On the increments of the principal value of Brownian local time

Endre CS^{}ÁKI^{1}

Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127,

H-1364, Hungary. E-mail: csaki@renyi.hu

Yueyun HU

Département de Mathématiques, Institut Galilée (L.A.G.A. UMR 7539) Université Paris XIII, 99 Avenue J-B Clément, 93430 Villetaneuse, France. E-mail: yueyun@math.univ-paris13.fr

Summary. Let W be a one-dimensional Brownian motion starting from 0. Define Y (t) = ∫_{0}^{t} \frac{ds}{W (s)} := lim_{ϵ→0} ∫_{0}^{t} 1_{(|W (s)|>ϵ)} \frac{ds}{W (s)} as Cauchy’s principal value related to local time. We prove limsup and liminf results for the increments of Y.

Running title. Principal value increments.

Keywords. Brownian motion, local time, principal value, large increments.

2000 Mathematics Subject Classification. 60J65 60J55 60F15

1 Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 037886 and T 043037.

1. 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 function f ≥ 0,

∫_{0}^{t} f(W (s)) ds = ∫_{−∞}^{∞} f(x)L(t, x) dx, t ≥ 0.

We are interested in the process

(1.1)  Y (t) := ∫_{0}^{t} \frac{ds}{W (s)},  t ≥ 0.

Rigorously speaking, the integral ∫_{0}^{t} ds/W (s) should be considered in the sense of Cauchy’s principal value, i.e., Y (t) is defined by

(1.2)  Y (t) := lim_{ε→0^{+}} ∫_{0}^{t} \frac{ds}{W (s)}1l_{\{|W (s)|≥ε\}} = ∫_{0}^{∞} \frac{L(t, x) − L(t, −x)}{x} dx.

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])

(1.3)  lim sup_{T →∞} \frac{Y (T)}{\sqrt{T log log T}} = \sqrt{8} ,  a.s.

This was extended in [10] to a Strassen-type [18] functional law of the iterated logarithm. Theorem B. (Csáki et al. [10]) With probability one the set

(1.4)  \{\frac{Y (xT)}{\sqrt{8T log log T}} , 0 ≤ x ≤ 1\}_{T ≥3}

is relatively compact in C[0, 1] with limit set equal to

(1.5)  S := {f ∈ C[0, 1] : f(0) = 0, f is absolutely continuous and ∫_{0}^{1}(f′(x))^{2} dx ≤ 1} .

Concerning Chung-type law of the iterated logarithm, we have the following result: Theorem C. (Hu [16])

(1.6)  lim inf_{T →∞} \sqrt{\frac{log log T}{T}} sup_{0≤s≤T} |Y (s)| = K_{1} ,  a.s.

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

(1.7)  \{ \begin{matrix}0 < a_{T} ≤ T, \\ T ↦ a_{T} and T ↦ T/a_{T} are both non-decreasing, \\ lim_{T →∞} \frac{log(T/a_{T})}{log log T} = ∞, \end{matrix}

we have

(1.8)  lim_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{a_{T} log(T/a_{T})}} = 2,  a.s.

Wen [19] studied the lag increments of Y and among others proved the following results. Theorem E. (Wen [19])

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

Under the conditions 0 < a_{T} ≤ T, a_{T} → ∞ as T → ∞, we have

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

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 (i)

(1.11)  lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{a_{T} (log \sqrt{T/a_{T}} + log log T )}} = \sqrt{8},  a.s.

(iia) If a_{T} > T (log T)^{−α} for some α < 2, then

(1.12)  lim inf_{T →∞} \sqrt{\frac{log log T}{a_{T}}} sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| = K_{2},  a.s.

(iib) If a_{T} ≤ T (log T)^{−α} for some α > 2, then

(1.13)  lim inf_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{a_{T} log(T/a_{T})}} = K_{3},  a.s.

with some positive constants K_{2}, K_{3}. If, moreover,

lim_{T →∞} \frac{log(T/a_{T})}{log log T} = ∞,

then K_{3} = 2. 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 (i)

(1.14)  lim inf_{T →∞} \frac{\sqrt{T log log T}}{a_{T}} inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| = K_{4},  a.s.

with some positive constant K_{4}. If, lim_{T →∞}(a_{T} /T) = 0, then K_{4} = 1/\sqrt{2}. (iia) If 0 < ρ ≤ 1, then

(1.15)  lim sup_{T →∞} \frac{inf_{0≤t≤T −ρT} sup_{0≤s≤ρT} |Y (t + s) − Y (t)|}{\sqrt{T log log T}} = ρ\sqrt{8},  a.s.

(iib) If

lim_{T →∞} \frac{a_{T} (log log T)^{2}}{T} = 0,

then

(1.16)  lim sup_{T →∞} \frac{\sqrt{T}}{a_{T} \sqrt{log log T}} inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| = K_{5},  a.s.

with some positive constant K_{5}.

Remark 1. The exact values of the constants K_{i}, i = 2, 3, 4, 5 are unknown. It seems difficult to determine the exact values of these constants. In the proofs we establish upper and lower bounds with possibly different constants. It follows however by 0-1 law for Brownian motion that the limsup’s and liminf’s considered here are non-random constants.

Remark 2. Plainly we recover some previous results on the path properties of Y by considering particular cases of Theorems 1.1 and 1.2. For instance, Theorems A and C follow from (1.11) and (1.12) respectively by taking a_{T} = T, and (1.8) follows from (1.11) combining with (1.13). However in Theorem 1.1(ii) and Theorem 1.2(ii) there are still small gaps in a_{T}.

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.

2. Facts

Let {W (t), t ≥ 0} be a standard Brownian motion and define the following objects:

(2.1)  g := sup{t : t ≤ 1, W (t) = 0}
(2.2)  B(s) := \frac{W (sg)}{\sqrt{g}},  0 ≤ s ≤ 1,
(2.3)  m(s) := \frac{|W (g + s(1 − g))|}{\sqrt{1 − g}},  0 ≤ s ≤ 1.

Here we summarize some well-known facts needed in our proofs.

Fact 2.1. (Biane and Yor [4])

(2.4)  \frac{ℙ(Y (1) ∈ dx)}{dx} = \sqrt{\frac{2}{π^{3}}} ∑_{k=0}^{∞}(−1)^{k} exp (−\frac{(2k + 1)^{2}x^{2}}{8} ) , x ∈ ℝ.

