Statistics and Probability Letters

The aim of this paper is to prove that wavelet leaders allow to get very fine properties of the trajectories of the Brownian motion: we show that the three well-known behaviors of its oscillations, namely to be ordinary, rapid and slow, are also present in the behavior of the size of its wavelet leaders.


Introduction
Let f : R → R be a continuous function. In order to quantify its regularity around some fixed point t ∈ R, it is natural to try to determine as sharply as possible the asymptotic behavior of the quantity Osc(f , I(t, ρ)) = sup t ′′ ,t ′ ∈I(t,ρ) when ρ → 0 + . This quantity is called the oscillation of f on I(t, ρ).
In the last decades, wavelet methods have become a very powerful tool to finely study regularity properties of functions (Meyer and Salinger, 1995;Jaffard, 2004) and to obtain numerical methods to study real-life signals (see among others Jaffard, 2004;Arneodo et al., 1995;Jaffard et al., 2007;Lashermes et al., 2008;Esser et al., 2017;Deliège et al., 2017). A compactly supported function ψ ∈ L 1 (R) whose first moment vanishes, i.e. ∫ R ψ(x) dx = 0 , is called a compactly supported wavelet. Let N be a positive integer such that the support of ψ is included in [−N, N]. The wavelet coefficients c J,K , (J, K ) ∈ N × Z, of the function f are defined by In what follows, we will use the notation c λ to denote the wavelet coefficient c J,K , where λ is the dyadic interval λ = λ J,K = [2 −J K , 2 −J (K + 1)), which provides the location of the wavelet ψ(2 J · −K ). We denote by Λ the set of all dyadic intervals of R and for any J ∈ N, we denote by Λ J the set of all dyadic intervals of size 2 −J . Amplitudes of wavelet coefficients c λ ′ located in some fixed interval λ (that is λ ′ ⊆ λ) can be very fluctuating from one scale to another. In order to avoid such a drawback, the so-called wavelet leaders, which among other things offer the advantage of stability, have been introduced in Jaffard et al. (2007). For every λ J,K ∈ Λ, the wavelet leader d λ J,K of f is defined by Remark 1.1. In a probabilistic framework, the supremum on N(λ J,K ) appearing in (5) can create correlation between wavelet leaders, even if it does not exist between wavelet coefficients. It might seem more natural to consider only the dyadic intervals λ ′ ⊆ λ J,K in the definition of the wavelet leaders so that the supremum at a given scale is taken on nonoverlapping intervals. Let us point out that the methodology of our article can easily be adapted to this setting. However we will work with the classical wavelet leaders defined as in (5) since they provide a characterization of pointwise regularity of functions (Jaffard, 2004).
Note that there exists a link between the notions of oscillation and wavelet leaders. First, for any t ∈ R and J ∈ N, let λ J (t) denote the unique dyadic interval of size 2 −J that contains t, and let d J (t) be the corresponding wavelet leader. From (4) and (1) , one has where c 0 = ∥ψ∥ L 1 (R) . As a consequence, one has that Conversely, Jaffard (1998) has shown that, under some general assumptions on f , the reverse inequality holds up to a logarithmic factor, i.e.
for some constant c 1 > 0. It can be derived from (6) and (7) that there is a priori a loss of information if one considers wavelet leaders instead of oscillations.
In the present paper, we treat the particular case of Brownian motion. Despite Inequality (7), a first numerical work  has led to the idea that, in this case, working with wavelet leaders instead of oscillations does not imply a logarithmic loss. We confirm in this paper that wavelet leaders are precise enough to reflect very fine properties of the trajectories of Brownian motion, namely the coexistence in them of slow, rapid and ordinary points. Section 2 contains useful recalls on the Brownian motion and the statement of the main result (Theorem 2.3). Section 3 is devoted to the proof of this result.

