Strong approximations of additive functionals of a planar Brownian motion

Endre CSÁKI^{1}, Antónia FÖLDES^{2} and Yueyun HU^{3}

Abstract: This paper is devoted to the study of the additive functional t → ∫_{0}^{t} f(W (s))ds, where f denotes a measurable function and W is a planar Brownian motion. Kasahara and Kotani [19] have obtained its second-order asymptotic behaviors, by using the skewproduct representation of W and the ergodicity of the angular part. We prove that the vector (∫_{0}^{·} f_{j}(W (s))ds)_{1≤j≤n} can be strongly approximated by a multi-dimensional Brownian motion time changed by an independent inhomogeneous Lévy process. This strong approximation yields central limit theorems and almost sure behaviors for additive functionals. We also give their applications to winding numbers and to symmetric Cauchy process.

Keywords. Additive functionals, strong approximation.

AMS Classification 2000. 60F15, 60J65.

1A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, P.O.B.

127, Budapest, H–1364, Hungary. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886 and T 043037 E-mail: csaki@renyi.hu

2City University of New York, 2800 Victory Blvd., Staten Island, New York 10314. Research supported by PSC-CUNY Research Grant 65685-00 34. E-mail: afoldes@gc.cuny.edu

3Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR–7599), Université Paris VI, case 188, 4 Place Jussieu, F–75252 Paris Cedex 05, France. E-mail: hu@ccr.jussieu.fr

1 Introduction

Let W = (W_{1}, W_{2}) be a planar Brownian motion, where W_{1} and W_{2} are two independent one-dimensional Brownian motions. Let f : ℝ^{2} → ℝ be a measurable locally integrable function. The additive functional, t → ∫_{0}^{t} f(W (s))ds, together with other functionals of planar Brownian motion such as windings and crossing numbers, have been a subject of many studies, see for instance Pitman and Yor [23], [25], Hu and Yor [15] for studies and references. Here, we will turn our attention to the additive functionals. The following two results describe respectively the first-order and the second-order asymptotic behaviors: Theorem A (Kallianpur-Robbins [17]) Let f_{1}, f_{2} ∈ L^{1}(ℝ^{2}) and f_{2} > 0, both having compact supports. Then as t → ∞,

\frac{∫_{0}^{t} f_{1}(W (s))ds}{∫_{0}^{t} f_{2}(W (s))ds} ⟶^{a.s.} \frac{C_{1}(f_{1})}{C_{1}(f_{2})},  (1.1)

\frac{1}{log t} ∫_{0}^{t} f_{1}(W (s))ds ⟶^{(d)} \frac{C_{1}(f_{1})}{2π} e,  (1.2)

where e denotes a standard exponential variable and C_{1}(f) =^{def} ∫_{ℝ^{2}} f(x)dx.

Ergodic results similar to (1.1) hold for a large class of recurrent Markov process, see e.g.

[2] for a general statement. The convergence in law (1.2) can be extended as the convergence in terms of processes, furthermore, the following result holds: Theorem B (Kasahara and Kotani [19]) Assume that f : ℝ^{2} → ℝ is a bounded function such that ∫ |f(x)||x|^{ν}dx < ∞ for some ν > 2. Then as λ → ∞,

(\frac{1}{λ} ∫_{0}^{e^{λt}} f(W (s))ds, t ≥ 0) ⟶^{(f.d.)} (\frac{C_{1}(f)}{2π}e(t), t ≥ 0),  (1.3)

where “ ⟶^{(f.d.)} ” means the convergence in the finite marginal sense, and (e(t), t ≥ 0) denotes an inhomogeneous Lévy process such that e(t) is an exponential variable with mean t. Moreover, if C_{1}(f) = 0, then

(\frac{1}{\sqrt{λ}} ∫_{0}^{e^{λt}} f(W (s))ds, t ≥ 0) ⟶^{(f.d.)} (C̃_{2}(f) β̃(e(t)), t ≥ 0), λ → ∞,

where β̃ is a standard one-dimensional Brownian motion, independent of e(·), and

C̃_{2}(f) = ( − \frac{1}{π^{2}} ∫_{ℝ^{2}} ∫_{ℝ^{2}} log |x − y|f(x)f(y)dxdy)^{1/2}.  (1.4)

The constant C̃_{2} was given in Kasahara [18] by evaluating the asymptotics of the resolvent (see also Touati [29] for more general Markov process). Let us briefly describe the idea of Kasahara and Kotani [19]: Identifying ℂ = ℝ^{2} and assuming without loss of generality that W (0) = 1, we recall the following skew-product representation (cf. [16], pp. 270):

W (t) = R(t) e^{i θ(t)} = exp (β(Ξ(t)) + iγ(Ξ(t))),  (1.5)
Ξ(t) = ∫_{0}^{t} \frac{ds}{R^{2}(s)},  (1.6)

where β and γ denote two independent real-valued Brownian motions both starting from 0. Then the additive functionals of the planar Brownian motion can be transferred to that of the Brownian motion (β_{t}, γ̇_{t}) on the cylinder ℝ × (ℝ/2πℤ), with ẋ =^{def} x(mod 2π); Therefore, we can make use of the ergodicity of γ̇ to solve the two-dimensional problem.

In this paper, our main goals are to unify Theorems A and B and to obtain the fluctuations in these convergences in law. This will be done by establishing a strong approximation of the vector of additive functionals (∫_{0}^{t} f_{j}(W (s))ds, 1 ≤ j ≤ n). Before stating our results, we remark that the Lévy process e(·) in Theorem B can be realized as

e(t) =^{def} ℓ(σ(t/2)),  t ≥ 0,  (1.7)

where (ℓ(t), t ≥ 0) denotes the process of local times at 0 of the one-dimensional Brownian motion β and σ(·) is the first passage process of β:

σ(x) =^{def} inf{s > 0 : β(s) > x},  x ≥ 0.  (1.8)

The inverse process of (e(t)) is called an extremal process, see Resnick [26] and Watanabe [31].

Theorem 1.1 Fix n ≥ 1. Let f_{1}, ..., f_{n} : ℝ^{2} → ℝ be n measurable real-valued functions. Assume that there exists some constants K > 0 and ν > \frac{5}{2} such that for all 1 ≤ j ≤ n,

sup_{|z|=r} |f_{j}(z)| ≤ \frac{K}{r^{2}(1 + | log r|)^{ν}} ,  r > 0.  (1.9)

Then, possibly in an enlarged probability space, we may define a version of the planar Brownian motion W, a ℝ^{n}-valued Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 and a process ẽ such that ẽ has the same law as e, Y and ẽ are independent and such that almost surely for any 1 ≤ j ≤ n and for all large t,

∫_{0}^{t} f_{j}(W (s))ds − \frac{C_{1}(f_{j})}{2π}e(log t) − C_{2}(f_{j})Y_{j}(ẽ(log t)) = o((log t)^{\frac{1}{2} −δ}), (1.10)
|e(log t) − ẽ(log t)| = o((log t)^{1−δ}),  (1.11)

where δ > 10^{−5} denotes some constant, and the covariance matrix of the n-dimensional Brownian motion Y is given by 𝔼(Y_{j}(1) Y_{k}(1)) = C_{3}(f_{j}, f_{k}), with

C_{2}(f) =^{def} ( − \frac{1}{π^{2}} ∫_{ℝ^{2}} dy ∫_{ℝ^{2}} dy′f(y)f(y′) log |y − y′|
+\frac{2}{π^{2}} C_{1}(f) ∫_{ℝ^{2}} dyf(y) log max(|y|, 1))^{1/2},  (1.12)
C_{3}(f_{j}, f_{k}) =^{def} \frac{1}{4} \frac{(C_{2}(f_{j} + f_{k}))^{2} − (C_{2}(f_{j} − f_{k}))^{2}}{C_{2}(f_{j}) C_{2}(f_{k})} .  (1.13)

The constant C_{2}(f) is well defined thanks to (3.27) and (3.8). Remark that e(r) is of order r and e(r) ≫ r^{1−δ} almost surely for all large r. This shows in particular the RHS of (1.11) is negligible with respect to e(log t) or to ẽ(log t).

It is essential that the Brownian motion Y and the inhomogeneous Lévy process ẽ are independent. But Y is not independent of the process e, which is defined from W in terms of (1.5) and (1.7). We also mention that it is impossible to choose Y independent of W, otherwise (1.10) would contradict the usual law of iterated logarithm.

Besides the unification of Theorems A and B, we deduce from (1.10) and (1.11) the central limit theorem for the ergodic result (1.1):

Corollary 1.2 Under the assumptions of Theorem 1.1 and assuming C_{1}(f_{2}) ≠ 0, we have

\sqrt{log t} (\frac{∫_{0}^{t} f_{1}(W (s))ds}{∫_{0}^{t} f_{2}(W (s))ds} − \frac{C_{1}(f_{1})}{C_{1}(f_{2})}) ⟶^{(d)} a(f_{1}, f_{2}) \frac{N}{\sqrt{e(1)}}, t → ∞,

where N denotes a standard Gaussian variable, independent of e(1) which is exponentially distributed with mean 1, and

a(f_{1}, f_{2}) = \frac{2π}{C_{1}^{2}(f_{2})} \sqrt{C_{1}^{2}(f_{1})C_{2}^{2}(f_{2}) + C_{1}^{2}(f_{2})C_{2}^{2}(f_{1}) − 2C_{1}(f_{1})C_{1}(f_{2})C_{2}(f_{1})C_{2}(f_{2})C_{3}(f_{1}, f_{2})}.

It is also interesting to compare (1.10) with the logarithmic average of Kallianpur and Robbins’ law obtained by Mörters ([22], Theorem 1.1).

Theorem 1.1 yields in particular the almost sure behaviors of the additive functionals, for instance, we can obtain the following laws of iterated logarithm:

Corollary 1.3 Let f : ℝ^{2} → ℝ satisfying (1.9) and such that C_{1}(f) = 0 and C_{2}(f) > 0. We have

lim sup_{t→∞} \frac{∫_{0}^{t} f(W (s))ds}{\sqrt{log t} log log log t} = \frac{C_{2}(f)}{\sqrt{2}},  a.s.  (1.14)

Let κ : ℝ_{+} → ℝ_{+} be a nondecreasing function such that \sqrt{log t}/κ(t) is a nondecreasing function. Then

ℙ( sup_{0≤s≤t} ∫_{0}^{s} f(W (u))du < \frac{\sqrt{log t}}{κ(t)}) = \{ \begin{matrix}0 \\ 1 \end{matrix} ⟺ ∫ ^{∞} \frac{dt}{t(log t)κ(t)} \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,
ℙ( sup_{0≤s≤t} | ∫_{0}^{s} f(W (u))du| < \frac{\sqrt{log t}}{κ(t)}) = \{ \begin{matrix}0 \\ 1 \end{matrix} ⟺ ∫ ^{∞} \frac{dt}{t(log t)κ^{2}(t)} \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} .

See also Touati [30] for Strassen-type limit theorems with random normalization.

Chen [7] and [8] obtained (1.14)-type results for a Harris’ recurrent Markov chain, see [3] for an interesting application. However it is not clear how to reduce the problem for the planar Brownian motion to a recurrent random walk problem in our settings.

The strong approximations of additive functionals of a one-dimensional diffusion process or a recurrent Markov chain have been extensively studied, see [14] for a survey and references. Let us also mention that Csáki and Földes [10] developed a general principle when the underlying Markov process is point-recurrent, this principle can not be applied here because every single point is polar for a planar Brownian motion.

