A joint functional law for the Wiener process

and principal value

by

Endre Csáki^{1}^{,}^{2}, Antónia Földes^{3} and Zhan Shi^{2}

Alfréd Rényi Institute of Mathematics, City University of New York & Université Paris VI

Summary. We present a joint functional iterated logarithm law for the Wiener process and the principal value of its local times.

Keywords. Wiener process, principal value of the local time, functional law of the iterated logarithm.

2000 Mathematics Subject Classification. 60F15; 60J55; 60J65.

1Research supported by the Hungarian National Foundation for Scientific Research, Grant Nos. T 029621 and T 037886.

2Research supported by the joint French–Hungarian Intergovernmental Grant “Balaton”, No. F-39/2000.

3Research supported by a PSC CUNY grant, No. 634680032.

1 Introduction

Let {W (t); t ⩾ 0} be a one-dimensional standard Wiener process with W (0) = 0, and let {L(t, x); t ⩾ 0, x ∈ ℝ} denote its local time process, jointly continuous in t and x. For any Borel function f ⩾ 0,

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

Put L(t, 0) = L(t) and

U_{t}(x) := \frac{W (xt)}{\sqrt{2t log log t}},
V_{t}(x) := \frac{L(xt)}{\sqrt{2t log log t}},  x ∈ [0, 1].

We consider x ↦ U_{t}(x) and x ↦ V_{t}(x) as elements of the space C = C[0, 1] of continuous functions with metric

d(f, g) = sup_{x∈[0,1]} |f(x) − g(x)|.

Recall the celebrated functional law of the iterated logarithm (FLIL) for W due to Strassen [15]:

Theorem A With probability one, the set {U_{t}}_{t⩾1} is relatively compact in C, with limit set equal to

S := {f ∈ C : f(0) = 0, f is absolutely continuous, with ∫_{0}^{1}(f′(x))^{2} dx ⩽ 1} .

Using that {L(t), t ⩾ 0} has the same distribution as {sup_{s∈[0,t]} W (s), t ⩾ 0}, one can easily obtain (cf. Csáki and Révész [7], Mueller [13], Chen [3]),

Theorem B With probability one, the set {V_{t}}_{t⩾1} is relatively compact in C, with limit set equal to

S_{M} := {g ∈ S : g is non-decreasing}.

In Csáki and Révész [7] a joint FLIL was given for the vector {(U_{t}(x), V_{t}(x)), x ∈ [0, 1]}_{t⩾1} on the space C^{(2)} := C × C with metric

d((f_{1}, g_{1}), (f_{2}, g_{2})) = sup_{x∈[0,1]} \sqrt{(f_{1}(x) − f_{2}(x))^{2} + (g_{1}(x) − g_{2}(x))^{2}}.

Theorem C With probability one, the set {(U_{t}, V_{t})}_{t⩾1} is relatively compact in C^{(2)}, with limit set equal to

S_{J}^{(2)} := {(f, g) : f ∈ S, g ∈ S_{M}, ∫_{0}^{1}(f′(x))^{2} + (g′(x))^{2}) dx ⩽ 1, f(x)g′(x) = 0 a.e.} .

We are interested in studying similar joint FLIL for the Wiener process and the process

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

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. (1.1)

Since x ↦ L(t, x) is Hölder continuous of order ν, for any ν < 1/2, the integral on the right hand side of (1.1) is well-defined.

The study of Cauchy’s principal value of Brownian local time goes back at least to Itô and McKean [12], and has become very active since the late 70s, due to applications in various branches of stochastic analysis. For a detailed account of various motivations, historical facts and general properties of principal values of local times, we refer to the recent collection of research papers in Yor [17], to Chapter 10 of the lecture notes by Yor [18], and to the survey paper by Yamada [16].

The process Y (·) defined in (1.1) is almost surely continuous, having zero quadratic variation. 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 the aforementioned zero quadratic variation property distinguishes 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. Hu and Shi [11] proved a law of the iterated logarithm for Y (·):

lim sup_{t→∞} \frac{Y (t)}{\sqrt{8t log log t}} = 1 a.s.

FLIL for Y was not known before. Here we show that similarly to Theorem C, a joint FLIL for W and Y holds. Introduce

Z_{t}(x) = \frac{Y (xt)}{\sqrt{8t log log t}}, 0 ⩽ x ⩽ 1.

