On the increments of the principal value of Brownian local time

Let $W$ be a one-dimensional Brownian motion starting from 0. Define $Y(t)= \int_0^t{\d s \over W(s)} := \lim_{\epsilon\to0} \int_0^t 1_{(|W(s)|>\epsilon)} {\d s \over W(s)} $ as Cauchy's principal value related to local time. We prove limsup and liminf results for the increments of $Y$.


Introduction
Let {W (t); t ≥ 0} be a one-dimensional standard Brownian motion with W (0) = 0, and let {L(t, x); t ≥ 0, x ∈ R} denote its jointly continuous local time process. That is, for any Borel We are interested in the process Rigorously speaking, the integral t 0 ds/W (s) should be considered in the sense of Cauchy's principal value, i.e., Y (t) is defined by Since x → L(t, x) is Hölder continuous of order ν, for any ν < 1/2, the integral on the extreme right in (1.2) is almost surely absolutely convergent for all t > 0. The process {Y (t), t ≥ 0} is called the principal value of Brownian local time.
It is easily seen that Y (·) inherits a scaling property from Brownian motion, namely, for any fixed a > 0, t → a −1/2 Y (at) has the same law as t → Y (t). Although some properties distinguish Y (·) from Brownian motion (in particular, Y (·) is not a semimartingale), it is a kind of folklore that Y behaves somewhat like a Brownian motion. For detailed studies and surveys on principal value, and relation to Hilbert transform see Biane and Yor [4], Fitzsimmons and Getoor [13], Bertoin [2], [3], Yamada [20], Boufoussi et al. [5], Ait Ouahra and Eddahbi [1], Csáki et al. [11] and a collection of papers [22] together with their references. Biane and Yor [4] presented a detailed study on Y and determined a number of distributions for principal values and related processes.
Concerning almost sure limit theorems for Y and its increments, we summarize the relevant results in the literature. It was shown in [17] that the following law of the iterated logarithm holds: Theorem A. (Hu and Shi [17]) This was extended in [10] to a Strassen-type [18] functional law of the iterated logarithm.
Concerning Chung-type law of the iterated logarithm, we have the following result: Theorem C. (Hu [16]) with some (unknown) constant K 1 > 0.
The large increments were studied in [7] and [8]: Theorem D. (Csáki et al. [7]) Under the conditions T → a T and T → T /a T are both non-decreasing, Wen [19] studied the lag increments of Y and among others proved the following results.
Theorem E. (Wen [19]) If a T is onto, then we have equality in (1.10).
In this note our aim is to investigate further limsup and liminf behaviors of the increments of Y .
Theorem 1.1. Assume that T → a T is a function such that 0 < a T ≤ T , and both a T and T /a T are non-decreasing. Then with some positive constants K 2 , K 3 . If, moreover, Theorem 1.2. Assume that T → a T is a function such that 0 < a T ≤ T , and both a T and T /a T are non-decreasing. Then The organization of the paper is as follows: In Section 2 some facts are presented needed in the proofs. Section 3 contains the necessary probability estimates. Theorem 1.1(i) and Theorem 1.1(iia,b) are proved in Sections 4 and 5, resp., while Theorem 1.2(i) and Theorem 1.2(iia,b) are proved in Sections 6 and 7, resp.
Throughout the paper, the letter K with subscripts will denote some important but unknown finite positive constants, while the letter c with subscripts denotes some finite and positive universal constants not important in our investigations. When the constants depend on a parameter, say δ, they are denoted by c(δ) with subscripts.

Facts
Let {W (t), t ≥ 0} be a standard Brownian motion and define the following objects: Here we summarize some well-known facts needed in our proofs.
Fact 2.4. (Csörgő and Révész [12]) Assume that T → a T is a function such that 0 < a T ≤ T , and both a T and T /a T are non-decreasing. Then Fact 2.7. (Grill [15]) Let β(t), γ(t) be positive functions slowly varying at infinity, such that Fact 2.7 the following estimate on d(T ) when T → ∞.
with some positive constants c 10 , c 11 .