The rest of this paper is organized as follows: In Section 2, we present some exponential moments related to a one-dimensional Brownian motion and a martingale representation; In Section 3, we state the corresponding results (Propositions 3.1 and 3.2) for the additive martingales on the cylinder, which imply in particular Theorem 1.1. We prove Propositions 3.1 and 3.2 in Section 4. Finally, some applications to winding numbers and Cauchy process are given in Section 5.

Throughout this paper, c, c′, c′′ > 0 denote some generic constants whose values may change from one paragraph to another one, whereas (C_{j}, 1 ≤ j ≤ 20) denote some more important constants which may depend on f_{j} and on ν. In the sequel, we write that f satisfies some condition, say (1.9), to mean that (1.9) holds for f in lieu of f_{j}, and the condition ν > \frac{5}{2} may be relaxed to ν > 1 or ν > 2, this will be stated explicitly in each case. For the sake of notational convenience, we shall sometimes write ξ_{t} instead of ξ(t).

2 One-dimensional Brownian motion

Let (ℓ(t, x), t ≥ 0, x ∈ ℝ) be the family of local times of the one-dimensional Brownian motion β. Let us write ℓ(t) ≡ ℓ(t, 0) and define

τ(r) =^{def} inf{t > 0 : ℓ(t) > r}, r ≥ 0.

2.1 Exponential moments

For the next result see Kazamaki ([20], pp. 9):

Lemma 2.1 Let (N_{t}, t ≥ 0) be a continuous real-valued local martingale with respect to the filtration (G_{t}). Denote by (⟨N⟩_{t}) its bracket. Then for any (G_{t})-stopping time T finite or not, we have

𝔼 exp (|N_{T} |) ≤ 2 \sqrt{𝔼 exp (2⟨N⟩_{T})}.

Recall Borell’s inequality for a Gaussian process (cf. [1], pp. 43, Theorem 2.1):

Lemma 2.2 Let {ξ(t), t ∈ Λ} be a centered Gaussian process with a.s. bounded sample paths, where Λ denotes some parameter set. Then C_{4} =^{def} 𝔼 sup_{t∈Λ} ξ(t) < ∞, and

ℙ(| sup_{t∈Λ} ξ(t) − C_{4}| > λ) ≤ 2 exp ( − \frac{λ^{2}}{2C_{5}} ), λ > 0,

with C_{5} =^{def} sup_{t∈Λ} 𝔼ξ^{2}(t).

Denote in this section by (B(x), x ∈ ℝ) a standard one-dimensional Brownian motion defined on ℝ. We have

Lemma 2.3 Let h : ℝ → ℝ be a measurable function such that

C_{6}(h) =^{def} ∫_{ℝ} |h(x)| |x| log log(|x| + \frac{1}{|x|} + 16)dx < ∞.

Then there exists some universal constant c ≥ 1 such that for all 0 ≤ a ≤ \frac{1}{4C_{6}} , we have

𝔼 exp (a ∫_{ℝ} |h(x)|B^{2}(x)dx) ≤ c.

In C_{6}(h), the term log log(|x| + \frac{1}{|x|} + 16) > 1 comes from the usual laws of iterated logarithm both for |x| → 0_{+} and for |x| → ∞.

Proof: Applying Lemma 2.2 twice to the Gaussian processes \{_{\frac{±B(x)}{\sqrt{|x| log log(|x|+ \frac{1}{|x|} +16)}}}, x ∈ ℝ\},

we obtain:

C_{7} =^{def} 𝔼m^{∗} < ∞,

where m^{∗} =^{def} sup_{x∈ℝ} _{\frac{|B(x)|}{\sqrt{|x| log log(|x|+ \frac{1}{|x|} +16)}}} < ∞, a.s. by the usual law of iterated logarithm at 0

and at ∞. Since 𝔼\frac{B^{2}(x)}{|x| log log(|x|+ \frac{1}{|x|} +16)} = \frac{1}{log log(|x|+ \frac{1}{|x|} +16)} < 1 we have

ℙ(m^{∗} > C_{7} + λ) ≤ 4 exp ( − \frac{λ^{2}}{2} ), λ > 0.

Remark that

∫_{ℝ} |h(x)|B^{2}(x)dx ≤ (m^{∗})^{2} ∫_{ℝ} |h(x)| |x| log log(|x| + \frac{1}{|x|} + 16)dx = C_{6}(h) (m^{∗})^{2}.

Hence 𝔼 exp (a ∫_{ℝ} |h(x)|B^{2}(x)dx) ≤ 𝔼 exp (a C_{6} (m^{∗})^{2}) ≤ 𝔼 exp (\frac{(m^{∗})^{2}}{4} ) = c < ∞. □

Proposition 2.4 Let h : ℝ → ℝ be a measurable function such that

∫_{ℝ} |h(x)| [1 + |x| log log(|x| + 16)]dx < ∞.  (2.1)

Then there exists some constant C_{8}(h) > 1 such that for all 0 ≤ a ≤ \frac{1}{C_{8}} and r > 0, we have

𝔼 exp (a ∫_{0}^{τ_{r}} |h(β_{s})|ds) ≤ C_{8} e^{C_{8} a r},  (2.2)
𝔼 exp (a | ∫_{0}^{τ_{r}} h(β_{s})ds − r ∫_{−∞}^{∞} h(x)dx|) ≤ C_{8} e^{C_{8} a^{2} r}.  (2.3)

Proof: According to Ray-Knight’s theorem (cf. [27], Chap. XI), x → ℓ(τ(r), x) is the square of a zero-dimensional Bessel process, which is the unique nonnegative solution of the stochastic equation

ℓ(τ(r), x) = r + 2 ∫_{0}^{x} \sqrt{ℓ(τ(r), y)} dB(y),  x ∈ ℝ,  (2.4)

for some one-dimensional Brownian motion (B(x), x ∈ ℝ). Remark that x → ℓ(τ(r), x) is stochastically smaller than x → (\sqrt{r}+B(x))^{2}, by using the comparison theorem of diffusions with different drift terms (cf. [27], Theorem IX.3.7). It follows that for 0 ≤ a ≤ \frac{1}{8C_{6}(h)}, we have from Lemma 2.3 that

𝔼 exp (a ∫_{0}^{τ_{r}} |h(β_{s})|ds) = 𝔼 exp (a ∫_{ℝ} |h(x)|ℓ(τ(r), x)dx)
≤ 𝔼 exp (a ∫_{ℝ} |h(x)|(\sqrt{r} + B(x))^{2}dx)
≤ 𝔼 exp (a ∫_{ℝ} |h(x)|2(r + B^{2}(x))dx)
≤ c exp (2a r ∫_{ℝ} |h(x)|dx ),  (2.5)

yielding (2.2). Define

H(x) =^{def} ∫_{x}^{∞} h(y)dy, x > 0; H(x) =^{def} − ∫_{−∞}^{x} h(y)dy, x ≤ 0.

Using the equation (2.4), we get

∫_{0}^{τ_{r}} h(β_{s})ds − r ∫_{−∞}^{∞} h(x)dx = ∫_{−∞}^{∞} h(x)(ℓ(τ(r), x)) − r)dx
= − ∫_{−∞}^{∞}(ℓ(τ(r), x) − r)dH(x)
= 2 ∫_{−∞}^{∞} H(x) \sqrt{ℓ(τ(r), x)} dB(x),

by integration by parts. It is elementary to check that H^{2} satisfies the condition of integrability:

∫_{ℝ} H^{2}(x) [1 + |x| log log(|x| + 16)]dx < ∞.

Using Lemma 2.1 with N(t) =^{def} 2a ∫_{−∞}^{t} H(x) \sqrt{ℓ(τ_{r}, x)} dB(x), we obtain that

𝔼 exp (a | ∫_{0}^{τ_{r}} h(β_{s})ds − r ∫_{−∞}^{∞} h(x)dx|) ≤ 2\sqrt{𝔼 exp (8 a^{2} ∫_{−∞}^{∞} H^{2}(x) ℓ(τ(r), x) dx)}.

Applying (2.5) to H^{2}(x) instead of h, we have that for 8a^{2} ≤ \frac{1}{8C_{6}(H^{2})} (the constant C_{6}(H^{2}) has been defined in Lemma 2.3),

𝔼 exp (a | ∫_{0}^{τ_{r}} h(β_{s})ds − r ∫_{−∞}^{∞} h(x)dx|) ≤ 2\sqrt{c} exp (8a^{2} r ∫_{ℝ} H^{2}(x)dx),

implying (2.3) by choosing a sufficiently large constant C_{8}. Finally, we shall also make use of the following simple fact (for example, by using (2.4)):

𝔼( ∫_{0}^{τ_{1}} \frac{ds}{(1 + |β_{s}|)^{ν}} )^{2} < ∞,  ν > \frac{3}{2}.  (2.6)
□

2.2 Martingale representation

Define

\overline{β}(t) = sup_{o≤s≤t} β(s), t ≥ 0.

Let (B_{t}) be the natural filtration generated by the Brownian motion β.

Lemma 2.5 Let r ≥ 2 and u ∈ ℝ. We have

𝔼(e^{iu\overline{β}(τ_{r})} | B_{t}) = 𝔼(e^{iu\overline{β}(τ_{r})}) + ∫_{0}^{t∧τ_{r}} ζ_{v}(r, u) dβ_{v}, t ≥ 0,

for some (B_{v})-predictable process ζ_{v}(r, u). Furthermore

|ζ_{v}(r, u)| ≤ 2(\frac{1_{(\overline{β}_{v}≥1)}}{\overline{β}_{v}} + 1_{(\overline{β}_{v}<1)}(1 + |u| log(1/\overline{β}_{v}))).

Proof: The two parameters r and u are fixed. Using the Markov property at t, we obtain

D_{t} =^{def} 𝔼(e^{iu\overline{β}(τ_{r})} | B_{t}) = e^{iu\overline{β}(τ_{r})}1_{(t≥τ_{r})} + 1_{(t<τ_{r})}ϕ(β_{t}, \overline{β}_{t}, r − ℓ_{t}),  (2.7)

with

ϕ(x, y, s) = 𝔼_{x}(e^{i u( y∨\overline{β}(τ_{s}))}), y ≥ 0 ∨ x, s > 0,

where y ∨ a = max(y, a) and 𝔼_{x} (resp: ℙ_{x}) denotes the expectation (resp: probability) with respect to the Brownian motion β starting from x. Write in this proof σ_{0} = inf{t ≥ 0 : β_{t} = 0} and define

η(a, s) = 𝔼_{0}(e^{i u (a∨\overline{β}(τ_{s}))}), a ≥ 0, s ≥ 0.

Therefore applying the strong Markov property at σ_{0}, 𝔼_{x}(e^{i u (y∨\overline{β}(τ_{s}))} |B_{σ_{0}}) = η(y ∨ \overline{β}(σ_{0}), s), we get

ϕ(x, y, s) = 𝔼_{x}(η(y ∨ \overline{β}(σ_{0}), s))
= 1_{(x≤0)}η(y, s) + 1_{(0<x≤y)}(η(y, s)(1 − \frac{x}{y}) + x ∫_{y}^{∞} \frac{da}{a^{2}}η(a, s)),  (2.8)

by using the fact that if x > 0, ℙ_{x}(\overline{β}(σ_{0}) ∈ da) = \frac{x}{a^{2}} 1_{(a≥x)}da. On the other hand, it is known (cf. [26], [31], [6] pp. 191) that

ℙ_{0}(\overline{β}(τ_{s}) ≤ a) = e^{−s/(2a)},  a, s > 0.  (2.9)

Hence

η(a, s) = e^{iua−s/(2a)} + ∫_{a}^{∞} \frac{db}{2b^{2}} s e^{iub−s/(2b)}.