Consequently we have the estimate: for δ > 0

(2.5)  c_{1} exp (−\frac{z^{2}}{8(1 − δ)}) ≤ ℙ(Y (1) ≥ z) ≤ exp (−\frac{z^{2}}{8} ) ,  z ≥ 1

with some positive constant c_{1} = c_{1}(δ). Moreover, g, {B(s), 0 ≤ s ≤ 1} and {m(s), 0 ≤ s ≤ 1} are independent, g has arcsine distribution, B is a Brownian bridge and m is a Brownian meander.

(2.6)  \begin{matrix}ℙ (∫_{0}^{1} \frac{dv}{m(v)} < z | m(1) = 0) \\ = ∑_{k=−∞}^{∞}(1 − k^{2}z^{2}) exp (−\frac{k^{2}z^{2}}{2} ) = \frac{8π^{2}\sqrt{2π}}{z^{3}} ∑_{k=1}^{∞} exp (−\frac{2k^{2}π^{2}}{z^{2}} ) , z > 0. \end{matrix}

(2.7)  ℙ(m(1) > x) = e^{−x^{2}/2},  x > 0.

Fact 2.2. (Yor [21, Exercise 3.4 and pp. 44]) Let Q_{x→0}^{δ} be the law of square of Bessel bridge from x to 0 of dimension δ > 0 during time interval [0, 1]. The process (m^{2}(1 − v), 0 ≤ v ≤ 1) conditioned on {m^{2}(1) = x} is distributed as Q_{x→0}^{3}. Furthermore, we have

(2.8)  Q_{x→0}^{δ} = Q_{0→0}^{δ} ∗ Q_{x→0}^{0}, ∀ δ > 0, x > 0,

where ∗ denotes convolution operator. Consequently, for any x > 0

(2.9)  ℙ (∫_{0}^{1} \frac{dv}{m(v)} < z | m(1) = x) ≥ ℙ (∫_{0}^{1} \frac{dv}{m(v)} < z | m(1) = 0) .

Fact 2.3. (Hu [16]) For 0 < z ≤ 1

(2.10)  c_{2} exp (−\frac{c_{3}}{z^{2}} ) ≤ ℙ( sup_{0≤s≤1} |Y (s)| < z) ≤ c_{4} exp (−\frac{c_{5}}{z^{2}} )

with some positive constants c_{2}, c_{3}, c_{4}, c_{5}. 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

(2.11)  lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |W (t + s) − W (t)|}{\sqrt{a_{T} (log(T/a_{T}) + log log T)}} = \sqrt{2},  a.s.

Fact 2.5. (Strassen [18]) If f ∈ S defined by (1.5), then for any partition x_{0} = 0 < x_{1} < … < x_{k} < x_{k+1} = 1 we have

(2.12)  ∑_{i=1}^{k+1} \frac{(f(x_{i}) − f(x_{i−1}))^{2}}{x_{i} − x_{i−1}} ≤ 1.

Fact 2.6. (Chung [6])

(2.13)  lim inf_{t→∞} \sqrt{\frac{log log t}{t}} sup_{0≤s≤t} |W (s)| = \frac{π}{\sqrt{8}},  a.s.

Define g(T) := max{s ≤ T : W (s) = 0}. A joint lower class result for g(T) and M(T) := sup_{0≤s≤T} |W (s)| reads as follows.

Fact 2.7. (Grill [15]) Let β(t), γ(t) be positive functions slowly varying at infinity, such that 0 < β(t) ≤ 1, 0 < γ(t) ≤ 1, β(t) is non-increasing, β(t)\sqrt{t} ↑ ∞, γ(t) is monotone, γ(t)t ↑ ∞, γ(t)/β^{2}(t) is monotone. Then

ℙ (M(T) ≤ β(T)\sqrt{T} , g(T) ≤ γ(T)T i.o.) = 0 or 1

according as I(β, γ) < ∞ or = ∞, where

I(β, γ) = ∫_{1}^{∞} \frac{1}{tβ^{2}(t)} (1 + \frac{β^{2}(t)}{γ(t)})^{−1/2} exp (−\frac{(4 − 3γ(t))π^{2}}{8β^{2}(t)} ) dt.

Now define d(T) := min{s ≥ T : W (s) = 0}. Since {d(T) > t} = {g(t) < T }, we deduce from Fact 2.7 the following estimate on d(T) when T → ∞.

Fact 2.8. With probability 1

d(T) = O(T (log T)^{3}), T → ∞.

3. Probability estimates

Lemma 3.1. For T ≥ 1, δ, z > 0 we have

(3.1)

ℙ ( sup_{0≤t≤T −1} sup_{0≤s≤1} |Y (t + s) − Y (t)| > z)

≤ c_{6} (\sqrt{T} exp (−\frac{z^{2}}{8(1 + δ)}) + T exp (−\frac{z^{2}}{2(1 + δ)}))

with some positive constant c_{6} = c_{6}(δ).

For the proof see Csáki et al. [7], Lemma 2.8.

Lemma 3.2. For T > 1, 0 < δ < 1/2, z > 1 we have

(3.2)

ℙ ( sup_{0≤t≤T −1}(Y (t + 1) − Y (t)) ≥ z)

≥ min (\frac{1}{2}, \frac{c\sqrt{T − 1}}{z} exp (−\frac{z^{2}}{8(1 − δ)})) − exp (−z^{2})

with some positive constant c_{7} = c_{7}(δ) > 0.