Our main result is Theorem 1.1 With probability one the set {(U_{t}, Z_{t})}_{t⩾1} is relatively compact in C^{(2)}, with limit set equal to

S̃_{J}^{(2)} = {(f, g) : f ∈ S, g ∈ S, ∫_{0}^{1}(f′(x))^{2} + (g′(x))^{2}) dx ⩽ 1, f(x)g′(x) = 0 a.e.} .

Some consequences are as follows.

Corollary 1.2 With probability one, the set {Z_{t}}_{t⩾1} is relatively compact in C, with limit set equal to S given in Theorem A.

Corollary 1.3 With probability one, the set {(U_{t}(1), Z_{t}(1))}_{t⩾1} is relatively compact in ℝ^{2} with limit set equal to

{(x, y) ∈ ℝ^{2} : |x| + |y| ⩽ 1}.

The organization of the paper is as follows: In Section 2 we present some preliminary results for the distribution of the Wiener process and principal value, as well as certain estimates for the increments of the processes concerned. In Section 3 we prove Theorem 1.1. In Section 4 we prove the Corollaries. Some further remarks and consequences are given in Section 5.

Throughout the paper, for any x ∈ ℝ, we denote by ℙ^{x} the probability under which the Wiener process W starts from W (0) = x (thus ℙ = ℙ^{0}); unimportant constants (which are finite and positive) are denoted by the letter c with subscript.

2 Preliminaries

2.1 Distribution results for Wiener process and principal value

First recall some results for principal value. Biane and Yor [1] proved the following result: Let {B(s), 0 ⩽ s ⩽ 1} be a Brownian bridge, then

\frac{d}{dx} ℙ (∫_{0}^{1} \frac{ds}{B(s)} < x) = \frac{|x|}{2} ∑_{n=1}^{∞} exp (−\frac{(2n − 1)^{2}x^{2}}{8} )
(2.1)  ⩾ \frac{|x|}{2} exp (−\frac{x^{2}}{8} ) .

It follows that for 0 < α < β

ℙ (∫_{0}^{1} \frac{ds}{B(s)} ∈ (α, β)) ⩾ 2 (exp (−\frac{α^{2}}{8} ) − exp (−\frac{β^{2}}{8} )) . (2.2)

It was proved in [5] (cf. (2.11), (2.14) and (2.16) there) that for any δ > 0 there exists c_{1}(δ) > 0 such that for all s > 0 and x > 0,

sup_{z∈ℝ}ℙ^{z}(|Y (s)| > x) ⩽ c_{1}(δ) exp (−\frac{x^{2}}{(8 + δ)s}) . (2.3)

Lemma 2.1 Let s > 0, λ > 0, δ > 0 and 0 < ε < 1. For (a, α, z) ∈ ℝ^{3}, define

I = I(a, α, z) := ℙ^{z} (a ⩽ W (s) ⩽ a + 2ελ, α ⩽ Y (s) ⩽ α + 4ελ) . (2.4)

Then

I ⩽ \frac{λ}{\sqrt{s}} exp (−\frac{(|a − z| − 2ελ)^{2} − 4ε^{2}λ^{2}}{2s} ) . (2.5)

Moreover, if |α| ⩾ 4ελ, then

I ⩽ c_{1}(δ) exp (−\frac{(|α| − 4ελ)^{2}}{(8 + δ)s} ) , (2.6)

where c_{1}(δ) is the constant in (2.3).

Proof: Observe that

I ⩽ ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ) = ℙ (\frac{a − z}{\sqrt{s}} ⩽ N ⩽ \frac{a − z + 2ελ}{\sqrt{s}}) ,

where N is a standard normal variable. Hence (2.5) follows from a straightforward Gaussian estimate.

Now for |α| ⩾ 4ελ, we have

I ⩽ ℙ^{z}(α ⩽ Y (s) ⩽ α + 4ελ) ⩽ ℙ^{z}(|Y (s)| ⩾ |α| − 4ελ),

which implies (2.6) by means of (2.3). □

For the lower estimates we prove several lemmas.