Observe from (2.8) that \frac{∂ϕ}{∂x} = 0 when x < 0. Elementary computations show that for x > 0,

\frac{∂ϕ}{∂x}(x, y, s) = −\frac{η(y, s)}{y} + ∫_{y}^{∞} \frac{da}{a^{2}}η(a, s)  (2.10)
= lim_{A→∞} ∫_{y}^{A} \frac{da}{a} \frac{∂η}{∂a}(a, s)
= i u lim_{A→∞} ∫_{y}^{A} \frac{da}{a} e^{iua− \frac{s}{2a}} .  (2.11)

Going back to (2.7) and applying Itô’s formula to the RHS of (2.7), we obtain that

D_{t} = D_{0} + ∫_{0}^{t∧τ_{r}} \frac{∂ϕ}{∂x}(β_{v}, \overline{β}_{v}, r − ℓ_{v}) dβ_{v} ≡ D_{0} + ∫_{0}^{t∧τ_{r}} ζ_{v}(r, u) dβ_{v},

the other terms vanish since (D_{t}) is a martingale. This gives that ζ_{v}(r, u) = \frac{∂ϕ}{∂x}(β_{v}, \overline{β}_{v}, r−ℓ_{v}). It remains to bound \frac{∂ϕ}{∂x}. Let 0 < x ≤ y. Using the fact that |η| ≤ 1 to (2.10) yields that

|\frac{∂ϕ}{∂x}| ≤ \frac{2}{y}, y > 0.

When 0 < y ≤ 1, we deduce from (2.11) that \frac{∂ϕ}{∂x}(x, y, s) − \frac{∂ϕ}{∂x}(x, 1, s) = iu ∫_{y}^{1} \frac{da}{a}e^{iua−s/(2a)}. Since |\frac{∂ϕ}{∂x}(x, 1, s)| ≤ 2, we have

|\frac{∂ϕ}{∂x}(x, y, s)| ≤ 2 + |u| ∫_{y}^{1} \frac{da}{a}e^{−s/(2a)} ≤ 2 + |u| log(1/y),

ending the proof. □

Lemma 2.6 Let ν > \frac{3}{2}. There exists some constant c > 0 such that for all r ≥ 2,

𝔼( ∫_{0}^{τ_{r}} \frac{ds}{(1 + |β_{s}|)^{ν}} \frac{1}{1 + \overline{β}_{s}} )^{2} ≤ c(log r)^{2}.

Proof: Observe that

∫_{0}^{τ_{r}} \frac{ds}{(1 + |β_{s}|)^{ν}} \frac{1}{1 + \overline{β}_{s}} ≤ ∑_{1≤j≤r} ∫_{τ_{j−1}}^{τ_{j}} \frac{ds}{(1 + |β_{s}|)^{ν}} \frac{1}{1 + \overline{β}(τ_{j−1})} =^{def} ∑_{1≤j≤r} \frac{ξ_{j}}{1 + \overline{β}(τ_{j−1})},

with obvious definition of ξ_{j}. The sequence (ξ_{j}) are i.i.d. and we have

𝔼(ξ_{j}^{2}) = 𝔼( ∫_{0}^{τ_{1}} \frac{ds}{(1 + |β_{s}|)^{ν}} )^{2} < ∞,

by virtue of (2.6). Thanks to the independence of ξ_{j} and B_{τ_{j−1}}, the sequence (\frac{(ξ_{j}−𝔼ξ_{j})}{1+\overline{β}(τ_{j−1})})_{j≥1} is a square-integrable martingale difference, hence

𝔼( ∑_{1≤j≤r} \frac{ξ_{j}}{1 + \overline{β}(τ_{j−1})})^{2} ≤ 2𝔼( ∑_{1≤j≤r} \frac{(ξ_{j} − 𝔼ξ_{j})}{1 + \overline{β}(τ_{j−1})})^{2} + 2(𝔼ξ_{1})^{2}𝔼( ∑_{1≤j≤r} \frac{1}{1 + \overline{β}(τ_{j−1})})^{2}
= 2Var(ξ_{1})𝔼 ∑_{1≤j≤r} \frac{1}{(1 + \overline{β}(τ_{j−1}))^{2}} + 2(𝔼ξ_{1})^{2}𝔼( ∑_{1≤j≤r} \frac{1}{1 + \overline{β}(τ_{j−1})})^{2}
≤ 4𝔼(ξ_{1})^{2} 𝔼( ∑_{1≤j≤r} \frac{1}{1 + \overline{β}(τ_{j−1})})^{2}.  (2.12)

Applying the strong Markov property at τ_{j−1}, we obtain that for l > j,

𝔼(\frac{1}{1 + \overline{β}(τ_{l−1})} | B_{τ_{j−1}}) ≤ 𝔼(\frac{1}{1 + \overline{β}(τ_{l−j})}) ≤ \frac{c′}{l − j} ,

by using the law of \overline{β}(τ_{l−j}) given in (2.9). This law also implies that 𝔼\frac{1}{(1+\overline{β}(τ_{j}))^{2}} ≤ \frac{c′}{j^{2}} for j ≥ 1. It follows that

𝔼( ∑_{1≤j≤r} \frac{1}{1 + \overline{β}(τ_{j−1})})^{2} ≤ ∑_{1≤j≤r} 𝔼\frac{1}{(1 + \overline{β}(τ_{j−1}))^{2}} + 2 ∑_{1≤j<l≤r} 𝔼\frac{1}{1 + \overline{β}(τ_{j−1})} 𝔼\frac{1}{1 + \overline{β}(τ_{l−j})}
≤ c′ ∑_{1≤j≤r} j^{−2} + 2(c′)^{2} ∑_{1≤j<l≤r} \frac{1}{j(l − j)}
≤ c′′(log r)^{2},  r ≥ 2,

which in view of (2.12) completes the proof. □

3 Additive martingales and additive functionals of Brownian motion on the cylinder

Let G denote the cylinder ℝ×ℝ/(2πℤ) endowed with the Haar measure dz = dxdθ, where z = (x, θ) ∈ G denotes a generic element of G. A Brownian motion X on the cylinder is a Feller process taking values in G, with homogeneous probability transition (p_{X}(t, (x, θ))dxdθ):

p_{X}(t, (x, θ)) = \frac{1}{2πt}e^{− \frac{x^{2}}{2t}} ∑_{k=−∞}^{∞} e^{− \frac{(θ+2πk)^{2}}{2t}} , (x, θ) ∈ ℝ × [0, 2π] ≡ G.

It is clear that X can be realized as X = (β, γ̇), where (β, γ) is a planar Brownian motion, i.e. β and γ are independent Brownian motions on the line, and ẏ =^{def} y(mod 2π). The main result in this section is a strong approximation of additive martingales on the cylinder.

Fix n ≥ 1. For 1 ≤ j ≤ n, let F^{(j)} : z ∈ G → (F_{1}^{(j)}(z), F_{2}^{(j)}(z)) ∈ ℝ^{2} be n measurable functions. Assume that there exist some constants ν > \frac{3}{2} and K > 0 such that for all 1 ≤ j ≤ n,

sup_{0≤θ≤2π} |F^{(j)}(x, θ)| ≤ \frac{K}{(1 + |x|)^{ν}} ,  x ∈ ℝ.  (3.1)

Define the martingales N^{(j)} from X:

N^{(j)}(t) =^{def} ∫_{0}^{t} F^{(j)}(X_{s})dX_{s} = ∫_{0}^{t} F_{1}^{(j)}(β_{s}, γ̇_{s})dβ_{s} + ∫_{0}^{t} F_{2}^{(j)}(β_{s}, γ̇_{s})dγ_{s}, t ≥ 0.

Recall (1.7) and (1.8) and that ℓ(·) denotes the local time at 0 of β. We have

Proposition 3.1 Assume (3.1) for some ν > \frac{3}{2}. Possibly in a larger probability space, we may define a version of X = (β, γ̇) a Brownian motion on the cylinder G and an ndimensional Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 and a process L such that L(·) has the same law as ℓ(·), Y and L are independent and such that almost surely for all 1 ≤ j ≤ n and all large t,

N_{t}^{(j)} − \sqrt{C_{9}(F^{(j)})} Y_{j}(L_{t}) = o(t^{\frac{1}{4} −δ}),  (3.2)
|ℓ_{t} − L_{t}| = o(t^{\frac{1}{2} −δ}),  (3.3)

where δ > 10^{−5} denotes some constant and the covariance matrix of the n-dimensional Brownian motion Y is given by 𝔼(Y_{j}(1)Y_{k}(1)) = C_{10}(F^{(j)}, F^{(k)}), with

C_{9}(F^{(j)}) =^{def} \frac{1}{2π} ∫_{ℝ} dx ∫_{0}^{2π} dθ((F_{1}^{(j)}(x, θ))^{2} + (F_{2}^{(j)}(x, θ))^{2})  (3.4)
C_{10}(F^{(j)}, F^{(k)}) =^{def} \frac{1}{4} \frac{C_{9}(F^{(j)} + F^{(k)}) − C_{9}(F^{(j)} − F^{(k)})}{\sqrt{C_{9}(F^{(j)}) C_{9}(F^{(k)})}}.

Similarly to Theorem 1.1, it is essential that the process L is independent from Y. To obtain Theorem 1.1, we also need an analogue of (3.2) such that (Y_{j}) are independent of (e(t)), where the process (e(t)) = (ℓ(σ(t/2))) is defined in (1.7).

Proposition 3.2 Assume (3.1) for some ν > \frac{3}{2}. On some suitable probability space, we may define a version of X = (β, γ̇), a n-dimensional Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 with the covariance matrix (C_{10}(F^{(j)}, F^{(k)}))_{1≤j,k≤n} and an inhomogeneous Lévy process ẽ(·) such that Y and ẽ(·) are independent, ẽ has the same law as e and such that almost surely for all large r and for all t ∈ [σ(r − 2 log r), σ(r + 2 log r)], we have

N_{t}^{(j)} − \sqrt{C_{9}(F^{(j)})} Y_{j}(ẽ(2r)) = o(r^{\frac{1}{2} −δ}),  (3.5)
|e(r) − ẽ(r)| = o(r^{1−δ}).  (3.6)

for some positive constant δ > 10^{−5}.

Let us postpone the proofs of Propositions 3.1 and 3.2 in Section 4. The rest of this section is devoted to a strong approximation of additive functionals on the cylinder and to the proof of Theorem 1.1.