Proof. Let us construct an increasing sequence of stopping times by η_{0} := 0 and

η_{k+1} := inf{t > η_{k} + 1 : W (t) = 0}, k = 0, 1, 2, …

Let

\begin{matrix}ν_{t} := min\{i ≥ 1 : η_{i} > t\} \\ Z_{i} := Y (η_{i−1} + 1) − Y (η_{i−1}), i = 1, 2, … \end{matrix}

Then (Z_{i}, η_{i} − η_{i−1})_{i≥1} are i.i.d. random vectors with

η_{i} − η_{i−1} =^{law} 1 + τ^{2}, Z_{i} =^{law} Y (1),

where τ has Cauchy distribution. Clearly, for t > 0,

sup_{0≤s≤t}(Y (s + 1) − Y (s)) ≥ max_{1≤i≤ν_{t}} Z_{i} = \overline{Z}_{ν_{t}},

with \overline{Z}_{k} := max_{1≤i≤k} Z_{i}. First consider the Laplace transform (λ > 0):

λ ∫_{0}^{∞} e^{−λu}ℙ (\overline{Z}_{ν_{u}} < z) du

= λ ∑_{k=1}^{∞} 𝔼 ∫_{0}^{∞} e^{−λu}1_{\{η_{k−1}≤u<η_{k}\}}1_{\{\overline{Z}_{k}<z\}} du

= ∑_{k=1}^{∞} 𝔼 ([e^{−λη_{k−1}} − e^{−λη_{k}}]1_{\{\overline{Z}_{k}<z\}})

= ∑_{k=1}^{∞} (𝔼[1_{\{\overline{Z}_{k}<z\}}e^{−λη_{k−1}}] − 𝔼[1_{\{\overline{Z}_{k}<z\}}e^{−λη_{k}}])

= ∑_{k=1}^{∞} (𝔼[1_{\{\overline{Z}_{k−1}<z\}}e^{−λη_{k−1}}] − 𝔼[1_{\{\overline{Z}_{k−1}<z, Z_{k}≥z\}}e^{−λη_{k−1}}] − 𝔼[1_{\{\overline{Z}_{k}<z\}}e^{−λη_{k}}])

= 1 − ∑_{k=1}^{∞} 𝔼[1_{\{\overline{Z}_{k−1}<z, Z_{k}≥z\}} e^{−λη_{k−1}}]

= 1 − ∑_{k=1}^{∞} 𝔼[1_{\{\overline{Z}_{k−1}<z\}}e^{−λη_{k−1}}] ℙ(Y (1) ≥ z)

= 1 − ∑_{k=1}^{∞} (𝔼[1_{{Z_{1}<z}}e^{−λη_{1}}] )^{k−1} ℙ(Y (1) ≥ z)

= 1 − \frac{ℙ(Y (1) ≥ z)}{1 − 𝔼[1_{\{Z_{1}<z\}}e^{−λη_{1}}]},

i.e.,

(3.3)  λ ∫_{0}^{∞} e^{−λu}ℙ (\overline{Z}_{ν_{u}} ≥ z) du = \frac{ℙ(Y (1) ≥ z)}{1 − 𝔼[1_{\{Z_{1}<z\}}e^{−λη_{1}}]}.

But (recalling that Z_{1} = Y (1))

1 − 𝔼[1_{{Z_{1}<z}}e^{−λη_{1}}] ≤ 1 − 𝔼(e^{−λη_{1}}) + ℙ(Y (1) ≥ z) and (cf. [14], 3.466/1)

1 − 𝔼e^{−λη_{1}} = 1 − \frac{1}{π} ∫_{−∞}^{∞} \frac{e^{−λ(1+x^{2})}}{1 + x^{2}} dx = \frac{2}{\sqrt{π}} ∫_{0}^{√\overline{λ}} e^{−x^{2}} dx ≤ 2\sqrt{λ},

hence

λ ∫_{0}^{∞} e^{−λu}ℙ (\overline{Z}_{ν_{u}} ≥ z) du ≥ \frac{ℙ(Y (1) ≥ z)}{2\sqrt{λ} + ℙ(Y (1) ≥ z)}.

On the other hand, for any u_{0} > 0 we have

\begin{aligned}λ ∫_{0}^{∞} e^{−λu}ℙ (\overline{Z}_{ν_{u}} ≥ z) du & = λ ∫_{0}^{u_{0}} e^{−λu}ℙ (\overline{Z}_{ν_{u}} ≥ z) du + λ ∫_{u_{0}}^{∞} e^{−λu}ℙ (\overline{Z}_{ν_{u}} ≥ z) du \\ & ≤ ℙ (\overline{Z}_{ν_{u_{0}}} ≥ z) + e^{−λu_{0}}. \end{aligned}

It turns out that

ℙ (\overline{Z}_{ν_{u_{0}}} ≥ z) ≥ \frac{ℙ(Y (1) ≥ z)}{2\sqrt{λ} + ℙ(Y (1) ≥ z)} − e^{−λu_{0}} ≥ min (\frac{1}{2}, \frac{ℙ(Y (1) ≥ z)}{4\sqrt{λ}}) − e^{−λu_{0}},

where the inequality

\frac{x}{y + x} ≥ min (\frac{1}{2}, \frac{x}{2y} ) , x > 0, y > 0

was used. Choosing u_{0} = T − 1, λ = z^{2}/u_{0}, and applying (2.5) of Fact 2.1, we finally get

(3.4)

ℙ ( sup_{0≤t≤T −1}(Y (t + 1) − Y (t)) ≥ z)

≥ min (\frac{1}{2}, \frac{c_{8}(δ)\sqrt{T − 1}}{z} exp (−\frac{z^{2}}{8(1 − δ)})) − exp (−z^{2}) .

This proves Lemma 3.2. ⊔ ⊓ Lemma 3.3. For T ≥ 2, 0 ≤ κ < 1 and δ, z > 0 we have

(3.6)  ℙ ( sup_{0≤t≤T −1}(Y (t + 1) − Y (t)) < z) ≤ \frac{5}{T^{κ/2}} + exp (−c_{9}T^{(1−κ)/2}e^{−(1+δ)z^{2}/8})

with some positive constant c_{9} = c_{9}(δ).

See Csáki et al. [7], Lemma 3.1. Lemma 3.4. For T > 1, 0 < z ≤ 1/2 we have

ℙ ( sup_{0≤t≤T −1} sup_{0≤s≤1} |Y (t + s) − Y (t)| < z) ≥ \frac{c_{10}}{\sqrt{T}} exp (−\frac{c_{11}}{z^{2}} )

with some positive constants c_{10}, c_{11}.

Proof. Define the events

A := \{ sup_{0≤s≤1} |Y (s)| < \frac{z}{4}, W (1) ≥ \frac{4}{z}, inf_{1≤u≤T} W (u) ≥ \frac{2}{z}\}

and

Ã:= { sup_{0≤t≤T −1} sup_{0≤s≤1} |Y (t + s) − Y (t)| < z} .

Then A ⊂ Ã, since if A occurs and t < 1, t + s ≤ 1, then

|Y (t + s) − Y (t)| ≤ 2 sup_{0≤s≤1} |Y (s)| ≤ \frac{z}{2} < z.