Lemma 2.2 For α > 0, β − α > 4, 0 < δ < 1 we have

ℙ(|W (1)| ⩽ 1, α ⩽ Y (1) ⩽ β) ⩾ c_{2}(δ) exp (−\frac{(α + 1)^{2}}{8(1 − δ)}) , (2.7)

where c_{2}(δ) is a constant depending only on δ.

Proof: Let

G := sup{t : t ⩽ 1, W (t) = 0},
B(s) := \frac{W (sG)}{\sqrt{G}},  s ∈ [0, 1].

It is known that (B(s), s ∈ [0, 1]), G and (\frac{W (G+s(1−G))}{\sqrt{1−G}}, s ∈ [0, 1]) are independent, and that (B(s), s ∈ [0, 1]) is a (standard) Brownian bridge.

We have

ℙ(|W (1)| ⩽ 1, α ⩽ Y (1) ⩽ β)
⩾ ℙ(|W (1)| ⩽ 1, α + 1 ⩽ Y (G) ⩽ β − 1, |Y (1) − Y (G)| ⩽ 1, G ⩾ 1 − δ)
= ∫_{1−δ}^{1} ℙ(|W (1)| ⩽ 1, α + 1 ⩽ Y (κ) ⩽ β − 1, |Y (1) − Y (κ)| ⩽ 1 | G = κ) ℙ(G ∈ dκ)
= ∫_{1−δ}^{1} ℙ(α + 1 ⩽ Y (κ) ⩽ β − 1 | G = κ)×
×ℙ(|W (1)| ⩽ 1, |Y (1) − Y (κ)| ⩽ 1 | G = κ) ℙ(G ∈ dκ).

Since under the condition G = κ, Y (κ)/\sqrt{κ} has the same distribution as ∫_{0}^{1} ds/B(s), where B is a Brownian bridge, we get from (2.2)

ℙ(α + 1 ⩽ Y (κ) ⩽ β − 1 | G = κ) ⩾ 2 (exp (−\frac{(α + 1)^{2}}{8κ} ) − exp (−\frac{(β − 1)^{2}}{8κ} ))
⩾ 2(1 − e^{−1}) exp (−\frac{(α + 1)^{2}}{8κ} )
⩾ 2(1 − e^{−1}) exp (−\frac{(α + 1)^{2}}{8(1 − δ)}) .

This gives (2.7), with

c_{2}(δ) := 2(1 − e^{−1})ℙ(|W (1)| ⩽ 1, |Y (1) − Y (G)| ⩽ 1, G ⩾ 1 − δ).

The lemma is proved. □

Now we introduce the notation

T_{b} := inf{t : t ⩾ 0, W (t) = b}. (2.8)

By the reflection principle, we have for all u > 0 and (a, z) ∈ ℝ^{2},

ℙ^{z}(T_{a} ⩽ u) = 2\overline{Φ} (\frac{|z − a|}{\sqrt{u}}) , (2.9)

where \overline{Φ}(x) := ℙ(N > x) is the standard Gaussian tail distribution function.

In the sequel we shall use the inequalities:

(2.10)  \overline{Φ}(x) ⩾ \frac{1}{(2π)^{1/2}} (\frac{1}{x} − \frac{1}{x^{3}} ) exp (−\frac{x^{2}}{2} ) ,  x ⩾ 1,
(2.11)  \overline{Φ}(x) ⩽ \frac{1}{2} exp (−\frac{x^{2}}{2} ) ,  x > 0.

(For (2.11), see Proposition II.1.8 of Revuz and Yor [14].)

Lemma 2.3 For s > 0, 0 < δ < 1, z ∈ ℝ we have

ℙ^{z}(T_{0} ⩽ δs, |Y (T_{0})| ⩽ 2\sqrt{s}) ⩾ c_{3}(δ) \overline{Φ} (\frac{||z| − \sqrt{s}|}{δ\sqrt{s}}) . (2.12)

Proof: By symmetry, it suffices to prove (2.12) for z > 0 (there is nothing to prove if z = 0). Assuming first z > \sqrt{s}, we have

ℙ^{z}(T_{0} ⩽ δs, |Y (T_{0})| ⩽ 2\sqrt{s})
⩾ ℙ^{z}(T_{0} − T\sqrt{s} ⩽ δ(1 − δ)s, T\sqrt{s} ⩽ δ^{2}s, Y (T_{0}) − Y (T\sqrt{s}) ⩽ \sqrt{s})
= ℙ^{\sqrt{s}}(T_{0} ⩽ δ(1 − δ)s, Y (T_{0}) ⩽ \sqrt{s}) ℙ^{z}(T\sqrt{s} ⩽ δ^{2}s),

where we used the fact that T\sqrt{s} ⩽ δ^{2}s implies Y (T\sqrt{s}) ⩽ T\sqrt{s}/\sqrt{s} ⩽ δ^{2}\sqrt{s} < \sqrt{s}.