3.1 Additive functionals of X

Let g : G → ℝ be a measurable function. First we define two constants related to g (when the integrals are well defined):

C_{11}(g) =^{def} \frac{1}{2π} ∫_{−∞}^{∞} dx ∫_{0}^{2π} dθ g(x, θ),  (3.7)
C_{12}(g) =^{def} ( − \frac{1}{2π^{2}} ∫_{G×G} dxdx′dθdθ′g(x, θ)g(x′, θ′) log |e^{x+iθ} − e^{x′+iθ′}|^{2}
+\frac{4}{π} C_{11}(g) ∫_{0}^{∞} ∫_{0}^{2π} x g(x, θ) dxdθ)^{1/2}.  (3.8)

The constant C_{12} is well defined, see (3.15) below. Now we will prove the following consequence of Proposition 3.1:

Corollary 3.3 Fix n ≥ 1. Let g_{1}, g_{2}, ..., g_{n} : G → ℝ be n measurable functions. Assume that there exist some constants ν > \frac{5}{2} and K > 0 such that for all 1 ≤ j ≤ n,

sup_{0≤θ≤2π} |g_{j}(x, θ)| ≤ \frac{K}{(1 + |x|)^{ν}} ,  x ∈ ℝ.  (3.9)

Then, possibly in a larger probability space, we may define a version of X = (β, γ̇), an n-dimensional Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 with covariance matrix (C_{13}(f_{j}, f_{k}))_{j,k≤n} and a process L such that L(·) has the same law as ℓ(·), Y and L are independent and such that almost surely for all 1 ≤ j ≤ n and all large t,

∫_{0}^{t} g_{j}(X(s))ds − C_{11}(g_{j}) ℓ(t) − C_{12}(g_{j}) Y_{j}(L_{t}) = o(t^{\frac{1}{4} −δ}),  (3.10)
|ℓ_{t} − L_{t}| = o(t^{\frac{1}{2} −δ}),  (3.11)

where δ > 10^{−5} denotes some constant and

C_{13}(g_{j}, g_{k}) =^{def} \frac{1}{4} \frac{(C_{12}(g_{j} + g_{k}))^{2} − (C_{12}(g_{j} − g_{k}))^{2}}{C_{12}(g_{j}) C_{12}(g_{k})} .

First let us introduce some notations. Let g be any of the functions g_{1}, …, g_{n} of Corollary 3.3 and define

\overline{g}(x) =^{def} \frac{1}{2π} ∫_{0}^{2π} g(x, θ)dθ,  h(x, θ) =^{def} g(x, θ) − \overline{g}(x),  (x, θ) ∈ G.  (3.12)

Note that ∫ dθh(x, θ) = 0 for any x ∈ ℝ and h satisfies (3.9). According to Kasahara and Kotani ([19], formula (2.1), pp. 141), we define

F (x, θ) = Ψ ∗ h(x, θ) = ∫_{−∞}^{∞} ∫_{0}^{2π} dx′dθ′Ψ(x − x′, θ − θ′)h(x′, θ′),  (3.13)

where Ψ ∗ h denotes the convolution of h with the function Ψ under the Haar measure, and Ψ is defined by

Ψ(x, θ) =^{def} − \frac{1}{2π} log |e^{iθ} − e^{−|x|}|^{2} = Ψ(x, −θ).  (3.14)

We need the following elementary estimates on the partial derivatives of F:

Lemma 3.4 Assume that g satisfies (3.9) with some ν > 1. Recall (3.8). We have

sup_{0≤θ≤2π} |∇F |(x, θ) ≤ \frac{c}{(1 + |x|)^{ν}} ,  x ∈ ℝ.
(C_{12}(g))^{2} = \frac{1}{2π} ∫_{−∞}^{∞} dx ∫_{0}^{2π} dθ((\frac{∂F}{∂x} (x, θ) + q(x))^{2} + \frac{∂F}{∂θ} (x, θ)^{2}), (3.15)

where

q(y) =^{def} 2 ∫_{y}^{∞} dx\overline{g}(x), y > 0;  q(y) =^{def} − 2 ∫_{−∞}^{y} dx\overline{g}(x), y ≤ 0.  (3.16)

Proof: Elementary calculations (cf. [19], pp. 136) show that

\frac{∂Ψ}{∂x}(x, θ) = −\frac{sgn(x)(cos θ − e^{−|x|})e^{−|x|}}{π|e^{iθ} − e^{−|x|}|^{2}} ∈ L^{1}(G, dxdθ),
\frac{∂Ψ}{∂θ}(x, θ) = −\frac{e^{−|x|} sin θ}{π|e^{iθ} − e^{−|x|}|^{2}} ∈ L^{1}(G, dxdθ).

It follows that

|\frac{∂F}{∂x} (x, θ)| = | ∫_{G} \frac{∂Ψ}{∂x}(x′, θ′)h(x − x′, θ − θ′)dx′dθ′|
≤ 2K ∫_{G} \frac{|\frac{∂Ψ}{∂x}(x′, θ′)|}{(1 + |x − x′|)^{ν}} dx′dθ′
≤ \frac{c}{(1 + |x|)^{ν}} .

The same holds for \frac{∂F}{∂θ}.

To show (3.15), we may assume without loss of generality that g is regular (for example g ∈ C^{2}), the general case follows from the usual approximation argument. Therefore, we have

\frac{1}{2}(\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂θ^{2}} )F = −h.  (3.17)

It follows from the periodicity on θ that the RHS of (3.15) equals

= \frac{1}{2π} ∫_{ℝ} dx ∫_{0}^{2π} dθ|∇F |^{2}(x, θ) + ∫_{ℝ} dxq^{2}(x)
= \frac{1}{π} ∫_{ℝ} dx ∫_{0}^{2π} dθh(x, θ)F (x, θ) + ∫_{ℝ} dxq^{2}(x) integration by parts (3.17)
= −\frac{1}{2π^{2}} ∫_{G×G} dxdx′dθdθ′h(x, θ)h(x′, θ′) log |e^{i(θ−θ′)} − e^{−|x−x′|}|^{2} + ∫_{ℝ} dxq^{2}(x)
= −\frac{1}{2π^{2}} ∫_{G×G} dxdx′dθdθ′g(x, θ)g(x′, θ′) log |e^{i(θ−θ′)} − e^{−|x−x′|}|^{2} + ∫_{ℝ} dxq^{2}(x), (3.18)

where the last equality follows from the elementary fact

∫_{0}^{2π} dθ log |e^{iθ} − r|^{2} = 0, ∀ 0 ≤ r < 1.

Since |e^{i(θ−θ′)} − e^{−|x−x′|}|^{2} = e^{−2 max(x,x′)} |e^{x+iθ} − e^{x′+iθ′}|^{2}, elementary computations show that the sum in (3.18) coincides with the RHS of (3.8), which is C_{12}^{2}. □

The following result is a first-order approximation of the additive functionals ∫_{0}^{t} g(X_{s})ds:

Lemma 3.5 Assume that g satisfies (3.9) with some ν > 2. For any ϵ > 0, we have

∫_{0}^{t} g(β_{s}, γ̇_{s})ds = C_{11}(g) ℓ(t) + o(t^{\frac{1}{4} +ϵ}), t → ∞, a.s.

where C_{11}(g) has been defined in (3.7).

Proof: We decompose the additive functionals ∫_{0}^{t} g(X_{s})ds as

∫_{0}^{t} g(X_{s})ds = ∫_{0}^{t} \overline{g}(β_{s})ds + ∫_{0}^{t} h(β_{s}, γ̇_{s})ds.  (3.19)

According to Csáki and Földes [10] (here we need ν > 2),

∫_{0}^{t} \overline{g}(β_{s})ds = C_{11} ℓ_{t} + o(t^{\frac{1}{4} +ϵ}), a.s.,

it remains to show that

∫_{0}^{t} h(β_{s}, γ̇_{s})ds = o(t^{\frac{1}{4} +ϵ}), a.s.

If g is in C^{2}, we apply Itô’s formula with (3.17):

F (β_{t}, γ̇_{t}) − F (β_{0}, γ̇_{0}) + ∫_{0}^{t} h(β_{s}, γ̇_{s})ds
= ∫_{0}^{t} \frac{∂F}{∂x} (β_{s}, γ̇_{s})dβ_{s} + ∫_{0}^{t} \frac{∂F}{∂θ} (β_{s}, γ̇_{s})dγ_{s} =^{def} M_{t},  (3.20)

is a martingale. Using the approximation of g by regular functions and Lemma 3.4, the equality in (3.20) also holds for all g satisfying (3.9). It turns out that

𝔼M_{t}^{2} = 𝔼 ∫_{0}^{t} |∇F |^{2}(β_{s}, γ̇_{s})ds
≤ c^{2}𝔼 ∫_{0}^{t} \frac{ds}{(1 + |β_{s}|)^{2ν}}
= c^{2} ∫_{ℝ} dx(1 + |x|)^{−2ν} 𝔼ℓ(t, x)
≤ c′\sqrt{t},  t > 0,

since 𝔼ℓ(t, x) ≤ 𝔼ℓ(t, 0) = \sqrt{\frac{2t}{π}}. Using Doob’s maximal inequality for martingales, we obtain

that for t_{n} =^{def} 2^{n},

ℙ( sup_{t≤t_{n}} |M_{t}| > tn^{\frac{1+ϵ}{4}} ) ≤ 2c′t_{n}^{−ϵ/2},

whose sums on n converges. The Borel-Cantelli lemma together with the monotonicity imply that M_{t} = o(t^{\frac{1}{4} +ϵ}), a.s. □ Proof of Corollary 3.3: For the notational convenience, we only consider the case n = 1 and g = g_{1}. To obtain the second order approximation, we first deduce from Tanaka’s formula that

∫_{0}^{t} \overline{g}(β_{s})ds = C_{11}(g) ℓ_{t} + ∫_{0}^{t} q(β_{s})dβ_{s} − ∫_{0}^{β_{t}} q(x) dx,  (3.21)

where q(·) is defined in (3.16). The term ∫_{0}^{β_{t}} q(x) dx is bounded due to the integrability: ∫_{ℝ} |q(x)|dx < ∞. This together with (3.19) and (3.20) implies that

∫_{0}^{t} g(X_{s})ds = C_{11}(g)ℓ_{t} + Q_{t} + F (X_{0}) − F (X_{t}) − ∫_{0}^{β_{t}} dxq(x)  (3.22)

with F (X_{0}) − F (X_{t}) − ∫_{0}^{β_{t}} q(x)dx = O(1) and

Q_{t} =^{def} M_{t} + ∫_{0}^{t} q(β_{s})dβ_{s}
= ∫_{0}^{t}(\frac{∂F}{∂x} (β_{s}, γ̇_{s}) + q(β_{s}))dβ_{s} + ∫_{0}^{t} \frac{∂F}{∂θ} (β_{s}, γ̇_{s})dγ_{s}.

Note from Lemma 3.4:

sup_{0≤θ≤2π} |∇F |(x, θ) + |q(x)| ≤ \frac{c}{(1 + |x|)^{ν−1}} ,

then we can apply Proposition 3.1 to (Q_{t}) and obtain Corollary 3.3, the constant C_{12}(g) follows from (3.15). □

3.2 Proof of Theorem 1.1

Just like as Corollary 3.3 was a consequence of Proposition 3.1, the next corollary follows from Proposition 3.2 in a similar way, hence we omit the details of its proof:

Corollary 3.6 Keeping all notations and assumptions of Corollary 3.3 we may define on a possibly larger probability space a version of X = (β, γ̇), an n-dimensional Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 with covariance matrix (C_{13}(g_{j}, g_{k}))_{j,k≤n} and a process ẽ such that Y and ẽ are independent, ẽ(·) has the same law as e(·) and such that almost surely for all large r and any t ∈ [σ(r − 2 log r), σ(r + 2 log r)], we have

∫_{0}^{t} g_{j}(X(s))ds − C_{11}(g_{j}) e(2r) − C_{12}(g_{j}) Y_{j}(ẽ(2r)) = o(r^{\frac{1}{2} −δ}),  (3.23)
|e(r) − ẽ(r)| = o(r^{1−δ}).  (3.24)

The factor 2 in e(·) comes from the fact that ℓ(σ(r)) = e(2r).