Brownian motion and statement of the main result
The Brownian motion is the unique real-valued centered Gaussian process B = {B(t)} t∈R defined on a probability space (Ω, A, P) with independent and stationary increments satisfying B(t) − B(s) ∼ N (0, |t − s|) for any t, s ∈ R and such that almost surely, B has continuous paths and B(0) = 0. From now on, if the value of B(t) has to be explicitly associated to an elementary event ω ∈ Ω, the notation B(t, ω) will be used. The following theorem (Kahane, 1985) summarizes some properties of the regularity of B: it shows the existence of three different possible behaviors of the oscillations of the Brownian motion.
Theorem 2.1. There exists an event Ω * ⊆ Ω of probability 1 such that for every ω ∈ Ω * and every non-empty open interval Moreover, for every ω ∈ Ω * , almost every t ∈ R is an ordinary point of B(·, ω), and for every t ∈ R, From now on, we assume that c J,K (resp. d J,K ), (J, K ) ∈ N × Z, represent the wavelet coefficients (resp. the wavelet leaders) of the Brownian motion B. Theorem 2.1 motivates the following definition.
Definition 2.2. Let ω ∈ Ω and t ∈ R. We say that Our main result stated below is a reminiscent of Theorem 2.1 in our context.

Theorem 2.3.
There exists an event Ω * 0 ⊆ Ω of probability 1 such that for every ω ∈ Ω * 0 and every non-empty open interval is a leader-ordinary point, t r (ω) is a leader-rapid point and t s (ω) is a leader-slow point of B(·, ω). Moreover, for every ω ∈ Ω * 0 , almost every t ∈ R is a leader-ordinary point of B(·, ω).
Remark 2.4. Theorem 2.1 together with the characterization (6) implies that for any ω ∈ Ω * , if t is an ordinary point of B(·, ω) (resp. a rapid or a slow point), then the upper limit appearing in the definition of the leader-ordinary points of B(·, ω) (resp. leader-rapid or leader-slow points) is bounded from above. The main difficulty is then to obtain the bound from below.

Proof of Theorem 2.3
The definition of B and Equalities (3) and (4) imply that, for every (J, Abry et al. (2003) for more details). Moreover, since the increments of B are independent and since the support of ψ is included in [−N, N], Equality (4) shows that the wavelet coefficients c J 1 ,K 1 , . . . , c Jn,Kn are independent as soon as In particular, the coefficients c J,K and c J,K ′ are independent if |K ′ − K | ≥ 2N. This leads us to define the following condition.
In the sequel, we fix an arbitrary sequence { ε λ : λ ∈ Λ } of real-valued N (0, 1) random variables verifying the property of Remark 3.2. Moreover, if (J, K ) ∈ N × Z and λ = λ J,K , then for any m ∈ N, we denote by S J,K ,m or S λ,m the finite set of cardinality 2 m whose elements are the dyadic intervals of scale J + m included in λ J,K . The following lemma allows to obtain a general lower bound for the size of the wavelet leaders of the Brownian motion.