By scaling, ℙ^{\sqrt{s}}(T_{0} ⩽ δ(1 − δ)s, Y (T_{0}) ⩽ \sqrt{s}) is a positive constant depending only on δ. In view of (2.9), we have proved (2.12) in case z > \sqrt{s}.

If 0 < z ⩽ \sqrt{s}, we have, by scaling,

ℙ^{z}(T_{0} ⩽ δs, |Y (T_{0})| ⩽ 2\sqrt{s}) = ℙ^{1} (T_{0} ⩽ \frac{δs}{z^{2}}, |Y (T_{0})| ⩽ \frac{2\sqrt{s}}{z})
⩾ ℙ^{1} (T_{0} ⩽ δ, |Y (T_{0})| ⩽ 2)
=: c_{4}(δ),

from which (2.12) follows. □

Lemma 2.4 Let s > 0, ε > 0, λ > 0, 0 < δ < 1, (α, z) ∈ ℝ^{2} be such that ελ > 8\sqrt{s}. Then we have

ℙ^{z}(|W (s)| ⩽ ελ, α ⩽ Y (s) ⩽ α + 4ελ)
(2.13)  ⩾ c_{5}(δ) exp (−\frac{(|α| + 2ελ)^{2}}{8s(1 − δ)^{2}} ) \overline{Φ} (\frac{||z| − \sqrt{s}|}{δ\sqrt{s}}) .

Proof: Define, for n ⩾ 1,

I_{λ,z}(α, n) := ℙ^{z}(|W (s)| ⩽ ελ, α ⩽ Y (s) ⩽ α + nελ).

Note that I_{λ,z}(α, n) is non-decreasing in n. Moreover,

I_{λ,z}(α, n) ⩾ ℙ^{z}(|W (s)| ⩽ ελ, α ⩽ Y (s) ⩽ α + nελ, T_{0} ⩽ δs)
= ∫_{0}^{δs} ℙ^{z}(|W (s)| ⩽ ελ, α ⩽ Y (s) ⩽ α + nελ | T_{0} = τ) ℙ^{z}(T_{0} ∈ dτ)
⩾ ∫_{0}^{δs} ℙ^{z}(A_{τ} ∩ B_{τ}(n) | T_{0} = τ) ℙ^{z}(T_{0} ∈ dτ),

where

A_{τ} := \{|Y (τ)| ⩽ 2\sqrt{s}\},
B_{τ}(n) := \{|W (s)| ⩽ ελ, α + 2\sqrt{s} ⩽ Y (s) − Y (τ) ⩽ α + nελ − 2\sqrt{s}\}.

Under the condition {W (0) = z, T_{0} = τ}, A_{τ} and B_{τ}(n) are independent, so that

I_{λ,z}(α, n) ⩾ ∫_{0}^{δs} ℙ^{z}(A_{τ} | T_{0} = τ) ℙ^{z}(T_{0} ∈ dτ) × inf_{τ∈(0, δs)} ℙ^{z}(B_{τ}(n) | T_{0} = τ).

By Lemma 2.3,

∫_{0}^{δs} ℙ^{z}(A_{τ} | T_{0} = τ) ℙ^{z}(T_{0} ∈ dτ) = ℙ^{z}(|Y (T_{0})| ⩽ 2\sqrt{s}, T_{0} ⩽ δs)
⩾ c_{3}(δ) \overline{Φ} (\frac{||z| − \sqrt{s}|}{δ\sqrt{s}}) ,

whereas according to Lemma 2.2, and by scaling,

ℙ^{z}(B_{τ}(1) | T_{0} = τ)
= ℙ(|W (s − τ)| ⩽ ελ, α + 2\sqrt{s} ⩽ Y (s − τ) ⩽ α + ελ − 2\sqrt{s})
⩾ ℙ (|W (1)| ⩽ 1, \frac{α + 2\sqrt{s}}{\sqrt{s − τ}} ⩽ Y (1) ⩽ \frac{α + ελ − 2\sqrt{s}}{\sqrt{s − τ}}) .