Proof of Theorem 1.1: We give the proof in the case n = 1. Write f ≡ f_{1}. Using the skew-product representation (1.5), we have

∫_{0}^{t} f(W (s))ds = ∫_{0}^{Ξ(t)} dve^{2β(v)}f(e^{β(v)+iγ(v)}) = ∫_{0}^{Ξ(t)} g(X_{v})dv,

where γ̇(s) =^{def} γ(s)(mod 2π), g(x, θ) =^{def} e^{2x}f(e^{x+iθ}) and

Ξ(t) = inf{u > 0 : ∫_{0}^{u} dse^{2β(s)} > t}.  (3.25)

Now we need the following result;

Lemma 3.7 (Shi [28]) For any s, t > 0, we have

ℙ(Ξ(t) ≤ σ(s)) ≤ 2 exp ( − \frac{t}{4}e^{−2s})
ℙ(Ξ(t) ≥ σ(s)) ≤ 4 exp ( − \frac{e^{2s}}{16t}),

where we recall that σ(s) =^{def} inf{u > 0 : β_{u} > s}.

Using the above Lemma and Borel-Cantelli, it is standard to obtain that almost surely for all large t,

σ(\frac{log t}{2} − log log t) ≤ Ξ(t) ≤ σ(\frac{log t}{2} + log log t).  (3.26)

Now Theorem 1.1 follows from Corollary 3.6 since we may define W through X by (1.5) and (3.25). Finally,

C_{1}(f) = 2πC_{11}(g),
C_{2}(f) = C_{12}(g),  (3.27)
C_{3}(f_{j}, f_{k}) = C_{13}(g_{j}, g_{k}),

with obvious definitions of g_{j}, g_{k} from f_{j}, f_{k}. Finally, we obtain (1.12) and (1.13) by change of variables. □ Proof of Corollary 1.2: It immediately follows from Theorem 1.1. □ Proof of Corollary 1.3: The proof goes in the same way as in Theorem 4.2 of [11]. The details are omitted. □

4 Proof of Propositions 3.1 and 3.2

Let us only consider the case n = 1 and Φ = F^{(1)} : z ∈ G → (Φ_{1}(z), Φ_{2}(z)) ∈ ℝ^{2}, the general case follows exactly in the same way, and we shall explain how to compute the correlation matrix when n ≥ 2. Assuming that Φ satisfies the condition (3.1), define

N_{t} =^{def} ∫_{0}^{t} Φ(X_{s})dX_{s} = ∫_{0}^{t} Φ_{1}(β_{s}, γ̇_{s})dβ_{s} + ∫_{0}^{t} Φ_{2}(β_{s}, γ̇_{s})dγ_{s},  t ≥ 0.  (4.1)

The goal is to approximate the continuous martingale N_{t} by a Brownian motion timechanged at ℓ(t) such that this Brownian motion is either independent of ℓ(·) (Proposition 3.1) or independent of (ℓ(σ(·))) (Proposition 3.2):

Dubins-Schwarz’ representation theorem of continuous martingale implies that

N_{t} = B(⟨N⟩_{t}),  (4.2)

with some one-dimensional Brownian motion B. It follows that

⟨N⟩_{t} = ∫_{0}^{t} |Φ|^{2}(X_{s})ds = ∫_{0}^{t}(Φ_{1}^{2}(X_{s}) + Φ_{2}^{2}(X_{s}))ds
= C_{9}(Φ) ℓ(t) + o(t^{\frac{1}{4} +ϵ}),  t → ∞,  a.s.,  (4.3)

by using Lemma 3.5. But we can not choose a Brownian motion B independent of ℓ(·) or independent of ℓ(σ(·)) at this stage. The independence will be obtained by using Berkes and Philipp’s lemma:

Lemma 4.1 ([4]) Let (η_{k}, k ≥ 1) be a sequence of random variables with values in ℝ^{d}, adapted with respect to some filtration (F_{k}). Let {g_{k}, k ≥ 1} be a sequence of characteristic functions of probability distributions G_{k} on ℝ^{d}. Suppose that for some nonnegative numbers ϵ_{k}, δ_{k} and Θ_{k} ≥ 10^{8}d,

𝔼|𝔼(e^{izη_{k}} | F_{k−1}) − g_{k}(z)| ≤ ϵ_{k}, ∀ z ∈ ℝ^{d}, |z| ≤ Θ_{k},

and

G_{k}(z : |z| > \frac{Θ_{k}}{4} ) ≤ δ_{k}.

Then without changing its distribution we can redefine the sequence {η_{k}, k ≥ 1} on a richer probability space together with a sequence of {Y_{k}, k ≥ 1} of independent random variables such that Y_{k} has characteristic function g_{k} and

ℙ(|η_{k} − Y_{k}| ≥ α_{k}) ≤ α_{k}

and

α_{k} = 16d\frac{log Θ_{k}}{Θ_{k}} + 4\sqrt{ϵ_{k}} Θ_{k}^{d} + δ_{k}.

Let (F_{t}) be the natural filtration generated by X and denote by 𝔼_{x,θ} the expectation with respect to the Brownian motion X starting from X_{0} = (x, θ) ∈ G. Let us present an exponential moment estimation:

Lemma 4.2 Fix ν > 2. Assume that g : G → ℝ is a measurable function such that for some constant b > 0,

sup_{0≤θ≤2π} |g(x, θ)| ≤ \frac{b}{(1 + |x|)^{ν}} , x ∈ ℝ.

There exists some positive constants C_{14}(g) > 1 such that for all 0 ≤ a ≤ \frac{1}{C_{14}} and r > 0, we have

𝔼_{0,θ} exp (a| ∫_{0}^{τ_{r}} g(β_{s}, γ̇_{s})ds − C_{11}(g)r|) ≤ C_{14} e^{C_{14} a^{2}r},  (4.4)
𝔼_{0,θ}( ∫_{0}^{τ_{r}} g(β_{s}, γ̇_{s})ds − C_{11}(g)r)^{2} ≤ C_{14} r,  r ≥ 1,  (4.5)

where C_{11} is defined in (3.7).

Proof of Lemma 4.2: Recall (3.12). Applying (2.3) to \overline{g}(·) implies that

𝔼 exp (a| ∫_{0}^{τ_{r}} \overline{g}(β_{s})ds − C_{11}(g)r|) ≤ c′ e^{c′ a^{2}r}, r > 0, 0 ≤ a ≤ \frac{1}{c′} ,

for some constant c′ > 1 depending on g. Recall the martingale (M_{t}) defined in (3.20). We have

∫_{0}^{τ_{r}} g(β_{s}, γ̇_{s})ds − C_{11}(g)r = ∫_{0}^{τ_{r}} \overline{g}(β_{s})ds − C_{11}(g)r + M(τ_{r}) + F (0, γ̇_{0}) − F (0, γ̇_{τ_{r}}).

Using successively Lemmas 2.1, 3.4 and (2.2), we obtain

𝔼_{0,θ} exp (a|M(τ_{r})|) ≤ 2\sqrt{𝔼_{0,θ} exp (2a^{2}⟨M⟩(τ_{r}))}

= 2\sqrt{𝔼_{0,θ} exp (2a^{2} ∫_{0}^{τ_{r}} |∇F |^{2}(β_{s}, γ̇_{s})ds)}

≤ 2\sqrt{𝔼 exp (2c^{2}a^{2} ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−2ν}ds)}
≤ 2\sqrt{C_{8}(ν)} e^{C_{8}(ν)c^{2}a^{2}r},  if 2c^{2}a^{2} ≤ \frac{1}{C_{8}(ν)},

where C_{8}(ν) ≥ 1 denotes the constant in (2.2) corresponding to the function h(x) = (1 + |x|)^{−2ν}. Let c′′ = max(c′, 2(c + 1)^{2}C_{8}(ν)). Then a ≤ \frac{1}{c′′} implies that 2c^{2}a^{2} ≤ \frac{1}{C_{8}(ν)}. Hence, we have shown that

𝔼_{0,θ} exp (a|M(τ_{r})|) ≤ c′′e^{c′′a^{2}r}, 0 ≤ a ≤ \frac{1}{c′′} .

The continuous function F (0, ·) is uniformly bounded by some constant, say c_{0}(F). It follows from Cauchy-Schwarz’ inequality that for all 0 ≤ a ≤ \frac{1}{2c′′}, we have

𝔼_{0,θ}e^{a| ∫_{0}^{τr} g(β_{s},γ̇_{s})ds−C_{11}(g)r|} ≤ e^{2c_{0}(F)a} \sqrt{𝔼_{0,θ}e^{2a| ∫_{0}^{τr} \overline{g}(β_{s})ds−C_{11}(g)r|}} \sqrt{𝔼_{0,θ}e^{2a|M(τ_{r})|}}
≤ e^{c_{0}(F)/c′′} \sqrt{c′c′′} e^{2(c′+c′′)a^{2}r}
≤ c′′′e^{c′′′a^{2}r},

with c′′′ =^{def} e^{c_{0}(F)/c′′}2(c′ + c′′) > 2c′′ > 2. Using the above estimate together with the elementary inequality: x^{2} ≤ 2 (c′′′)^{2} r exp (\frac{|x|}{c′′′\sqrt{r}}), we obtain that for all 0 ≤ a ≤ \frac{1}{c′′′},

𝔼_{0,θ}| ∫_{0}^{τ_{r}} g(β_{s}, γ̇_{s})ds − C_{11}(g)r| ≤ 2(c′′′)^{3} e^{1/c′′′} r < 4(c′′′)^{3}r.

Finally, we choose C_{14} = 4(c′′′)^{3} and both (4.4) and (4.5) are satisfied. □

We shall use several times the following estimates: Lemma 4.3 Assuming that Φ satisfies (3.1) for some ν > 1. There exists some constant C_{15}(Φ) > 1 such that for all r ≥ 1, |u| ≤ \frac{r^{1/4}}{C_{15}} , we have

𝔼_{0,θ}e^{\frac{2u^{2}}{r} |⟨N,N⟩_{τr} −C_{9}(Φ)r|} ≤ C_{15},  (4.6)
𝔼_{0,θ}|e^{\frac{u^{2}}{r} (⟨N,N⟩_{τr} −C_{9}(Φ)r)} − 1| ≤ \frac{C_{15}u^{2}}{\sqrt{r}} ,  (4.7)

for any θ ∈ [0, 2π], and C_{9}(Φ) has been defined in (3.4). Proof: By the condition on Φ,

|\frac{d⟨N, β⟩_{s}}{ds} | ≤ K (1 + |β_{s}|)^{−ν},  (4.8)
\frac{d⟨N, N⟩_{s}}{ds} = |Φ(X_{s})|^{2} ≤ K^{2} (1 + |β_{s}|)^{−2ν}.  (4.9)

Applying Lemma 4.2 to g = |Φ|^{2} and a = 2u^{2}/r ≤ 2/C_{15}^{2} ≤ 1/C_{14}(|Φ|^{2}) implies (4.6). Using the elementary fact that for x ∈ ℝ, |e^{x} − 1| ≤ |x|e^{|x|}, (4.6) and Cauchy-Schwarz’ inequality, we have

𝔼_{0,θ}|e^{\frac{u^{2}}{r} (⟨N,N⟩_{τr} −C_{9}(Φ)r)} − 1|
≤ \frac{u^{2}}{r} \sqrt{𝔼_{0,θ}(⟨N, N⟩_{τ_{r}} − C_{9}(Φ)r)^{2} 𝔼_{0,θ}e^{\frac{2u^{2}}{r} (⟨N,N⟩_{τr} −C_{9}(Φ)r)}}
≤ \sqrt{C_{14}C_{15}} \frac{u^{2}}{\sqrt{r}},

by virtue of (4.5). □

The main technical lemmas are the following Lemmas 4.4 and 4.5: Lemma 4.4 Assume that Φ satisfies (3.9) with ν > \frac{3}{2}. For u, v ∈ ℝ with |u| ≤ r^{1/4}/C_{15} and |v| ≤ r, we have that for any θ ∈ [0, 2π],

|𝔼_{0,θ}e^{iu \frac{N(τr)}{\sqrt{r}} +iv \frac{\overline{β}(τr)}{r}} − e^{−C_{9}(Φ)u^{2}/2} 𝔼e^{iv\overline{β}(τ_{1})}| ≤ \frac{c|u| log r}{\sqrt{r}} + \frac{cu^{2}}{\sqrt{r}} .