If A occurs and t < 1, s ≤ 1, 1 < t + s ≤ T, then

|Y (t + s) − Y (t)| ≤ Y (t + s) − Y (1) + |Y (t) − Y (1)| ≤ ∫_{1}^{t+s} \frac{du}{W (u)} + \frac{z}{2} < z.

Moreover, if A occurs and 1 ≤ t, s ≤ 1, t + s ≤ T, then

|Y (t + s) − Y (t)| = ∫_{t}^{t+s} \frac{du}{W (u)} ≤ \frac{z}{2} < z.

Hence A ⊂ Ã as claimed. But by the Markov property of W,

(3.8) ℙ(A) = ∫_{4/z}^{∞} ℙ ( sup_{0≤s≤1} |Y (s)| < \frac{z}{4} | W (1) = x) ℙ ( inf_{1≤u≤T} W (u) ≥ \frac{2}{z} | W (1) = x) φ(x) dx,

where φ denotes the standard normal density function.

Using reflection principle and x ≥ 4/z, z ≤ 1/2, we get

(3.9)  \begin{matrix}ℙ ( inf_{1≤u≤T} W (u) ≥ \frac{2}{z} | W (1) = x) = 2Φ (\frac{x − 2/z}{\sqrt{T − 1}} ) − 1 \\ ≥ 2Φ (\frac{2}{z\sqrt{T − 1}}) − 1 ≥ 2Φ (\frac{4}{\sqrt{T}} ) − 1 ≥ \frac{c_{12}}{\sqrt{T}} , \end{matrix}

with some constant c > 0, where Φ(·) is the standard normal distribution function. Hence

(3.10)  ℙ(Ã) ≥ ℙ(A) ≥ \frac{c_{12}}{\sqrt{T}} ℙ ( sup_{0≤s≤1} |Y (s)| ≤ \frac{z}{4}, W (1) ≥ \frac{4}{z}) .

To get a lower bound of the probability on the right-hand side, define g, (m(v), 0 ≤ v ≤ 1), (B(u), 0 ≤ u ≤ 1) by (2.1), (2.2) and (2.3), respectively. Recall (see Fact 2.1 ) that these three objects are independent, g has arc sine distribution, m is a Brownian meander and B is a Brownian bridge. Moreover, (g, m, B) are independent of sgn(W (1)) which is a Bernoulli variable. Observe that

sup_{0≤s≤g} |Y (s)| = \sqrt{g} sup_{0≤s≤1} |∫_{0}^{s} \frac{du}{B(u)}| ,
sup_{g≤s≤1} |Y (s)| = |Y (1) − Y (g)| = \sqrt{1 − g} ∫_{0}^{1} \frac{dv}{m(v)},
|W (1)| = \sqrt{1 − g} m(1).

Then

ℙ ( sup_{0≤s≤1} |Y (s)| ≤ \frac{z}{4}, W (1) ≥ \frac{4}{z})

≥ ℙ ( sup_{0≤s≤g} |Y (s)| ≤ \frac{z}{8}, Y (1) − Y (g) ≤ \frac{z}{8}, W (1) ≥ \frac{4}{z})

≥ ℙ (\sqrt{g} sup_{0≤s≤1} |∫_{0}^{s} \frac{du}{B(u)}| ≤ \frac{z}{8}, \sqrt{1 − g} ∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8}, \sqrt{1 − g} m(1) ≥ \frac{4}{z}, W (1) > 0, g < z^{2})

≥ ℙ ( sup_{0≤s≤1} |∫_{0}^{s} \frac{du}{B(u)}| ≤ \frac{1}{8}, ∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8}, m(1) ≥ \frac{4}{z\sqrt{1 − z^{2}}} , W (1) > 0, g < z^{2})

= ℙ ( sup_{0≤s≤1} |∫_{0}^{s} \frac{du}{B(u)}| ≤ \frac{1}{8}) ℙ (∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8}, m(1) ≥ \frac{4}{z\sqrt{1 − z^{2}}} ) ℙ(W (1) > 0)ℙ(g < z^{2})

≥ c_{13}zℙ (∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8}, m(1) ≥ \frac{4}{z\sqrt{1 − z^{2}}} )

= c_{13}z ∫_{4/(z\sqrt{1−z^{2}})}^{∞} ℙ (∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8} | m(1) = x) ℙ(m(1) ∈ dx).

It follows from Facts 2.1 and 2.2 that for x > 0, z > 0

(3.11)  ℙ (∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8} | m(1) = x) ≥ ℙ (∫_{0}^{1} \frac{dv}{m(v)} ≤ \frac{z}{8} | m(1) = 0) ≥ \frac{c_{14}}{z^{3}} exp (−\frac{c_{15}}{z^{2}} )

and

(3.12)  ℙ (m(1) > \frac{4}{z\sqrt{1 − z^{2}}} ) = exp (−\frac{8}{z^{2}(1 − z^{2})}) .

Putting (3.10), (3.11), (3.12) together, we get (3.7). ⊔ ⊓ Lemma 3.5. For T > 1, 0 < z ≤ 1/2, 0 < δ ≤ 1/2 we have

(3.13)

ℙ ( inf_{0≤t≤T −1} sup_{0≤s≤1} |Y (t + s) − Y (t)| < z)

≤ c_{16} (exp (−\frac{(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} ) + exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} ) + exp (\frac{c_{17}}{z^{2}} − \frac{c_{18}z^{2}}{T} e^{c_{19}/z^{2}}))

with some positive constants c_{16}, c_{17} = c_{17}(δ), c_{18} = c_{18}(δ), c_{19} = c_{19}(δ).

Proof. Consider a positive integer N to be given later, h = (T − 1)/N, t_{k} = kh, k = 0, 1, 2, …, N. Then for 0 < δ ≤ 1/2 we have

ℙ ( inf_{0≤t≤T −1} sup_{0≤s≤1} |Y (t + s) − Y (t)| < z)

≤ ℙ ( inf_{0≤k≤N} sup_{0≤s≤1} |Y (t_{k} + s) − Y (t_{k})| ≤ (1 + δ)z) + ℙ ( sup_{0≤t≤T −1} sup_{0≤s≤h} |Y (t + s) − Y (t)| > δz)

=: P_{1} + P_{2}.