Assume α ⩾ 0 for the moment. By Lemma 2.2,

ℙ^{z}(B_{τ}(1) | T_{0} = τ) ⩾ c_{2}(δ) exp (−\frac{(α + ελ)^{2}}{8s(1 − δ)^{2}} ) ,

which yields

I_{λ,z}(α, 1) ⩾ c_{6}(δ) exp (−\frac{(α + ελ)^{2}}{8s(1 − δ)^{2}} ) \overline{Φ} (\frac{||z| − \sqrt{s}|}{δ\sqrt{s}}) , α ⩾ 0, (2.14)

with c_{6}(δ) := c_{3}(δ)c_{2}(δ). Since I_{λ,z}(α, 4) ⩾ I_{λ,z}(α, 1), this yields (2.13) in case α ⩾ 0. To treat the case α ⩽ −ελ, we observe that

I_{λ,z}(α, 4) ⩾ ℙ^{z}(|W (s)| ⩽ ελ, α ⩽ Y (s) ⩽ α + ελ)
= ℙ^{−z}(|W (s)| ⩽ ελ, −α − ελ ⩽ Y (s) ⩽ −α),

the last identity following via replacing W by −W. This gives I_{λ,z}(α, 4) ⩾ I_{λ,−z}(−α − ελ, 1). Since −α − ελ ⩾ 0, we are entitled to apply (2.14) to deduce (2.13).

It remains to study the situation α ∈ (−ελ, 0). In this case,

I_{λ,z}(α, 4) ⩾ ℙ^{z}(|W (s)| ⩽ ελ, α + ελ ⩽ Y (s) ⩽ α + 2ελ) = I_{λ,z}(α + ελ, 1),

which yields (2.13) in view of (2.14).

Lemma 2.4 is proved. □

Lemma 2.5 For s > 0, ε > 0, λ > 0, (a, z) ∈ ℝ^{2} such that ε^{2}λ^{2} ⩾ 2s, az > 0, and

|z| > \frac{s}{2ελ} + 3ελ, |a| > \frac{s}{2ελ} + 3ελ

we have

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ) ⩾ \frac{1}{2} exp (−\frac{(|a − z| + 2ελ)^{2}}{2s} ) . (2.15)

Proof: It suffices to prove the lemma for z > \frac{s}{2ελ} + ελ and a > \frac{s}{2ελ} + ελ (then by symmetry, it will also cover the case a < 0 and z < 0). We have,

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ)
⩾ ℙ^{z} ( inf_{0⩽u⩽s} W (u) > \frac{s}{2ελ}, a ⩽ W (s) ⩽ a + 2ελ)
= ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ)

−ℙ^{z} ( inf_{0⩽u⩽s} W (u) ⩽ \frac{s}{2ελ}, a ⩽ W (s) ⩽ a + 2ελ) .

By the reflection principle,

ℙ^{z} ( inf_{0⩽u⩽s} W (u) ⩽ \frac{s}{2ελ}, a ⩽ W (s) ⩽ a + 2ελ)

= ℙ^{z} (\frac{s}{ελ} − a − 2ελ ⩽ W (s) ⩽ \frac{s}{ελ} − a)
⩽ ℙ (\frac{W (s)}{\sqrt{s}} ⩽ −\frac{a + z − \frac{s}{ελ}}{\sqrt{s}} )
⩽ \frac{1}{2} exp (−\frac{(a + z − \frac{s}{ελ})^{2}}{2s} ) ,

the last inequality following from (2.11). On the other hand,

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ) = ℙ (\frac{a − z}{\sqrt{s}} ⩽ \frac{W (s)}{\sqrt{s}} ⩽ \frac{a − z + 2ελ}{\sqrt{s}})
⩾ \frac{2ελ}{\sqrt{2πs}} exp (−\frac{(|a − z| + 2ελ)^{2}}{2s} )
⩾ exp (−\frac{(|a − z| + 2ελ)^{2}}{2s} ) .