Proof: Recall that (F_{t}) is the natural filtration generated by (β, γ). Lemma 2.5 implies that

D_{t} =^{def} 𝔼(e^{iv \frac{\overline{β}(τr)}{r}} | F_{t}) = D_{0} + ∫_{0}^{t∧τ_{r}} ζ_{s}(r, \frac{v}{r}) dβ_{s},

with D_{0} = 𝔼e^{iv \frac{\overline{β}(τr)}{r}} = 𝔼e^{iv\overline{β}(τ_{1})} by the Brownian scaling property, and

|ζ_{s}(r, \frac{v}{r})| ≤ 2(\frac{1_{(\overline{β}_{s}≥1)}}{\overline{β}_{s}} + 1_{(\overline{β}_{s}<1)}(1 + \frac{|v|}{r} log(1/\overline{β}_{s}))).

Let

R_{t} = exp (i\frac{u}{\sqrt{r}}N_{t} + \frac{u^{2}}{2r}(⟨N⟩_{t} − C_{9}(Φ)r)) = e^{−C_{9} u^{2}/2} + \frac{iu}{\sqrt{r}} ∫_{0}^{t} R_{s}dN_{s}.

Define R_{t}^{∗} =^{def} sup_{0≤s≤t} |R_{s}|. Then by Lemma 4.3,

𝔼(R_{τ_{r}}^{∗})^{4} ≤ 𝔼e^{\frac{2u^{2}}{r} (⟨N,N⟩_{τr} −C_{9}r)} ≤ C_{15},  |u| ≤ \frac{r^{1/4}}{C_{15}} .  (4.10)

We have d⟨R, D⟩_{s} = i\frac{u}{\sqrt{r}}R_{s}ζ_{s}(r, \frac{v}{r})Φ_{1}(X_{s})ds, hence

|\frac{d⟨R, D⟩_{s}}{ds} | ≤ \frac{2c|u|}{\sqrt{r}} R_{τ_{r}}^{∗} (1 + |β_{s}|)^{−ν} × \{ \begin{matrix}(1 + \overline{β}_{s})^{−1} & if \overline{β}_{s} > 1, \\ 1 + \frac{|v|}{r} log(1/\overline{β}_{s}) & if \overline{β}_{s} ≤ 1. \end{matrix}  (4.11)

First we prove that

|𝔼_{0,θ}(R_{τ_{r}} D_{τ_{r}}) − R_{0} D_{0}| ≤ \frac{c|u| log r}{\sqrt{r}}.  (4.12)

To this end, we remark that

|𝔼_{0,θ}(R_{τ_{r}} D_{τ_{r}}) − R_{0}D_{0}| = |𝔼_{0,θ}⟨R, D⟩_{τ_{r}}|
≤ \frac{2K|u|}{\sqrt{r}} (𝔼_{0,θ}R_{τ_{r}}^{∗} ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−ν} \frac{1_{(\overline{β}_{s}≥1)}}{\overline{β}_{s}} ds

+𝔼_{0,θ}R_{τ_{r}}^{∗} ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−ν}1_{(\overline{β}_{s}<1)}(1 + \frac{|v|}{r} log(1/\overline{β}_{s}))ds)

=^{def} \frac{2K|u|}{\sqrt{r}}(J_{1} + J_{2}),

with obvious definitions of J_{1} and J_{2}. By using Cauchy-Schwarz’ inequality and Lemma 2.6, we have

J_{1} ≤ \sqrt{𝔼_{0,θ}(R_{τ_{r}}^{∗})^{2}} \sqrt{𝔼_{0,θ}( ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−ν} \frac{ds}{1 + \overline{β}_{s}} )^{2}} ≤ c′ log r, r ≥ 2.

Recall (1.8) for σ(x). Again from Cauchy-Schwarz’ inequality,

J_{2} ≤ (C_{15})^{1/4} \sqrt{𝔼( ∫_{0}^{σ(1)} \frac{ds}{(1 + |β_{s}|)^{ν}} (1 + log(1/\overline{β}_{s})))^{2}} = c′′ < ∞,

where we may obtain the square integrability of ∫_{0}^{σ_{1}} \frac{ds}{(1+|β_{s}|)^{ν}} (1 + log(1/\overline{β}_{s})) by using the following argument: for n ≥ 1,

∫_{σ(e^{−n})}^{σ(1)} \frac{ds}{(1 + |β_{s}|)^{ν}} (1 + log(1/\overline{β}_{s})) ≤ ∑_{k=1}^{n}(1 + k) ∫_{σ(e^{−k})}^{σ(e^{−(k−1)})} \frac{ds}{(1 + |β_{s}|)^{ν}} .

It is easy to obtain that the second moment of the above sum (of independent variables) is uniformly bounded with respect to n, hence c′′ is finite and (4.12) follows. Finally, we have

|𝔼_{0,θ}e^{iu \frac{N(τr)}{\sqrt{r}} +iv \frac{\overline{β}(τr)}{r}} − 𝔼_{0,θ}R_{τ_{r}}D_{τ_{r}} | ≤ 𝔼_{0,θ}|e^{\frac{u^{2}}{2r} (⟨N⟩_{τr} −C_{9}r)} − 1| ≤ \frac{cu^{2}}{r^{1/2}} ,

by (4.7). This together with (4.12) completes the proof. □

Recall the Fourier transform for the stable law τ_{1}: 𝔼e^{ivτ_{1}} = exp(−\sqrt{|v|}(1−i sgn(v))), v ∈ ℝ.

Lemma 4.5 Assume that Φ satisfies (3.9) with ν > \frac{3}{2}. There exists some constant C_{16}(g) > 0 such that for any θ ∈ [0, 2π], |u| ≤ \frac{r^{1/4}}{C_{15}} , v ∈ ℝ and r ≥ 2, we have

|𝔼_{0,θ}e^{i \frac{u}{\sqrt{r}} N(τ_{r})+i \frac{v}{r^{2}} τ_{r}} − e^{− \frac{C_{9}(Φ)u^{2}}{2}} exp(−\sqrt{|v|}(1 − i sgn(v)))| ≤ C_{16} (\frac{u^{2}}{\sqrt{r}} + \frac{|u||v|^{1/2}}{\sqrt{r}} ). (4.13)

Proof: Let

w =^{def} (1 − i sgn(v))\frac{\sqrt{|v|}}{r}.

Observe that the process

S_{t} =^{def} exp (i\frac{v}{r^{2}} t − w(|β_{t}| − ℓ_{t} + r)) = S_{0} − w ∫_{0}^{t} S_{s} sgn(β_{s})dβ_{s}, t ≥ 0,

is a martingale such that S_{τ_{r}} = e^{ivτ_{r}/r^{2}}. Furthermore, sup_{0≤s≤τ_{r}} |S_{s}| ≤ 1. Using the martingale (R_{t}) introduced in the proof of Lemma 4.4 and (4.8), we obtain that

|𝔼_{0,θ} (R_{τ_{r}}S_{τ_{r}}) − R_{0}S_{0}| = |𝔼_{0,θ} ⟨R, S⟩_{τ_{r}}|
≤ K \frac{|u|}{\sqrt{r}}|w| 𝔼_{0,θ} ( R_{τ_{r}}^{∗} ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−ν} ds)

≤ K \frac{|u|}{\sqrt{r}}|w| \sqrt{𝔼_{0,θ} (R_{τ_{r}}^{∗})^{2}} \sqrt{𝔼( ∫_{0}^{τ_{r}}(1 + |β_{s}|)^{−ν} ds)^{2}}

≤ c \frac{|u|\sqrt{|v|}}{\sqrt{r}},

by means of (4.10) and (2.6). Therefore we have shown that

|𝔼_{0,θ} (e^{iu \frac{N(τr)}{\sqrt{r}} + \frac{u^{2}}{2r} (⟨N,N⟩_{r}−C_{9}r)}e^{iv \frac{τr}{r^{2}}} ) − e^{−C_{9} \frac{u^{2}}{2}} exp(−(1 − i sgn(v))\sqrt{|v|})| ≤ c \frac{|u|\sqrt{|v|}}{\sqrt{r}},

which together with (4.7) implies Lemma 4.5. □ Proof of Proposition 3.2: Let t_{k} = k^{ρ} with ρ > 20 and k ≥ 100. Define

η_{k} = (\frac{N(τ_{t_{k}}) − N(τ_{t_{k−1}})}{\sqrt{t_{k} − t_{k−1}}}, \frac{sup_{τ_{t_{k−1}}≤s≤τ_{t_{k}}} β_{s}}{t_{k} − t_{k−1}} ) ≡ (η_{k}^{(1)}, η_{k}^{(2)}).

The strong Markov property implies that for z = (u, v) ∈ ℝ^{2}, we have

𝔼(e^{iz·η_{k}} | F_{τ_{t_{k−1}}}; γ̇(τ_{t_{k−1}}) = θ) = 𝔼_{0,θ}(e^{i \frac{u}{\sqrt{r}} N(τ_{r})+i \frac{v}{r} \overline{β}_{τr}} ),

with r = t_{k} − t_{k−1}. Let

g(u, v) = e^{−C_{9} \frac{u^{2}}{2}} 𝔼e^{iv\overline{β}(τ_{1})}, u, v ∈ ℝ,

be the joint Fourier transform of a Gaussian variable N (0, C_{9}) and an independent copy of \overline{β}(τ_{1}), whose law has been given in (2.9).

Applying Lemmas 4.4 and 4.1 with Θ_{k} = k^{\frac{ρ−1}{16}} , we get that for all |u|, |v| ≤ Θ_{k},

𝔼|𝔼(e^{iz·η_{k}} | F_{τ_{t_{k−1}}}) − g(z)| ≤ Kk^{− \frac{3(ρ−1)}{8}} .

It follows from the Gaussian tail and the distribution of \overline{β}(τ_{1}) given in (2.9) that

G_{k}(z : |z| ≥ \frac{Θ_{k}}{4} ) ≤ K k^{−(ρ−1)/16}.

