The Laws of Chung and Hirsch for Cauchy's Principal Values Related to Brownian Local Times

Two Chung-type and Hirsch-type laws are established to describe the liminf asymp-totic behaviours of the Cauchy's principal values related to Brownian local times. These results are generalized to a class of Brownian additive functionals.


Introduction
Let (B(t), t ≥ 0) be a one-dimensional Brownian motion starting from 0, and denote (L x t , t ≥ 0, x ∈ R) a continuous version of its local times such that x → L x t is Hölder continuous of order β for every β < 1/2 uniformly in t on each compact interval, and for every bounded Borel function f : R → R, For every −∞ < α < 3/2, we define an additive functional X α (t) ≡ p.v. t 0 ds/ B(s) α as Cauchy's principal values related to (L x t ) ( x α def = |x| α sgn(x) for x ∈ R) by: (1.1) (in the case α < 1, the integral t 0 ds/ B(s) α is absolutely convergent).The principal values have been a subject of many recent works.For motivations and studies on X α and related topics, see e.g.Biane and Yor [4], Fitzsimmons and Getoor [14,15] and Bertoin [1,3], Csáki et al. [7,8,10], Yamada [34] and Yor [36] together with their references.We only mention the facts that the process X α admits a (Hölder) continuous version, and inherits from the Brownian motion the self-similarity of order (1 − α/2).
In constrast with the deep understanding of the limsup asymptotic behaviors of X α , relatively little is known about its liminf properties, to our best knowledge.This paper aims at giving two Chung-type and Hirsch-type laws for X α .The first one reads as follows: When 1 < α < 3/2, we have The exact value of C 2 (α) remains unknown.When α < 1, Theorem 1.1 yields some interesting examples which a priori do not involve principal values: 3), we recover the following Chung's law due to Khoshnevisan and Shi [22] for the integrated Brownian motion: More generally, let θ ≥ 0, we obtain: (1. for some constant C2 (−θ) > 0.
It is also of interest to describe how small sup 0≤s≤t X α (s) can be: Let f (t) > 0 be a nondecreasing function, we have where here and in the sequel, i.o.means "infinitely often" as the relevant index goes to infinity.

Remark
In view of the invariance principle for additive functionals established by Csáki et al. [8,Theorem 1.3], the above two LILs also hold for the principal values related to the simple symmetric random walk on Z.
It is worth noticing that the above results hold for a more general class of additive functionals, see Section 5.The proofs of Theorems 1.1 and 1.2 rely on the estimates of P( sup 0≤s≤1 |X α (s)| < ) and P( sup 0≤s≤1 X α (s) < ), which we shall give in Section 3 with aid of Brownian excursions and Biane and Yor [4]'s representation of stable processes.In Section 4, we prove Theorems 1.1 and 1.2.
Throughout the whole paper, we adopt the notation that f (x) ∼ g(x) (resp: For notational convenience, we write in the rest of this paper

Define
The following important fact is due to Biane and Yor [4], see also Fitzsimmons and Getoor [14] and Bertoin [1,3] for generalizations to Lévy processes.

Small Deviations
The main results of this section are the following Propositions 3.1 and 3.2: (3.1) Throughout the proofs, we shall constantly use the simple observation that X α is monotone on each Brownian excursion interval (τ s− , τ s ).We begin with the proof of Proposition 3.1:

Proof of Proposition 3.1 (lower bound).
Only small needs to be considered.Pick up a large r > 0 whose value will be determined later.It follows from (2.3) and the self-similarity that by choosing r = 2C 5 −ν and using the fact that τ 1 law = 1/N 2 for a standard Gaussian law N .This implies the desired lower bound in the case α ≤ 1.
In the case 1 < α < 3/2, we prove the lower bound by showing: To this end, recall the definition (2.4) of m.Let p be the Brownian bridge: The Brownian path-decomposition (cf.[29, Exercise (XII.3.8)])shows that the quadruplet (p, m, g 1 , sgn (B(1))) are independent.Since the Brownian motion B does not change the sign over (g 1 , 1), it follows that where Y denotes the principal value related to the local times of the Brownian bridge p, which can be defined in the same way of (1.1): It follows from (3.5) and from the independence between Y , m, g 1 and sgn(B(1)) that Applying Lemma 2.2 and using Lévy's first arcsine law: we obtain (3.4), as desired.