Proof. Define the events
Then A ⊂ A, since if A occurs and t < 1, t + s ≤ 1, then If A occurs and t < 1, s ≤ 1, 1 < t + s ≤ T , then Moreover, if A occurs and 1 ≤ t, s ≤ 1, t + s ≤ T , then Hence A ⊂ A as claimed. But by the Markov property of W , where ϕ denotes the standard normal density function.
Then for 0 < δ ≤ 1/2 we have By scaling and Lemma 3.1 To bound P 1 , we denote by d(t) := inf{s ≥ t : W (s) = 0} the first zero of W after t. Consider means that the Brownian motion W does not change sign over [t k , t k + 1 − δ), then and it follows that Let W (s) = W (d(t k ) + s) for s ≥ 0 and Y (s) be the associated principal values. Observe

By scaling and Fact 2.3 we have
Therefore, we obtain: .

3) of Theorem
A. Now we assume that a T /T ≤ ρ < 1, with some constant ρ for all T > 0.

Proof of Theorem 1.1(ii)
First assume that By Theorem C, proving the lower bound in (1.12).
To get an upper bound, note that by scaling, (3.7) of Lemma 3.4 is equivalent to Let t k and θ k be defined by (4.4) and (4.5), resp., as in the proof of Theorem 1.1(i) and for any ε > 0 and for δ > 0 such that α/2 + c 11 /δ 2 < 1, define the events Then for any ε. Put, as before, T k = θ k−1 + t k . For large enough k by (4.7) and (4.8) we have a T k ≤ (1 + ε)a t k , a.s. and T k − a T k ≤ θ k−1 + (1 + ε)t k − (1 + ε)a t k , a.s. Thus given any ε > 0, we have for large k By Theorem A, Fact 2.8, (4.7), (5.1) and simple calculation, a.s.
To get a lower bound under (5.7), observe that by scaling, (3.6) of Lemma 3.3 is equivalent to for a ≤ T , 0 ≤ κ < 1, 0 < δ, 0 < z. Using (5.7) we get further In the case when (1.7) holds, (1.13) was proved in [7]. In other cases the proof is similar. Let T k = e k and define the events with some constant C 1 to be given later. By (5.9) For given α > 2, choose small ε > 0, κ = 2/α + ε, One can easily see that with these choices k P(F k ) < ∞, consequently for ε can be choosen arbitrary small.
Since sup 0≤t≤T −a T sup 0≤s≤a T |Y (t + s) − Y (t)| is increasing in T , we obtain a lower bound in (1.13). This together with the 0-1 law for Brownian motion complete the proof of Theorem

Proof of Theorem 1.2(i)
If a T = T , then (1.14) is equivalent to Theorem C. Now assume that ρ := lim T →∞ a T /T < 1.
First we prove the lower bound, i.e.

Define the events
Let T k = e k and put T = T k+1 , a = a T k , into (6.2). The constant C 2 will be choosen later. Denoting the terms on the right-hand side of (6.2) by I 1 , I 2 , I 3 , resp., we have < ∞. So we show that for appropriate choice of C 2 we have also First consider the case 0 < ρ > 0. Choosing a positive δ one can select C 2 < min( √ c 21 , c 22 ρ ) and it is easy to verify that k I (k) j < ∞, j = 1, 2, hence also k P(G k ) < ∞.
In the case ρ = 0 choose Borell-Cantelli lemma and interpolation between T k 's finish the proof of (6.1). We have also verified that in the case ρ = 0 one can choose C 2 = 1/ √ 2, since δ can be choosen arbitrary small. Now we turn to the proof of the upper bound, i.e.
with some constant C 3 .
If ρ > 0, then According to the law of the iterated logarithm, with probability one there exists a sequence {T i , i ≥ 1} such that lim i→∞ T i = ∞ and But Fact 2.4 implies that for ε > 0 (6.6) |W (λ T i ) − W (s)| ≤ 2(1 + ε)εT i log log T i , λ T i ≤ s ≤ λ T i + εT i , i ≥ 1.
The lower bound Assuming W (λ T ) > 0, we get for all large T .
The case when W (λ T ) < 0 is similar. This shows the upper bound in (1.16).