Hence we may construct a sequence {Z_{k} = (Z_{k}^{(1)}, Z_{k}^{(2)}), k ≥ 1} of i.i.d. variables and a version of {η_{k} = (η_{k}^{(1)}, η_{k}^{(2)}), k ≥ 1} in a sufficiently large probability space such that the two sequences (Z_{k}^{(1)}) and (Z_{k}^{(2)}) are independent and that

Z_{k}^{(1)} =^{law} N (0, C_{9}(Φ)),  Z_{k}^{(2)} =^{law} \overline{β}(τ_{1}),
ℙ(|η_{k} − Z_{k}| ≥ α_{k}) ≤ α_{k},
α_{k} ≤ c k^{−(ρ−1)/16} log k.

The Borel-Cantelli lemma yields that almost surely for all large k, |η_{k} − Z_{k}| ≤ α_{k}. Hence

N(τ_{t_{n}}) = ∑_{k=1}^{n} η_{k}^{(1)} \sqrt{t_{k} − t_{k−1}} = ∑_{k=1}^{n} Z_{k}^{(1)} \sqrt{t_{k} − t_{k−1}} + Φ_{n}^{(1)},  (4.14)
\overline{β}(τ_{t_{n}}) = max_{1≤k≤n} (η_{k}^{(2)}(t_{k} − t_{k−1})) = max_{1≤k≤n} (Z_{k}^{(2)}(t_{k} − t_{k−1})) + Φ_{n}^{(2)},  (4.15)

where the error terms Φ_{n}^{(1)} and Φ_{n}^{(2)} can be estimated as follows: almost surely as n → ∞,

|Φ_{n}^{(1)}| ≤ ∑_{1}^{n} α_{k}\sqrt{t_{k} − t_{k−1}} + O(1) ≤ O(n^{\frac{7ρ+9}{16}} log n) ≤ O(t_{n}^{\frac{7}{16} + \frac{9}{16ρ}} log t_{n}),
|Φ_{n}^{(2)}| ≤ max_{1≤k≤n}(α_{k}(t_{k} − t_{k−1})) + O(1) ≤ O(t_{n}^{\frac{15}{16} − \frac{15}{16ρ}} log t_{n}).

Lemma 4.6 ([4]) Let S_{i}, i = 1, 2, 3 be separable Banach spaces. Let F be a distribution on S_{1} × S_{2} and let G be a distribution on S_{2} × S_{3} such that the second marginal of F equals the first marginal of G. Then there exist three random variables Z_{1}, Z_{2} and Z_{3} defined on some probability space such that

(Z_{1}, Z_{2}) =^{law} F, (Z_{2}, Z_{3}) =^{law} G.

Using repeatedly Lemma 4.6, we may rewrite (4.14) and (4.15) as follows: Possibly in an enlarged probability space, we may define a version of (N(τ_{r}), \overline{β}(τ_{r})) and a Brownian motion Y and a process T_{r} such that Y and T are independent, T has the same law as \overline{β}(τ_{·}) and

|N(τ_{t_{n}}) − \sqrt{C_{9}(Φ)} Y (t_{n})| ≤ O(t_{n}^{\frac{7}{16} + \frac{9}{16ρ}} log t_{n}),  (4.16)
|\overline{β}(τ_{t_{n}}) − T (t_{n})| ≤ O(t_{n}^{\frac{15}{16} − \frac{15}{16ρ}} log t_{n}).  (4.17)

Recall the following result on the increments of a standard Brownian motion (cf. [12], Theorem 1.2.1): For a non-decreasing function 0 < a_{t} ≤ t such that t/a_{t} ↑ +∞, we have

lim sup_{t→∞} \frac{1}{\sqrt{2a_{t}(log(t/a_{t}) + log log t)}} sup_{0≤s≤t−a_{t}} sup_{0≤v≤a_{t}} |Y (s + v) − Y (s)| = 1,  a.s. (4.18)

Using (4.2), (4.3) and (4.18), we obtain that almost surely for all large k (t_{k} = k^{ρ}), we have

sup_{τ(t_{k})≤s≤τ(t_{k+1})} |N(s) − N(τ(t_{k}))| = sup_{⟨N⟩(τ(t_{k}))≤u≤⟨N⟩(τ(t_{k+1}))} |B(u) − B(⟨N⟩(τ(t_{k})))|
≤ 3\sqrt{C_{9} (t_{k+1} − t_{k}) log log k}
≤ t_{k}^{\frac{1}{2} − \frac{1}{2ρ}} log k.

The same holds for the Brownian motion Y:

sup_{τ(t_{k})≤s≤τ(t_{k+1})} |Y (ℓ_{s}) − Y (t_{k})| ≤ t_{k}^{\frac{1}{2} − \frac{1}{2ρ}} log k.

It follows that almost surely for all large r, τ_{t_{k}} ≤ r < τ_{t_{k+1}}, we have t_{k} ≤ ℓ_{r} ≤ 2\sqrt{r log log r} and

N(r) = N(τ_{t_{k}}) + O(t_{k}^{\frac{1}{2} − \frac{1}{2ρ}} log k) = \sqrt{C_{9}(Φ)} Y (t_{k}) + O(t_{k}^{\frac{1}{2} −ϵ})
= \sqrt{C_{9}(Φ)} Y (ℓ(r)) + O(r^{\frac{1}{4} − \frac{ϵ}{2}} ), a.s.,  (4.19)

with ϵ < min(\frac{1}{16} − \frac{9}{ρ}, \frac{1}{2ρ}).

We also need the increments of the process e(t) = ℓ(σ(t/2)). Remark that for r > 10,

ℙ(ℓ(σ(r + 2r^{1−ϵ/2})) − ℓ(σ(r − 2r^{1−ϵ/2})) ≥ r^{1−ϵ/3}) ≤ ℙ(ℓ(σ(4r^{1−ϵ/2})) ≥ r^{1−ϵ/3}) = e^{−r^{1/6}/8},

since ℓ(σ(t)) is exponentially distributed with mean 2t. It is routine to apply the Borel-Cantelli lemma and the monotonicity and obtain that almost surely for all large r,

ℓ(σ(r + r^{1−ϵ/2})) − ℓ(σ(r − r^{1−ϵ/2})) ≤ r^{1−ϵ/3}.  (4.20)

Similarly, we have that almost surely for all large r,

ℓ(σ(r + 2 log r)) − ℓ(σ(r − 2 log r)) ≤ 16 log^{2} r.  (4.21)

Let Γ =^{def} T^{−1} denote the inverse process of T and define ẽ(r) = Γ(r/2). Hence ẽ(·) has the same law as e(·). Using (4.17) and (4.20), we can show that for all large r,

|Γ_{r} − ℓ(σ_{r})| ≤ r^{1−ϵ/6}.

In fact, for large r, there exists a n such that \overline{β}(τ_{t_{n−1}}) ≤ r < \overline{β}(τ_{t_{n}}), then t_{n−1} ≤ ℓ(σ_{r}) < t_{n}. By (4.17), r < \overline{β}(τ_{t_{n}}) < T (t_{n}) + t_{n}^{1−ϵ}. Since T^{−1} has the same law as ℓ(σ(·)), we deduce from (4.20) that T_{r}^{−1} ≤ T^{−1}(T_{t_{n}} + t_{n}^{1−ϵ}) < t_{n} + (T (t_{n}))^{1−ϵ/3} < t_{n} + t_{n}^{1−ϵ/4} since t_{n}^{1−ϵ} < (T_{t_{n}})^{1−ϵ/2}. Similarily, we have T_{r}^{−1} > t_{n−1} − t_{n−1}^{1−ϵ/4}. Hence

|Γ_{r} − ℓ(σ_{r})| ≤ t_{n} − t_{n−1} + 2t_{n}^{1−ϵ/4} ≤ 3t_{n}^{1−ϵ/4} ≤ r^{1−ϵ/6}.

Assembling (4.18) and (4.21) and applying (4.19), we have that almost surely for all large r and for all σ(r − 2 log r) ≤ t ≤ σ(r + 2 log r),

N_{t} = \sqrt{C_{9}(Φ)} Y (ℓ(σ_{r})) + O(r^{(1−ϵ)/2}) = \sqrt{C_{9}(Φ)} Y (Γ_{r}) + o(r^{\frac{1}{2} − \frac{ϵ}{14}} ),

proving Proposition 3.6 for the case n = 1, and we may choose ρ = 160, ϵ = \frac{1}{400} and δ = ϵ/16 > 10^{−5}. □ Proof of Proposition 3.1: The proof goes in the same way as Proposition 3.2, by considering the sequence of vectors

(\frac{N(τ_{t_{k}}) − N(τ_{t_{k−1}})}{\sqrt{t_{k} − t_{k−1}}}, \frac{τ_{t_{k}} − τ_{t_{k−1}}}{(t_{k} − t_{k−1})^{2}} ), t_{k} = k^{ρ}, ρ > 10,

and by applying Lemma 4.5. The details are omitted.

Let us compute the correlation matrix. Assume (3.1) for F^{(1)}, ..., F^{(n)}. Define N_{j} the martingale from F^{(j)} in the same way as N was defined from Φ. Then by using Lemma 3.5, we have

⟨N_{j}, N_{k}⟩(t) = C_{17}(F^{(j)}, F^{(k)}) ℓ(t) + o(t^{1/4+ϵ}), a.s..,

where

C_{17}(F^{(j)}, F^{(k)}) =^{def} \frac{1}{2π} ∫_{ℝ} dx ∫_{0}^{2π} dθ [F_{1}^{(j)}(X_{s})F_{1}^{(k)}(X_{s}) + F_{2}^{(j)}(X_{s})F_{2}^{(k)}(X_{s})]
= \frac{1}{4}[C_{9}(F^{(j)} + F^{(k)}) − C_{9}(F^{(j)} − F^{(k)})],

Therefore, the correlation matrix is given by

C_{10}(F^{(j)}, F^{(k)}) =^{def} \frac{C_{17}(F^{(j)}, F^{(k)})}{\sqrt{C_{9}(F^{(j)})C_{9}(F^{(k)})}} = \frac{1}{4} \frac{C_{9}(F^{(j)} + F^{(k)}) − C_{9}(F^{(j)} − F^{(k)})}{\sqrt{C_{9}(F^{(j)})C_{9}(F^{(k)})}}.

□

5 Some applications

5.1 Winding numbers

Recall (1.5). The process θ(·) describes the total winding angle of W around the origin. Define

Π(t) =^{def} ∫_{0}^{t} f(|W (s)|) dθ(s), t ≥ 0,

Where f : ℝ^{+} → ℝ is a function satisfying

|f(x)| ≤ \frac{K}{(1 + | log x|)^{ν}} , x > 0

with some K > 0 and ν > \frac{3}{2}.

Using the skew-product representation (1.5) and (1.6), we have

Π(t) = ∫_{0}^{Ξ(t)} f(e^{β_{s}}) dγ_{s}.

In view of (3.26), we deduce from Proposition 3.2 that in a suitable probability space

Π(t) = (∫_{0}^{∞} \frac{f^{2}(r)}{r} dr)^{1/2} Y (ẽ(log t)) + o((log t)^{\frac{1}{2} −δ}, a.s.,

for a one-dimensional Brownian motion Y, independent of the process ẽ. This allows us to obtain the upper and lower functions for Π(t) as in Corollary 1.3, the details are omitted.

Let 0 < r_{1} < r_{2} < ∞, then the particular function f(x) = 1_{(r_{1}≤x≤r_{2})} gives winding in a ring. See Messulam and Yor [21] for studies of convergence in law, Shi [28] (case r_{1} = 0 or r_{2} = ∞) and Dorofeev [13] for upper and lower functions.