By scaling and Lemma 3.1

P_{2} = ℙ ( sup_{0≤t≤(T −1)/h} sup_{0≤s≤1} |Y (t + s) − Y (t)| > \frac{δz}{\sqrt{h}})

≤ c_{6} (\sqrt{\frac{T − 1}{h} + 1} exp (−\frac{δ^{2}z^{2}}{8h(1 + δ)}) + (\frac{T − 1}{h} + 1) exp (−\frac{δ^{2}z^{2}}{2h(1 + δ)}))

≤ 2c_{6}(N + 1) exp (−\frac{δ^{2}z^{2}}{8h(1 + δ)}) .

To bound P_{1}, we denote by d(t) := inf{s ≥ t : W (s) = 0} the first zero of W after t. Consider those k for which sup_{0≤s≤1} |Y (t_{k} + s) − Y (t_{k})| ≤ (1 + δ)z. If, moreover, d(t_{k}) ≥ t_{k} + 1 − δ, which means that the Brownian motion W does not change sign over [t_{k}, t_{k} + 1 − δ), then

(1 + δ)z ≥ |Y (t_{k} + 1 − δ) − Y (t_{k})| = ∫_{0}^{1−δ} \frac{ds}{|W (t_{k} + s)|} ≥ \frac{1 − δ}{sup_{0≤s≤T} |W (s)|},

and it follows that

P_{1} ≤ ℙ ( sup_{0≤s≤T} |W (s)| > \frac{(1 − δ)}{z(1 + δ)})

+ ℙ (∃k ≤ N : sup_{0≤s≤1} |Y (t_{k} + s) − Y (t_{k})| ≤ (1 + δ)z; d(t_{k}) < t_{k} + 1 − δ)

≤ 4 exp (−\frac{(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} )

+ ∑_{k=0}^{N} ℙ ( sup_{0≤s≤1} |Y (t_{k} + s) − Y (t_{k})| ≤ (1 + δ)z; d(t_{k}) < t_{k} + 1 − δ) .

Let Ŵ(s) = W (d(t_{k}) + s) for s ≥ 0 and Ŷ(s) be the associated principal values. Observe that on {sup_{0≤s≤1} |Y (t_{k} + s) − Y (t_{k})| ≤ (1 + δ)z; d(t_{k}) < t_{k} + 1 − δ}, we have sup_{0≤u≤δ} |Ŷ(u) + (Y (d(t_{k})) − Y (t_{k}))| < (1 + δ)z, and |Y (d(t_{k})) − Y (t_{k})| ≤ (1 + δ)z which implies that

sup_{0≤u≤δ} |Ŷ(u)| < 2(1 + δ)z.

By scaling and Fact 2.3 we have

ℙ ( sup_{0≤u≤δ} |Ŷ(u)| < 2(1 + δ)z) ≤ c_{4} exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} ) .

Therefore, we obtain:

P_{1} ≤ 4 exp (−\frac{(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} ) + c_{4}(N + 1) exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} ) .

Hence

P_{1} + P_{2} ≤ 4 exp (−\frac{(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} ) + c_{4}(N + 1) exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} )

+2c_{6}(N + 1) exp (−\frac{δ^{2}z^{2}}{8h(1 + δ)}) .

By taking N = [e^{c_{5}δ/(4(1+δ)^{2}z^{2})}] + 1, we get P_{1} + P_{2}

≤ c_{16} (exp (−\frac{(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} ) + exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} ) + exp (\frac{c_{17}}{z^{2}} − \frac{c_{18}z^{2}}{T} e^{c_{19}/z^{2}}))

with relevant constants c_{16}, c_{17}, c_{18}, c_{19}, proving (3.13). ⊔ ⊓

4. Proof of Theorem 1.1(i)

The upper estimation, i.e.

(4.1)  lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{8a_{T} (log \sqrt{T/a_{T}} + log log T )}} ≤ 1,  a.s.

follows easily from Wen’s Theorem E.

Now we prove the lower bound, i.e.

(4.2)  lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{8a_{T} (log \sqrt{T/a_{T}} + log log T )}} ≥ 1,  a.s.

In the case when a_{T} = T, (4.2) follows from the law of the iterated logarithm (1.3) of Theorem A. Now we assume that a_{T} /T ≤ ρ < 1, with some constant ρ for all T > 0.

By scaling, (3.2) of Lemma 3.2 is equivalent to

(4.3)

ℙ ( sup_{0≤t≤T −a}(Y (t + a) − Y (t)) ≥ z\sqrt{a})

≥ min (\frac{1}{2}, \frac{c_{7}\sqrt{T/a − 1}}{z} exp (−\frac{z^{2}}{8(1 − δ)})) − exp (−z^{2})

for 0 < a < T, 0 < δ < 1/2, z > 1.

Define the sequences

(4.4) t_{k} := e^{7k log k}, k = 1, 2, …

and θ_{0} := 0,

(4.5) θ_{k} := inf{t > T_{k} : W (t) = 0}, k = 1, 2, …,

where T_{k} := θ_{k−1} + t_{k}. For 0 < δ < min(1/2, 1 − ρ) define the events

A_{k} := { sup_{0≤t≤t_{k}(1−δ)−a_{t_{k}}}(Y (θ_{k−1} + t + a_{t_{k}}) − Y (θ_{k−1} + t)) ≥ (1 − δ)β_{k}} , k = 1, 2, …

with

β_{k} := \sqrt{8a_{t_{k}} (log \sqrt{\frac{t_{k}}{a_{t_{k}}}} + log log t_{k})}.

Applying (4.3) with T = t_{k}(1 − δ), a = a_{t_{k}}, z = (1 − δ)\sqrt{8(log \sqrt{t_{k}/a_{t_{k}}} + log log t_{k})}, we have for k large

ℙ(A_{k}) = ℙ ( sup_{0≤t≤t_{k}(1−δ)−a_{t_{k}}}(Y (t + a_{t_{k}}) − Y (t)) ≥ (1 − δ)β_{k})

≥ min (\frac{1}{2}, \frac{b_{k}}{(log t_{k})^{1−δ}} ) − \frac{1}{(log t_{k})^{8(1−δ)^{2}}}

with

b_{k} = \frac{c_{7}\sqrt{t_{k}(1 − δ)/a_{t_{k}} − 1}}{(t_{k}/a_{t_{k}})^{(1−δ)/2}\sqrt{log \sqrt{t_{k}/a_{t_{k}}} + log log t_{k}}} ≥ \frac{c_{20}}{\sqrt{log k}} .