Proof of Proposition 3.1 (upper bound).
Let us introduce the ranked Brownian excursion lengths: For t > 0, denote by the ordered lengths of the countable excursion intervals of B over [0, t] (including the incomplete meander length t − g t ).Therefore, n≥1 V n (t) = t.For detailed studies on excursion lengths, see Pitman and Yor [27,28] together with their references.Denote by {(a i , b i ), i ≥ 1} the corresponding excursions intervals of B over [0, Therefore the processes {e i (•), i ≥ 1} are i.i.d., with the common law of a Brownian normalized excursion e defined by (2.5).Furthermore, these normalized excursions {e i , i ≥ 1}, the meander m defined by (2.4) and the excursion lengths {V i (1), i ≥ 1} are mutually independent.Recall ν ≡ 1/(2 − α).By the change of variable, we have that for i ≥ 1, Now, we distinguish three different cases: Case α = 0: It follows from (3.9) and (3.10) that (ν = 1/2) Using the Laplace transform of 1/V 1 (1) (cf.Pitman and Yor [28, Proposition 7 and Corollary 12]), we obtain by applying the analytical continuation that E exp 1/(2V 1 (1)) < ∞ (in fact, 1/V 1 (1) has a tail of exponential decay, a fact already known for the simple random walk (cf.Csáki et al. [9]).It follows that yielding the desired upper bound in the case α = 0.
To obtain the upper bound, we consider an independent exponentially distributed variable e, with parameter 1/2.According to the result of Brownian path-decomposition at g e (see e.g.Yor [35, Proposition 3.2]), we obtain: where we have used the Markov property at τ /2 to obtain (3.23).It follows from (2.2) and scaling that the above integral of (3.23) is less than with some constants C 13 , C 14 > 0. It follows that for some constant C 15 = C 15 (α) > 0. We omit here the details.We shall make use of (3.3) to prove that lim inf t→∞ log log t t To this end, fix c > (C 5 ) 2−α and choose a small δ such that 0 < δ < (c 2/(2−α) C −2 5 − 1)/2.Define , with y + def = max(y, 0) for y ∈ R. Let (F t , t ≥ 0) be the natural filtration generated by the Brownian motion B. Remark that A n is F tn -measurable.Let us prove that n , P(A n | F t n−1 ) = ∞, a.s., (4.3) which, according to Lévy's version of Borel-Cantelli's lemma (cf.[30]), yields that P A n is realized infinitely often as n → ∞ = 1.(4.4) To arrive to (4.3), let B(u) def = B(u + d t n−1 ) for u ≥ 0. Then B is a standard Brownian motion, independent of F dt n−1 .Define X α (•) from B the same way X α (•) does from B. Therefore, for some large n 0 .The convergence part of Borel-Cantelli's lemma yields that almost surely for all large n, we have d t n−1 ≤ 1 n t n .Applying (1.2) to X and to −X, we establish that for all large n, This fact together with (4.4) show that almost surely, there are infinite n such that the event A n realizes; and on A n , . Hence (4.2) is proven by letting δ → 0 and c → (C 5 ) 2−α .In view of (4.1) and (4.2), (1.3) follows from the usual Kolmogorov's 0-1 law for Brownian motion, with Proof of Theorem 1.2.By using scaling and the upper bound of Proposition 3.2, the easy part of Borel-Cantelli's lemma immediately implies the convergence part of Theorem 1.2.We omit the details.
To show the divergence part of Theorem 1.2, we can suppose without any loss of generality that for some large t 0 , ( log t) 1/ν ≤ f (t) ≤ ( log t) 4/ν , t≥ t 0 , (4.5) see e.g.Erdős [13] for a rigorous justification.Recall ν ≡ 1/(2 − α).Consider large n.Define Let us show that P F n is realized infinitely often as n → ∞ > 0. (4.6) Assuming that we have proven (4.6), we deduce from our definition of F n , t n and f (t) that P sup 0≤s≤t X α (s) < t 1/(2ν) /f (t) , i. o. > 0. Kolmogorov's 0-1 law implies that this probability equals in fact 1 and proves the divergence part of Theorem 1.2.
It remains to show (4.6). to this end, using scaling and Proposition 3.2 that implies: whose sum on n diverges thanks to the divergence of the integral of (1.7).We shall estimate the second moment P(F i ∩ F j ) for large j > i ≥ i 0 .Define B(t) def = B(t + τ t i ) for t ≥ 0 be a Brownian motion independent of F i .Define Xα and τ from B the same way X α and τ do from B. Observe that on F i ∩ F j , we have sup 0≤s≤t This remark together with (2.2) yield:  [23]) yields (4.6) and completes the whole proof.

Some Generalizations
Notice that the main ingredients to obtain Theorems 1.1 and 1.2 are the self-similarity and the fact that the process X α is monotone over each excursion interval (τ r− , τ r ), it turns out that the same method can be applied to the processes of the following type: where Λ ν > 0 denotes a stable variable of index ν, Λν is an independent copy of Λ ν , and we have used the Ray-Knight theorem to obtain (5.2) (for details, see Pitman and Yor [26, pp.435]).The density function of the ratio Λ ν / Λν is explicitly known (I learn it from M. Yor, see Lamperti [25, (3.17)] for an equivalent statement; see also Barlow, Pitman and Yor: "Une extension multidimensionnelle de la loi de l'arc sinus" (1989) Séminaire de Probabilités XXIII): (5.From these, we can make use of the same method together the known estimates for stable processes (see Bertoin [2,Chap. VIII]) to obtain the following results.The proofs are omitted.= ∞ < ∞ , where 0 < ρ < 1 is given by (5.3).

of Theorems 1.1 and 1.2 Proof of Theorem 1.1.
Let us only prove(1.3), the rest can be shown in the same way.Using the upper bound of Proposition 3.1, it is routine (see e.g.
2, yielding the desired upper bound part of Proposition 2.2.4 Proofs for some constant C 16 = C 16 (ν) > 0. From (4.7) and (4.8), it is elementary to show there exists some constantC 17 > 0 such that lim inf n→∞ 1≤i,j≤n P(F i ∩ F j ) 1≤i≤n P(F i ) 2 ≤ C 17 < ∞,which according to Kochen and Stone's version of Borel-Cantelli's lemma (see