5.2 Additive functionals of a Cauchy process

Let (C(t), t ≥ 0) be a symmetric Cauchy process on ℝ, which means a Lévy process with marginal distribution

ℙ(C(t) ∈ dx) = \frac{t dx}{π(t^{2} + x^{2})},  x ∈ ℝ,  t > 0.

Several interesting geometric quantities such as level crossings have been studied in [24] and [5]. Let f : ℝ → ℝ. Kasahara [18] studied the additive functional ∫_{0}^{·} f(C(s))ds and obtained its second-order behavior similar to Theorem B. The goal of this paragraph is to obtain an analogue Theorem 1.1 for Cauchy process, by using Spitzer’s representation of C(·): Let W = (W_{1}, W_{2}) be the planar Brownian motion starting from (1, 0). Denote by L_{2}(·) the local time at 0 of W_{2} and τ_{2}(·) the inverse process of L_{2}(·). Then the process

C(t) =^{def} W_{1}(τ_{2}(t)), t ≥ 0,

is a symmetric Cauchy process starting from 1 (the starting point does not influence our result). Recall that X is the Brownian motion on the cylinder: X_{s} = (β_{s}, γ̇_{s}), s ≥ 0.

Lemma 5.1 Let f : ℝ → ℝ be a measurable function such that for some constants K > 0 and ν > 2, we have

|f(x)| ≤ \frac{K}{|x| (1 + | log |x||)^{ν}} ,  x ∈ ℝ\\{0\}.  (5.1)

Then

∫_{0}^{t} f(C(s))ds

= C_{18}(f) ℓ(Ξ(τ_{2}(t))) + ∫_{0}^{Ξ(τ_{2}(t))} ((\frac{∂F_{0}}{∂x} (X_{s}) + q_{0}(β_{s}))dβ_{s} + \frac{∂F_{0}}{∂θ} (X_{s})dγ_{s})

+F_{0}(X_{0}) − F_{0}(X(Ξ(τ_{2}(t)))) − ∫_{0}^{β(Ξ(τ_{2}(t)))} dxq_{0}(x),

where, using the function Ψ given in (3.14), we define

C_{18}(f) =^{def} \frac{1}{2π} ∫_{ℝ} f(x) dx
F_{0}(x, θ) =^{def} ∫_{ℝ} dx′e^{x′}(f(e^{x′})Ψ(x − x′, θ) + f(−e^{x′})Ψ(x − x′, θ − π)), x ∈ ℝ, θ ∈ [0, 2π]
q_{0}(x) =^{def} \frac{1}{π} ∫_{|y|≥e^{x}} f(y)dy, if x > 0; q_{0}(x) =^{def} − \frac{1}{π} ∫_{|y|<e^{x}} f(y)dy, if x ≤ 0.

Proof: Assume without loss of generality that f ∈ C^{2}(ℝ → ℝ) has compact support. Let f_{ϵ} : ℝ^{2} → ℝ defined by

f_{ϵ}(x_{1}, x_{2}) =^{def} f(x_{1}) \frac{1}{ϵ} 1_{(0<x_{2}<ϵ)}, x_{1}, x_{2} ∈ ℝ.

Define g_{ϵ} : G → ℝ by

g_{ϵ}(x, θ) =^{def} e^{2x}f_{ϵ}(e^{x+iθ}) ≡ e^{2x}f_{ϵ}(e^{x} cos θ, e^{x} sin θ), (x, θ) ∈ G.

Therefore using Spitzer’s representation and change of variable,

∫_{0}^{t} f(C(s)) ds = ∫_{0}^{τ_{2}(t)} f(W_{1}(u)) dL_{2}(u)
= lim_{ϵ→0} ∫_{0}^{τ_{2}(t)} f_{ϵ}(W_{1}(u), W_{2}(u)) du  (approximation of local time)
= lim_{ϵ→0} ∫_{0}^{Ξ(τ_{2}(t))} g_{ϵ}(X_{s}) ds.  (1.5) and (1.6)

Recall (3.12), (3.13), (3.16), (3.19) and (3.20). We define h_{ϵ}, F_{ϵ}, q_{ϵ} and M_{ϵ} from g_{ϵ} in the same way as h, F, q, M was defined from g. Then (3.22) holds for g_{ϵ} instead of g, which implies the Lemma by letting ϵ → 0. □

Applying (3.26) gives that almost surely for all large t, σ(log t − 3 log log t) ≤ Ξ(τ_{2}(t)) ≤ σ(log t + 3 log log t), since \frac{t^{2}}{log t} ≤ τ_{2}(t) ≤ t^{2} log^{3} t. Applying Proposition 3.2 to the above lemma and using (4.21), we obtain

Proposition 5.2 Assume that f_{1}, ...f_{n} : ℝ → ℝ are n measurable functions such that for some constants K > 0 and ν > \frac{5}{2}, we have

|f_{j}(x)| ≤ \frac{K}{|x| (1 + | log |x||)^{ν}} , 1 ≤ j ≤ n, x ∈ ℝ\\{0\}.

Then we may define a version of a Cauchy process C(·), an n-dimensional Brownian motion Y = (Y_{1}, ..., Y_{n}) starting from 0 with covariance matrix (C_{20}(f_{j}, f_{k}))_{1≤j,k≤n} and two inhomogeneous Lévy process e and ẽ such that Y and ẽ are independent, ẽ has the same law as e and such that almost surely for all large t, we have

∫_{0}^{t} f_{j}(C(s))ds − C_{18}(f_{j}) e(2 log t) − C_{19}(f_{j}) Y_{j}(ẽ(2 log t)) = o((log t)^{\frac{1}{2} −δ}),
|e(log t) − ẽ(log t)| = o((log t)^{1−δ}),

for some positive constant δ > 0 and

C_{19}(f) =^{def} ( − \frac{1}{2π^{2}} ∫_{ℝ} dx ∫_{ℝ} dx′f(x)f(x′) log |x − x′|^{2}

+\frac{2}{π^{2}} C_{18}(f) ∫_{|x|≥1} f(x) log |x|dx)^{1/2},

C_{20}(f_{j}, f_{k}) =^{def} \frac{1}{4} \frac{(C_{19}(f_{j} + f_{k}))^{2} − (C_{19}(f_{j} − f_{k}))^{2}}{C_{19}(f_{j}) C_{19}(f_{k})}

Acknowledgements

Cooperation between the authors was supported by the joint French–Hungarian Intergovernmental Grant “Balaton” (grant no. F-39/00). The authors are indebted to Marc Yor, and the unknown referee for useful remarks.

References

[1] Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990.

[2] Azéma, J., Duflo, M. and Revuz, D.: Mesure invariante sur les classes récurrentes des processus de Markov. Z. Wahrsch. verw. Geb. \underline{8} (1967) 157–181.

[3] Berkes, I., Chen, X. and Horváth, L.: Central limit theorems for logarithmic averages. Studia Sci. Math. Hungar. \underline{38} (2001), 79–96.

[4] Berkes, I. and Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. \underline{7} (1979), no. 1, 29–54.

[5] Burdzy, K., Pitman, J.W. and Yor, M.: Some asymptotic laws for crossings and excursions. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987) Astérisque \underline{157-158} (1988), 59–74.

[6] Borodin, A.N. and Salminen, P.: Handbook of Brownian Motion—Facts and Formulae. Probability and its Applications. Birkhäuser Verlag, Basel, 1996.

[7] Chen, X.: How often does a Harris recurrent Markov chain recur? Ann. Probab. \underline{27} (1999), no. 3, 1324–1346.

[8] Chen, X.: On the limit laws of the second order for additive functionals of Harris recurrent Markov chains. Probab. Th. Rel. Fields \underline{116} (2000), no. 1, 89–123.

[9] Csáki, E., Csörgő, M., Földes, A. and Révész, P.: Brownian local time approximated by a Wiener sheet. Ann. Probab. \underline{17} (1989), no. 2, 516–537.

[10] Csáki, E. and Földes, A.: On asymptotic independence of partial sums. Asymptotic Methods in Probability and Statistics (Ottawa, ON, 1997) 373–381, North-Holland, Amsterdam, 1998.

[11] Csáki, E., Földes, A. and Révész, P.: A strong invariance principle for the local time difference of a simple symmetric planar random walk. Studia Sci. Math. Hungar. \underline{34} (1998), no. 1-3, 25–39.

[12] Csörgő, M. and Révész, P.: Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest and Academic Press, New York, 1981.

[13] Dorofeev, E.A.: Upper functions for plane Brownian windings. Bernoulli \underline{4} (1998), no. 4, 461–475.

[14] Földes, A.: Asymptotic independence and strong approximation. A survey. Period. Math. Hungar. \underline{41} (2000), no. 1-2, 121–147.

[15] Hu, Y. and Yor, M.: Asymptotic studies of Brownian functionals. Bolyai Society Mathematical Studies \underline{9} (1999), 187–217.

[16] Itô, K. and McKean, H.P.: Diffusion Processes and their Sample Paths. (Second edition) Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974

[17] Kallianpur, G. and Robbins, H.: Ergodic property of the Brownian motion process. Proc. Nat. Acad. Sci. U.S.A. \underline{39} (1953) 525–533.

[18] Kasahara, Y.: Another limit theorem for slowly increasing occupation times. J. Math. Kyoto Univ. \underline{24} (1984), no. 3, 507–520.

[19] Kasahara, Y. and Kotani, S.: On limit processes for a class of additive functionals of recurrent diffusion processes. Z. Wahrsch. Verw. Gebiete \underline{49} (1979), no. 2, 133–153.

[20] Kazamaki, N.: Continuous Exponential Martingales and BMO. Lecture Notes Math. \underline{1579}. Springer-Verlag, Berlin, 1994.

[21] Messulam, P. and Yor, M.: On D. Williams’ ”pinching method” and some applications. J. London Math. Soc. (2) 26 (1982), no. 2, 348–364.

[22] Mörters, P.: Almost sure Kallianpur-Robbins laws for Brownian motion in the plane. Probab. Th. Rel. Fields \underline{118}

(2000) pp 49–64.

[23] Pitman, J. and Yor, M.: Asymptotic laws of planar Brownian motion. Ann. Probab. \underline{14} (1986) 733–779.

[24] Pitman, J. and Yor, M.: Level crossings of a Cauchy process. Ann. Probab. \underline{14} (1986) 780–792.

[25] Pitman, J. and Yor, M.: Further asymptotic laws of planar Brownian motion. Ann. Probab. \underline{17} (1989) 965–1011.

[26] Resnick, S.I.: Inverses of extremal processes. Adv. Appl. Probab. \underline{6} (1974), 392–406.

[27] Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. Third edition. Springer-Verlag, Berlin, 1999.

[28] Shi, Z.: Windings of Brownian motion and random walks in the plane. Ann. Probab. \underline{26} (1998), no. 1, 112–131.

[29] Touati, A.: Théorèmes limites pour des processus de Markov récurrents. C. R. Acad. Sci. Paris Sér. I Math. \underline{305} (1987), no. 19, 841–844.

[30] Touati, A.: Loi fonctionnelle du logarithme itéré pour les processus de Markov récurrents. Ann. Probab. \underline{18} (1990), no. 1, 140–159.

[31] Watanabe, S.: A limit theorem for sums of i.i.d. random variables with slowly varying tail probability. In: Multivariate Analysis, V (Proc. Fifth Internat. Sympos., Univ. Pittsburgh, Pittsburgh, Pa., 1978) pp. 249–261, North-Holland, Amsterdam-New York, 1980.