Since a + z − s/(ελ) ⩾ |a − z| + 2ελ, we obtain (2.15). □ Lemma 2.6 For s > 0, ε > 0, λ > 0, (a, z) ∈ ℝ^{2} such that az < 0, |a| > 2ελ + \sqrt{s} and min(ελ/2, |z|) > \sqrt{s}, we have

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ) ⩾ c_{7}(δ) \overline{Φ} (\frac{|a − z| + 2ελ}{(1 − δ)\sqrt{s}}) . (2.16)

Proof: First we show for a > \sqrt{u}, ελ > 2\sqrt{u},

P (u) := ℙ(a ⩽ W (u) ⩽ a + 2ελ, |Y (u)| ⩽ 2\sqrt{u}) ⩾ c_{8}(δ) exp (−\frac{a^{2}}{2(1 − δ)u}) . (2.17)

Define G\sqrt{u} := sup\{t ⩽ u : W (t) = \sqrt{u} \}. Then

P (u) ⩾ ∫_{0}^{δu} ℙ(a ⩽ W (u) ⩽ a + 2ελ, |Y (u)| ⩽ 2\sqrt{u} | G\sqrt{u} = v) ℙ(G\sqrt{u} ∈ dv)
⩾ ∫_{0}^{δu} ℙ(|Y (v)| ⩽ \sqrt{u} | G\sqrt{u} = v) ℙ(a ⩽ W (u) ⩽ a + 2ελ | G\sqrt{u} = v) ℙ(G\sqrt{u} ∈ dv).

Under the condition \{G\sqrt{u} = v\}, \{M(r) := \frac{W (v+r(u−v))−\sqrt{u}}{\sqrt{u−v}}, r ∈ [0, 1]\} is a standard Brownian meander, and from the well-known identity (Biane and Yor [1]) ℙ(M(1) ⩽ x) = 1 − exp(−x^{2}/2), we get that, for v ∈ [0, δu], a > \sqrt{u} and ελ > 2\sqrt{u},

ℙ(a ⩽ W (u) ⩽ a + 2ελ | G\sqrt{u} = v) = ℙ (\frac{a − \sqrt{u}}{\sqrt{u − v}} ⩽ M(1) ⩽ \frac{a − \sqrt{u} + 2ελ}{\sqrt{u − v}})
= exp (−\frac{(a − \sqrt{u})^{2}}{2(u − v)} ) − exp (−\frac{(a − \sqrt{u} + 2ελ)^{2}}{2(u − v)} )
⩾ c_{9} exp (−\frac{(a − \sqrt{u})^{2}}{2(u − v)} )
⩾ c_{9} exp (−\frac{a^{2}}{2(1 − δ)u}) ,

where c_{9} > 0 is an absolute constant. Hence

P (u) ⩾ c_{10}(δ) exp (−\frac{a^{2}}{2(1 − δ)u}) ,

with

c_{10}(δ) := c_{9} ∫_{0}^{δu} ℙ(|Y (v)| ⩽ \sqrt{u} | G\sqrt{u} = v) ℙ(G\sqrt{u} ∈ dv)
= c_{9} ℙ(G\sqrt{u} ⩽ δu, |Y (G\sqrt{u})| ⩽ \sqrt{u}),

which, by scaling, does not depend on u. This yields (2.17).

We now start proving (2.16). Let ελ > 2\sqrt{s}. Let T_{0} and T_{−\sqrt{s}} be as in (2.8). It suffices to prove (2.16) for z < −\sqrt{s} and a > \sqrt{s} (then by symmetry, it will also cover the case z > \sqrt{s}, a < −2ελ − \sqrt{s}). Since |Y (T_{−\sqrt{s}})| ⩽ \sqrt{s} under ℙ^{z} (recalling that z < −\sqrt{s}), we have

ℙ^{z} (a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ)
⩾ ℙ^{z} (T_{0} − T_{−\sqrt{s}} ⩽ δs, |Y (T_{0}) − Y (T_{−\sqrt{s}})| ⩽ \sqrt{s},
a ⩽ W (s) ⩽ a + 2ελ, |Y (s) − Y (T_{−\sqrt{s}})| ⩽ 2\sqrt{s}) .