Hence ∑_{k} ℙ(A_{k}) = ∞ and since A_{k} are independent, Borel-Cantelli lemma yields

ℙ(A_{k} i.o.) = 1.

It follows that

(4.6)  lim sup_{k→∞} \frac{sup_{0≤t≤t_{k}(1−δ)−a_{t_{k}}} (Y (θ_{k−1} + t + a_{t_{k}}) − Y (θ_{k−1} + t))}{\sqrt{8a_{t_{k}} (log \sqrt{\frac{t_{k}}{a_{t_{k}}}} + log log t_{k})}} ≥ 1 − δ,  a.s.

It can be seen (cf. [9]) that we have almost surely for large enough k

t_{k} ≤ T_{k} ≤ t_{k} (1 + \frac{1}{k} ) ,

consequently

(4.7) lim_{k→∞} \frac{t_{k}}{T_{k}} = 1, a.s.

Since by our assumptions

\frac{t_{k}}{T_{k}} ≤ \frac{a_{t_{k}}}{a_{T_{k}}} ≤ 1,

we have also

(4.8) lim_{k→∞} \frac{a_{t_{k}}}{a_{T_{k}}} = 1, a.s.

On the other hand, for any δ > 0 small enough we have almost surely for large k

a_{T_{k}} ≤ (1 + δ)a_{t_{k}} ≤ t_{k}δ + a_{t_{k}},

thus

T_{k} − a_{T_{k}} ≥ T_{k} − t_{k}δ − a_{t_{k}},

consequently (4.9)

sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)|

≥ sup_{0≤t≤t_{k}(1−δ)−a_{t_{k}}}(Y (θ_{k−1} + t + a_{t_{k}}) − Y (θ_{k−1} + t)),

hence we have also

(4.10) lim sup_{k→∞} \frac{sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)|}{\sqrt{8a_{t_{k}} (log \sqrt{\frac{t_{k}}{a_{t_{k}}}} + log log t_{k})}} ≥ 1 − δ, a.s.

and since δ > 0 can be arbitrary small, (4.2) follows by combining (4.7), (4.8), (4.9) and (4.10). ⊔ ⊓

5. Proof of Theorem 1.1(ii)

First assume that

(5.1) a_{T} > \frac{T}{(log T)^{α}} for some α < 2.

By Theorem C, (5.2)

lim inf_{T →∞} \sqrt{\frac{log log T}{a_{T}}} sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|

≥ lim inf_{T →∞} \sqrt{\frac{log log a_{T}}{a_{T}}} sup_{0≤s≤a_{T}} |Y (s)| ≥ K_{1}, a.s.,

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

(5.3)  ℙ ( sup_{0≤t≤T −a} sup_{0≤s≤a} |Y (s + t) − Y (t)| < z\sqrt{a}) ≥ c_{10}\sqrt{\frac{a}{T}} exp (−\frac{c_{11}}{z^{2}} )

for T ≥ a, 0 < z ≤ 1/2.

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

E_{k} := \{ sup_{0≤t≤(1+ε)t_{k}−a_{t_{k}(1+ε)}} sup_{0≤s≤a_{t_{k}(1+ε)}} |Y (θ_{k−1} + t + s) − Y (θ_{k−1} + t)| ≤ δ\sqrt{\frac{a_{t_{k}}}{log log t_{k}}} \} .

Then putting T = (1 + ε)t_{k}, a = a_{(1+ε)t_{k}}, z = δ/\sqrt{log log t_{k}}, into (5.3), we get

ℙ(E_{k}) = ℙ ( sup_{0≤t≤(1+ε)t_{k}−a_{t_{k}(1+ε)}} sup_{0≤s≤a_{t_{k}(1+ε)}} |Y (t + s) − Y (t)| ≤ δ\sqrt{\frac{a_{t_{k}}}{log log t_{k}}} )

≥ c_{10}\sqrt{\frac{a_{t_{k}}}{t_{k}}} exp(−(c_{11}/δ^{2}) log log((1 + ε)t_{k})) ≥ \frac{c_{10}}{(log t_{k})^{α/2+c_{11}/δ^{2}}} ,

hence ∑_{k} ℙ(E_{k}) = ∞, and since E_{k} are independent, we have ℙ(E_{k} i.o.) = 1, i.e.

(5.4) lim inf_{k→∞} \sqrt{\frac{log log t_{k}}{a_{t_{k}}}} sup_{0≤t≤(1+ε)t_{k}−a_{t_{k}(1+ε)}} sup_{0≤s≤a_{t_{k}(1+ε)}} |Y (θ_{k−1} + t + s) − Y (θ_{k−1} + t)| ≤ δ, a.s.

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

(5.5)

sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)|

≤ 2 sup_{0≤t≤θ_{k−1}} |Y (t)| + sup_{0≤t≤(1+ε)t_{k}−a_{t_{k}(1+ε)}} sup_{0≤s≤a_{t_{k}(1+ε)}} |Y (θ_{k−1} + t + s) − Y (θ_{k−1} + t)|.

By Theorem A, Fact 2.8, (4.7), (5.1) and simple calculation,

(5.6)

sup_{0≤t≤θ_{k−1}} |Y (t)| = O(θ_{k−1} log log θ_{k−1})^{1/2}

= O(t_{k−1}(log t_{k−1})^{3} log log t_{k−1})^{1/2} = o (\frac{a_{t_{k}}}{log log t_{k}} )^{1/2} , a.s.

as k → ∞. Assembling (5.4), (5.5) and (5.6), we get

lim inf_{k→∞} \sqrt{\frac{log log t_{k}}{a_{t_{k}}}} sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)|

= lim inf_{k→∞} \sqrt{\frac{log log T_{k}}{a_{T_{k}}}} sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)| ≤ δ,  a.s.

which together with (5.2) yields (1.12).

Now assume that

(5.7)  a_{T} ≤ \frac{T}{(log T)^{α}}  for some α > 2.

By Theorem 1.1(i),

(5.8)

lim inf_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{a_{T} log(T/a_{T})}}

≤ lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{a_{T} log(T/a_{T})}}

≤ lim sup_{T →∞} \frac{sup_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|}{\sqrt{\frac{2αa_{T}}{α+2} (log \sqrt{T/a_{T}} + log log T )}} ≤ 2\sqrt{\frac{α + 2}{α}},

i.e., an upper bound in (1.13) follows.