Lemma 3.3.
There exists an event Ω * 1 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 1 and every t ∈ R, one has Proof. Let us fix (J, K ) ∈ N × Z. For any m ∈ N and any S ∈ S J,K ,m , there is a unique finite sequence (I n ) 0≤n≤m of dyadic intervals which is decreasing in the sense of the inclusion and satisfies I 0 = λ J,K , I m = S and I n ∈ S J,K ,n for all n ∈ {1, . . . , m}. Next, we consider the sequence (T n ) 1≤n≤m of dyadic intervals constructed as follows: for every n ∈ {1, . . . , m}, T n is the unique dyadic interval of S J,K ,n such that I n−1 = T n ∪ I n . Note that, since the sequence (I n ) 0≤n≤m is decreasing, this construction ensures that the intervals (T n ) 1≤n≤m are pairwise disjoint. Let us also note that, for every n ∈ {1, . . . , m}, one has T n ∈ N ( . Moreover, for every n ∈ {1, . . . , m}, there is a dyadic interval T ′ n ∈ S Tn,⌊log 2 (N)⌋+2 such where T ′ n = λ jn,kn . Consequently, by assumption, the corresponding Gaussian random variables (ε T ′ n ) 1≤n≤m are independent.
In the sequel, the set {T ′ n : 1 ≤ n ≤ m} is denoted by T ′ J,K ,m (S).
Let c 0 = 2 −3/2 √ π. Then, the probability p 0 that a real-valued N (0, 1) random variable belongs to the interval (−c 0 , c 0 ) belongs to (0, 1/2). For all S ∈ S J,K ,m , we denote by B J,K ,m (S) the Bernoulli random variable defined as Notice that, using the definition of p 0 and the independence property of the random variables ε T ′ for T ′ ∈ T ′ have a probability equal to 1.
Let us now consider ω ∈ Ω * 1 and t ∈ R. We fix J ∈ N and K = ⌊2 J t⌋, so that λ J,K = λ J (t). Since for every m ∈ N, G J,K ,m takes values in {0, . . . , 2 m }, (13) implies that there are infinitely many m such that B J,K ,m (S) = 0 for every S ∈ S J,K ,m , i.e. using (12), there exists T ′ ∈ T ′ J,K ,m (S) such that |ε T ′ | ≥ c 0 . In particular, we have this result for S = λ J+m (t). In this case, T ′ ∈ S λ,⌊log 2 (N)⌋+2 with λ ∈ N(λ J+m (t)). Hence (11) is satisfied. □ Proposition 3.4. Let Ω * 1 denote the event of probability 1 of Lemma 3.3. For every ω ∈ Ω * 1 and every t ∈ R, one has Proof. Note that using (5) and (10), one has As we will see in the proof of Theorem 2.3, this result allows to get the existence of leader-slow points. Let us now focus on leader-ordinary points. First, let us recall the following classical lemma which provides asymptotic estimates on the tail behavior of a standard Gaussian distribution.
Lemma 3.5. If ε is a real-valued N (0, 1) random variable, then one has Lemma 3.6. There exists an event Ω * 2 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 2 and almost every t ∈ R, one has Proof. Within this proof, we will use the same notations as in the proof of Lemma 3.3. Let us fix t ∈ R. If J ∈ N, we set K = ⌊2 J t⌋. For every m ∈ N, we consider the dyadic interval S = λ J+m (t) ∈ S J,K ,m and the associated sequence (T ′ n ) 1≤n≤m of dyadic intervals. Next, we set } By construction, the Gaussian random variables ε T ′ n , 1 ≤ n ≤ m, are independent. Therefore, one has ) ) m where ε ∼ N (0, 1). Let us set C = 1/2 (2π −1 ) 1/2 > 0. Using Lemma 3.5 and the fact that log(1 − x) ≤ −x if x ∈ (0, 1), there exists M ∈ N such that for any m > M, we have ) ≥ 1 − exp(−m γ ) for γ ∈ (0, 1/2). Consequently, one has in particular In view of the fact that the events E 2 M ,2 M (t), M ∈ N, are independents, it follows from the Borel-Cantelli lemma that Hence for a fixed t ∈ R, almost surely, there are infinitely many j ∈ N such that Fubini's theorem implies then that there is an event Ω * 2 ⊆ Ω of probability 1 on which for almost every t ∈ R, (15) holds true. □ Proposition 3.7. Let Ω * 2 denote the event of probability 1 of Lemma 3.6. For every ω ∈ Ω * 2 and almost every t ∈ R, one has Proof. It suffices to proceed as in the proof of Proposition 3.4, using Lemma 3.6. □ Let us end with a result which will be useful for rapid points.
Lemma 3.8. There exists an event Ω * 3 ⊆ Ω of probability 1 such that, for every ω ∈ Ω * 3 and every non-empty open interval Proof. To avoid making the notations heavier, we suppose that A = (0, 1). The proof can be easily adapted in the general case. The conclusion follows then by covering R with all open intervals with rational endpoints.
Let us first show that almost every t ∈ R is a leader-ordinary point of B(·, ω). Using Theorem 2.1, we know that almost every t ∈ R is an ordinary point of B(·, ω). Together with Remark 2.4 and Proposition 3.7, this implies that for almost every t ∈ R, In particular, there exist leader-ordinary points of B(·, ω) in A. Secondly, Proposition 3.9 shows that there exists t r (ω) ∈ A such that lim sup Using Proposition 3.4, we obtain that t s (ω) is a leader-slow point of B(·, ω). □