By the strong Markov property at times T_{−\sqrt{s}} and T_{0}, we get:

ℙ^{z} (a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ)
(2.18) ⩾ ∫_{0}^{δs} ( ∫_{0}^{s−y} P (s − h − y) ℙ^{z}(T_{−\sqrt{s}} ∈ dh)) ℙ^{−\sqrt{s}} (T_{0} ∈ dy, |Y (T_{0})| ⩽ \sqrt{s}) .

By (2.17),

P (s − h − y) ⩾ c_{8}(δ) exp (−\frac{a^{2}}{2(1 − δ)(s − h − y)})

⩾ 2c_{8}(δ) \overline{Φ} (\frac{a}{\sqrt{(1 − δ)(s − h − y)}})
= 2c_{8}(δ) ℙ^{−z−\sqrt{s}} (W (s − h − y) > \frac{a}{\sqrt{1 − δ}} − z − \sqrt{s}) ,

the second inequality being a consequence of (2.11). Therefore, for y ∈ [0, δs],

∫_{0}^{s−y} P (s − h − y) ℙ^{z}(T_{−\sqrt{s}} ∈ dh)

⩾ 2c_{8}(δ) ∫_{0}^{s−y} ℙ^{−z−\sqrt{s}} (W (s − h − y) > \frac{a}{\sqrt{1 − δ}} − z − \sqrt{s}) ℙ^{z} (T_{−\sqrt{s}} ∈ dh)
= 2c_{8}(δ) ℙ (W (s − y) > \frac{a}{\sqrt{1 − δ}} − z − \sqrt{s})
= 2c_{8}(δ) \overline{Φ} ( \frac{\frac{a}{\sqrt{1−δ}} − z − \sqrt{s}}{\sqrt{s − y}})
⩾ 2c_{8}(δ) \overline{Φ} (\frac{a − z}{(1 − δ)\sqrt{s}}) ,

(recalling that z < 0). Plugging this into (2.18), we get

ℙ^{z} (a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ⩽ 2ελ)

⩾ 2c_{8}(δ) \overline{Φ} (\frac{a − z}{(1 − δ)\sqrt{s}}) ℙ^{−\sqrt{s}} (T_{0} ⩽ δs, |Y (T_{0})| ⩽ \sqrt{s})
= c_{11}(δ) \overline{Φ} (\frac{a − z}{(1 − δ)\sqrt{s}}) ,

where c_{11}(δ) := 2c_{8}(δ) ℙ^{−1}(T_{0} ⩽ δ, |Y (T_{0})| ⩽ 1). This yields (2.16). □ Lemma 2.7 For s > 0, ε > 0, λ > 0, (a, z) ∈ ℝ^{2} such that ελ > 2\sqrt{s} and |a| > 2ελ + \sqrt{s}, we have

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ≤ 2ελ)
(2.19)  ⩾ c_{12}(δ) exp (−\frac{a^{2}}{2(1 − δ)^{2}s}) \overline{Φ} (\frac{||z| − \sqrt{s}|}{δ\sqrt{s}}) ,

with a constant c_{12}(δ) > 0. Proof: Again, it suffices to treat the case a > \sqrt{s}. In this case, we have

ℙ^{z}(a ⩽ W (s) ⩽ a + 2ελ, |Y (s)| ≤ 2ελ)

⩾ ∫_{0}^{δs} ℙ^{z}(|Y (T_{0})| ≤ 2\sqrt{s}, T_{0} ∈ dh) ℙ(a ≤ W (s − h) ≤ a + 2ελ, |Y (s − h)| ≤ 2\sqrt{s − h})
⩾ c_{13}(δ) exp (−\frac{a^{2}}{2(1 − δ)^{2}s}) ℙ^{z}(|Y (T_{0})| ≤ 2\sqrt{s}, T_{0} ≤ δs),

hence (2.19) follows from Lemma 2.3. □

2.2 Increments

Recall the results for the increments of Wiener process (cf. [9]) and principal value (cf. [4]). As T → ∞, we have almost surely

(2.20)  sup_{0⩽t⩽T −a_{T}} sup_{0⩽s⩽a_{T}} |X(t + s) − X(t)| = O (\sqrt{a_{T} (log(T/a_{T}) + log log T)} ) ,

and for fixed T, as δ → 0 we have almost surely

(2.21)  sup_{0⩽t⩽T} sup_{0⩽s⩽δ} |X(t + s) − X(t)| = O(\sqrt{δ log(1/δ)}).