To get a lower bound under (5.7), observe that by scaling, (3.6) of Lemma 3.3 is equivalent to

ℙ ( sup_{0≤t≤T −a}(Y (t + a) − Y (t)) < z\sqrt{a}) ≤ 5 (\frac{a}{T} )^{κ/2} + exp (−c_{9} (\frac{T}{a} )^{(1−κ)/2} e^{−(1+δ)z^{2}/8})

for a ≤ T, 0 ≤ κ < 1, 0 < δ, 0 < z. Using (5.7) we get further

(5.9)

ℙ ( sup_{0≤t≤T −a}(Y (t + a) − Y (t)) < z\sqrt{a})

≤ \frac{5}{(log T)^{ακ/2}} + exp (−c_{9} (log T)^{α(1−κ)/2}) e^{−(1+δ)z^{2}/8}) .

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

F_{k} = \{ sup_{0≤t≤T_{k}−a_{T_{k}}}(Y (t + a_{T_{k}}) − Y (t)) ≤ C_{1}\sqrt{a_{T_{k}} log \frac{T_{k}}{a_{T_{k}}}} \}

with some constant C_{1} to be given later. By (5.9)

ℙ(F_{k}) ≤ \frac{5}{k^{ακ/2}} + exp (−c_{9}k^{α((1−κ)/2−(1+δ)C_{1}^{2}/8)}) .

For given α > 2, choose small ε > 0, κ = 2/α + ε,

C_{1} = 2\sqrt{\frac{α − 2 − 2ε(1 + α)}{(1 + ε)α}}.

One can easily see that with these choices ∑_{k} ℙ(F_{k}) < ∞, consequently

lim inf_{k→∞} \frac{sup_{0≤t≤T_{k}−a_{T_{k}}} (Y (t + a_{T_{k}}) − Y (t))}{\sqrt{a_{T_{k}} log \frac{T_{k}}{a_{T_{k}}}}} ≥ C_{1}, a.s.,

implying also

lim inf_{k→∞} \frac{sup_{0≤t≤T_{k}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)|}{\sqrt{a_{T_{k}} log \frac{T_{k}}{a_{T_{k}}}}} ≥ 2\sqrt{\frac{α − 2}{α}}, a.s.,

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 1.1(ii). ⊔ ⊓

6. 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.

(6.1)  lim inf_{T →∞} \frac{\sqrt{T log log T}}{a_{T}} inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| ≥ c,  a.s.

By scaling, (3.13) of Lemma 3.5 is equivalent to

(6.2)

ℙ ( inf_{0≤t≤T −a} sup_{0≤s≤a} |Y (t + s) − Y (t)| < z)

≤ c_{16} (exp (−\frac{a(1 − δ)^{2}}{2(1 + δ)^{2}z^{2}T} ) + exp (−\frac{c_{5}δ}{4(1 + δ)^{2}z^{2}} ) + exp (\frac{c_{17}}{z^{2}} − \frac{c_{18}az^{2}}{T} e^{c_{19}/z^{2}}))

for a < T, 0 < z ≤ 1/2, 0 < δ ≤ 1/2.

Define the events

G_{k} = { inf_{0≤t≤T_{k+1}−a_{T_{k}}} sup_{0≤s≤a_{T_{k}}} |Y (t + s) − Y (t)| < z_{k}} k = 1, 2, …

Let T_{k} = e^{k} and put T = T_{k+1}, a = a_{T_{k}},

z = z_{k} = C_{2}\sqrt{\frac{a_{T_{k}}}{T_{k+1} log log T_{k+1}}}

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

ℙ(G_{k}) ≤ c_{16}(I_{1}^{(k)} + I_{2}^{(k)} + I_{3}^{(k)}),

where

I_{1}^{(k)} = exp (−\frac{c_{21}}{C_{2}^{2}} log log T_{k+1}) ,

I_{2}^{(k)} = exp (−\frac{c_{22}T_{k}}{C_{2}^{2}a_{T_{k}}} log log T_{k+1}) ,

I_{3}^{(k)} = exp (\frac{c_{23}T_{k} log log T_{k+1}}{C_{2}^{2}a_{T_{k}}} − \frac{c_{24}C_{2}^{2}a_{T_{k}}^{2}}{T_{k}^{2} log log T_{k+1}} (log T_{k+1})^{\frac{c_{25}T_{k}}{C^{2}_{2}a_{T_{k}}}} )

with some constants c_{21} = c_{21}(δ), c_{22} = c_{22}(δ), c_{23}, c_{24}, c_{25}.

One can see easily that for any choice of positive C_{2} and for all possible a_{T} (satisfying our conditions) we have ∑_{k} I_{3}^{(k)} < ∞. So we show that for appropriate choice of C_{2} we have also ∑_{k} I_{j}^{(k)} < ∞, j = 1, 2.

First consider the case 0 < ρ > 0. Choosing a positive δ one can select C_{2} < min(\sqrt{c_{21}}, \sqrt{\frac{c_{22}}{ρ}}) and it is easy to verify that ∑_{k} I_{j}^{(k)} < ∞, j = 1, 2, hence also ∑_{k} ℙ(G_{k}) < ∞.

In the case ρ = 0 choose C_{2} < (1 − δ)/((1 + δ)\sqrt{2}). With this choice we have ∑_{k} I_{1}^{(k)} < ∞ for arbitrary δ > 0. Since lim_{k→∞}(T_{k}/a_{T_{k}}) = ∞, we have also ∑_{k} I_{2}^{(k)} < ∞ and ∑_{k} ℙ(G_{k}) < ∞. 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/\sqrt{2}, since δ can be choosen arbitrary small.

Now we turn to the proof of the upper bound, i.e.

(6.3)  lim inf_{T →∞} \frac{\sqrt{T log log T}}{a_{T}} inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| ≤ C_{3},  a.s.

with some constant C_{3}.

If ρ > 0, then

inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| ≤ sup_{0≤s≤a_{T}} |Y (s)| ≤ sup_{0≤s≤T} |Y (s)|

and hence (6.3) with some positive constant C_{3} follows from Theorem C.

If ρ = 0, then let for any ε > 0

(6.4)  λ_{T} := inf{t : |W (t)| = sup_{0≤s≤T (1−ε)} |W (s)|}.

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

(6.5)  |W (λ_{T_{i}})| ≥ \sqrt{2T_{i}(1 − ε) log log T_{i}}.

