {1; : : : ; Mm−1 }. In this case, I 00 veriÿes assumptions of Case 1a, which entail for any m¿2 (m  ) Y 1 + i 2 2 Mm (I ) Mm (I 00 ) = 64 (2.10) |I |1− −2
m : Mm Mm 1 − i i=1

Case 2b: I 0 contains at least two intervals Im−1; k with k ∈ {1; : : : ; Mm−1 }. Therefore, I 0 may be written as a disjoint union of two intervals I1 and I2 such that I1 = Im−1; k1 for k1 ∈ {1; : : : ; Mm−1 }, and I2 has endpoints belonging to {tjm−1 (i); 16i6mjm−1 }. The proof of statement H (m) for any m¿1, is made by induction. We suppose that H (m − 1) is true for m¿2. A few calculations and (1ii) show that  1−2 −2m 1 + m |I1 | Mm (I1 ) 1−2 −2m 64 |I | Mm (1 − m )2 |I2 | − Lm−1 (m  2 ) 2 (1=6)m Y 1 + i 64 |I |1− −2
m : 1 + m 1 − i

(2.11)

i=1

Next, combining Lemma 2.6, (1iii), with a few computation we get (m  1−2 −2
m 2 )  1 + (1=2)m Y 1 + i |I2 | Mm (I2 ) 1−2 −2
m 64 |I | Mm 1 + m 1 − i |I | i=1

(m  2 ) 2 1 + (5=6)m Y 1 + i 64 |I |1− −2
m : 1 + m 1 − i

(2.12)

i=1

The combination of (2.11) and (2.12) entail H (m). Since H (1) is true, we conclude that (2.8) is true for any m¿1 and I of Case 2b. (b2) Case 3: If for some m¿1 and any k ∈ {1; : : : ; Mm−1 }, I ∩ Im−1; k = ∅, we have Mm (I )6Mm (I 0 ) = 0 and H (m) is veriÿed for any m¿1. Finally, in view of (2.10) – (2.12) we conclude that the statement H (m) is true for any m¿1 and therefore, Lemma 2.4 is veriÿed for any I ⊆ [0; 1=2], with
= 1=2, c = 1 − 2 − Á, d = 8 and m(I ) = 1. Using Lemma 2.1 applied to  = m , Â = Â(m ), j = jm easily entails that K ⊆ L(f), and therefore, dim K6dim L(f): Recall that C0 [0; 1] denotes the set of all continuous functions f on [0; 1] such that f(0) = 0 and that H designs the reproducing kernel Hilbert space of the Wiener measure PW on C0 [0; 1]. Let Á ∈ (0; 1=4) and deÿne the set SÁ = {f ∈ S: 2Á6|f|2H 61 − 2Á} ⊆ S: Since H is separable (see e.g. Adler, 1990, p. 65; Lifshits, 1995, p. 86), choose G = {gj : j¿1} ⊆ S such that G is dense in (S; |:|H ) with for any f ∈ SÁ , M(f; Á) 6= ∅ and limj → ∞ min16k6j; k ∈ M(f; Á) |f − gk |H = 0. Here, we set M(f; Á) := {j¿1: |gj |H 6|f|H ; Á ¡ |gj |2H } for f ∈ SÁ . These last conditions and the compactness of S on (C0 [0; 1]; k:k) entail that G is dense in (S; k:k). Besides, for each sequence j ↓ 0, there exists integers 1 62 6 · · · such that for any j¿1 and f ∈ SÁ , M(f; Á) ∩ {1; : : : ; j } 6= ∅ with min16i6j ;i ∈ M(f;Á) kf − gi k ¡ j : Now, set Fj = {gi : 16i6j }. For any f ∈ SÁ , let {fj : j¿1} denote the sequence of functions such that ∀j¿1, fj = gi for gi ∈ Fj fulÿlling kf − gi k ¡ j and i ∈ M(f; Á) ∩ {1; : : : ; j }: Hence, conditions on sets

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Fj described on paragraph 2 follows easily. Making use of (2.2) with
= 1=2, c = 1 − 2 − Á, d = 8 and m(I ) = 1, we obtain with probability 1, dim − |f|2H − Á for any f ∈ SÁ : Next, choose a sequence SL(f)¿1 Án {Án : n¿1} decreasing to 0 and remark that S = S \ {0} to obtain with probability 1 dim L(f)¿1 − |f|2H , ∀f ∈ S \ {0} with |f|H 6= 1: The conclusion of the proof follows easily. References Adler, R.J., 1990. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institution of Mathematics and Statistics Notes, Monograph Series, Vol. 12. Hayward, CA. Deheuvels, P., Lifshits, M.A., 1993. Strassen-type functional laws for strong topologies. Probab. Theory Related Fields 97, 151–167. Deheuvels, P., Lifshits, M.A., 1997. On the Hausdor dimension of the set generated by exceptional oscillations of a Wiener process. Studia Sci. Math. Hungar. 33, 75–110. Deheuvels, P., Mason, D.M., 1992. Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20, 1248–1287. Deheuvels, P., Mason, D.M., 1994. Random fractals generated by oscillations of processes with stationary and independent increments. Probab. Banach Spaces, Sandjberg 9, 73–89. Deheuvels, P., Mason, D.M., 1995. On the fractal nature of empirical increments. Ann. Probab. 23, 355–387. Falconer, K.J., 1990. Fractal Geometry. Mathematical Foundations and Applications. Wiley, New York. KomlÃos, J., Major, P., TusnÃady, G., 1975a. Weak convergence and embedding. Colloq. Math. Soc. JÃanos Bolyai 11, 149–165. KomlÃos, J., Major, P., TusnÃady, G., 1975b. An approximation of partial sums of independent r.v.’s and the sample d.f. I. Z. Wahrsch. Verw. Gebiete 32, 111–131. KomlÃos, J., Major, P., TusnÃady, G., 1976. An approximation of partial sums of independant r.v.’s and the sample d.f. II. Z. Wahrsch. Verw. Gebiete 34, 33–58. Lifshits, M.A., 1995. Gaussian Random Functions. Mathematics and its Applications, Vol. 322. Kluwer, Dordrecht. Orey, S., Taylor, S.J., 1974. How often on a Brownian path does the law of the iterated logarithm fail?. Proc. London Math. Soc. 28, 174–192. Shorack, G.R., Wellner, J.A., 1986. Empirical Processes with Applications to Statistics. Wiley, New York. Strassen, V., 1964. An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3, 211–226. Stute, W., 1982. The oscillation behaviour of empirical process. Ann. Probab. 10, 86–107.