But Fact 2.4 implies that for ε > 0

(6.6)  |W (λ_{T_{i}}) − W (s)| ≤ \sqrt{2(1 + ε)εT_{i} log log T_{i}}, λ_{T_{i}} ≤ s ≤ λ_{T_{i}} + εT_{i}, i ≥ 1.

Now assume that W (λ_{T_{i}}) > 0. The case when W (λ_{T_{i}}) < 0 is similar. Then (6.5) and (6.6) imply

(6.7)  W (s) ≥ (\sqrt{1 − ε} − \sqrt{ε(1 + ε)}) \sqrt{2T_{i} log log T_{i}}, λ_{T_{i}} ≤ s ≤ λ_{T_{i}} + εT_{i}.

ρ = 0 implies that a_{T} ≤ εT for any ε > 0 and large enough T, hence we have from (6.7) for large i

sup_{0≤s≤a_{T_{i}}}(Y (λ_{T_{i}} + s) − Y (λ_{T_{i}})) = Y (λ_{T_{i}} + a_{T_{i}}) − Y (λ_{T_{i}}) = ∫_{λ_{T_{i}}}^{λ_{T_{i}} +a_{T_{i}}} \frac{ds}{W (s)}

≤ \frac{a_{T_{i}}}{(\sqrt{1 − ε} − \sqrt{ε(1 + ε)}) \sqrt{2T_{i} log log T_{i}}} .

Since ε > 0 is arbitrary, (6.3) follows with C_{3} = 1/\sqrt{2}. This completes the proof of Theorem 1.2(i). ⊔ ⊓

7. Proof of Theorem 1.2(ii)

If ρ = 1, then (1.15) is equivalent to (1.3) of Theorem A. So we may assume that 0 < ρ < 1. First we prove the upper bound

(7.1)  lim sup_{T →∞} \frac{inf_{0≤t≤T −ρT} sup_{0≤s≤ρT} |Y (t + s) − Y (t)|}{\sqrt{8T log log T}} ≤ ρ,  a.s.

Let k be the largest integer for which kρ < 1 and put x_{i} = iρ, i = 0, 1, …, k, x_{k+1} = 1. It suffices to show that if f ∈ S defined by (1.5), then

min_{1≤i≤k+1} |f(x_{i}) − f(x_{i−1})| ≤ ρ.

Assume on the contrary that

|f(x_{i}) − f(x_{i−1})| > ρ, ∀i = 1, 2, …, k + 1.

Then

∑_{i=1}^{k+1} \frac{(f(x_{i}) − f(x_{i−1}))^{2}}{x_{i} − x_{i−1}} > ∑_{i=1}^{k} \frac{ρ^{2}}{ρ} + \frac{ρ^{2}}{1 − kρ} = kρ + \frac{ρ^{2}}{1 − kρ} ≥ 1,

contradicting (2.12) of Fact 2.5. This proves (7.1).

The lower bound

(7.2)  lim sup_{T →∞} \frac{inf_{0≤t≤T −ρT} sup_{0≤s≤ρT} |Y (t + s) − Y (t)|}{\sqrt{8T log log T}} ≥ ρ,  a.s.

follows from the fact that by Theorem B the function f(x) = x, 0 ≤ x ≤ 1 is a limit point of

\frac{Y (xt)}{\sqrt{8T log log T}}

and for this function

min_{0≤x≤1−ρ} |f(x + ρ) − f(x)| = ρ.

This completes the proof of Theorem 1.2(iia). ⊔ ⊓ Now assume that

(7.3)  lim_{T →∞} \frac{a_{T} (log log T)^{2}}{T} = 0.

Define λ_{T} as in (6.4). Then according to Chung’s LIL (cf. Fact 2.6)

(7.4)  |W (λ_{T})| ≥ \frac{π}{\sqrt{8}}(1 − ε)\sqrt{\frac{T}{log log T}}

for every T sufficiently large. But according to Fact 2.4,

sup_{0≤s≤a_{T}} |W (λ_{T} + s) − W (λ_{T})|

≤ \sqrt{(2 + ε)a_{T} (log(T/a_{T}) + log log T)} ≤ \sqrt{\frac{(2 + ε)εT}{log log T}} .

Assuming W (λ_{T}) > 0, we get

W (λ_{T} + s) ≥ W (λ_{T}) − \sqrt{\frac{(2 + ε)εT}{log log T}} ≥ c\sqrt{\frac{T}{log log T}} .

Hence

inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| ≤ Y (λ_{T} + a_{T}) − Y (λ_{T})

= ∫_{0}^{a_{T}} \frac{ds}{W (λ_{T} + s)} ≤ \frac{a_{T}}{c} \sqrt{\frac{log log T}{T}}

for all large T.

The case when W (λ_{T}) < 0 is similar. This shows the upper bound in (1.16).

For the lower bound we use Fact 2.6: with probability one

(7.5)  g_{T} ≤ \frac{T}{(log log T)^{2}} , max_{0≤u≤T} |W (u)| ≤ \frac{π}{\sqrt{2}}\sqrt{\frac{T}{log log T}} i.o.

According to Theorem 1.2(i) for every large T we have for any ε > 0 and sufficiently large T

(7.6)

inf_{0≤t≤T (log log T)^{−2}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)|

≥ \frac{(K_{4} − ε)a_{T}}{\sqrt{(\frac{T}{(log log T)^{2}} + a_{T} ) log log T}} ≤ \frac{(K_{4} − ε)a_{T}}{\sqrt{(1 + ε)T log log T}} .

On the other hand, if T (log log T)^{−2} ≤ t ≤ T − a_{T}, then by (7.5)

|Y (t + a_{T}) − Y (t)| = ∫_{t}^{t+a_{T}} \frac{ds}{|W (s)|} ≥ \frac{a_{T} \sqrt{2 log log T}}{π\sqrt{T}} .

Combining (7.6) and (7.7) we get for ε > 0 and all large T

inf_{0≤t≤T −a_{T}} sup_{0≤s≤a_{T}} |Y (t + s) − Y (t)| ≥ min (\frac{K_{4} − ε}{\sqrt{1 + ε}}, \frac{\sqrt{2}}{π}) \frac{a_{T} \sqrt{log log T}}{T} .

This shows the lower bound in (1.16). The proof of Theorem 1.2(iib) is complete by applying the 0-1 law for Brownian motion. ⊔ ⊓

Acknowledgements

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

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