Here in (2.20) and (2.21) X can be either W or Y.

3 Proof of Theorem 1.1

According to (2.20) for Y,

lim_{δ→0} sup_{t⩾1} sup_{0⩽x, x′⩽1, |x−x′|⩽δ} |Z_{t}(x) − Z_{t}(x′)| → 0,  a.s.

Now the relative compactness of {Z_{t}}_{t⩾1} in C follows from the Arzelà–Ascoli theorem. This fact and Theorem A imply that {(U_{t}, Z_{t})} is relatively compact in C^{(2)}. Our further proof will consist of two steps:

(1) With probability one any (f, g) ∉ S̃_{J}^{(2)} is not a limit point.

(2) With probability one every (f, g) ∈ S̃_{J}^{(2)} is a limit point.

Proof of (1): Obviously, if either f ∉ S, or g(0) ≠ 0, then (f, g) cannot be a limit point almost surely. So from now on we assume that f ∈ S and g(0) = 0. Let x_{0} ∈ (0, 1] be a point, where f(x_{0}) ≠ 0. Since f is continuous, there exists an interval (x_{1}, x_{2}) ⊂ [0, 1] such that x_{0} ∈ (x_{1}, x_{2}] and f(x) ≠ 0 for all x ∈ (x_{1}, x_{2}). We show that if (f, g) is a limit point, then g is constant in (x_{1}, x_{2}). Since (f, g) is a limit point, there exists a sequence {t_{n}}_{n⩾1} such that lim_{n→∞} t_{n} = ∞ and

|W (xt_{n})| ⩾ c_{14} \sqrt{2t_{n} log log t_{n}}, x ∈ (x_{1}, x_{2})

for some c_{14} > 0 and for every x ∈ (x_{1}, x_{2})

|\frac{Y (xt_{n}) − Y (x_{0}t_{n})}{\sqrt{8t_{n} log log t_{n}}} | = \frac{1}{\sqrt{8t_{n} log log t_{n}}} |∫_{x_{0}t_{n}}^{xt_{n}} \frac{ds}{W (s)}|
⩽ \frac{|xt_{n} − x_{0}t_{n}|}{4c_{14} t_{n} log log t_{n}} → 0 = g(x) − g(x_{0}),

as n → ∞. So g(x) = g(x_{0}) for every x ∈ (x_{1}, x_{2}). So if (f, g) is a limit point and g is absolutely continuous (which is not guaranteed so far), then we must have f(x)g′(x) = 0 a.e.

To this end, we need a lemma.

Lemma 3.1 Let (f, g) be such that f ∈ S, g(0) = 0 and either g is not absolutely continuous or f(x)g′(x) = 0 a.e., and

∫_{0}^{1} ((f′(x))^{2} + (g′(x))^{2}) dx > 1, (3.1)

holds. Then there exists a partition x_{0} = 0 < x_{1} < … < x_{k−1} < x_{k} = 1 of [0, 1] such that for any δ > 0 small enough, we have

Λ_{k} := ∑_{i=1}^{k} (\frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} 1l_{\{g_{i}−g_{i−1}=0\}} + \frac{8}{8 + δ} \frac{(g_{i} − g_{i−1})^{2}}{x_{i} − x_{i−1}} ) > 1, (3.2)

where f_{i} := f(x_{i}) and g_{i} := g(x_{i}).

Proof: If g is not absolutely continuous, then we can clearly find a partition x_{0} = 0 < x_{1} < … < x_{j−1} < x_{j} = 1 of [0, 1] such that

∑_{i=1}^{k} \frac{(g_{i} − g_{i−1})^{2}}{x_{i} − x_{i−1}} > 1 + \frac{δ}{8},

so we have also (3.2). If g is absolutely continuous and (3.1) holds, then we can find a partition x_{0} = 0 < x_{1} < … < x_{j−1} < x_{j} = 1 of [0, 1] such that

∑_{i=1}^{j} (\frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} + \frac{(g_{i} − g_{i−1})^{2}}{x_{i} − x_{i−1}} ) > 1

holds. Moreover, for any small enough δ > 0, we have also

∑_{i=1}^{j} (\frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} + \frac{8}{8 + δ} \frac{(g_{i} − g_{i−1})^{2}}{x_{i} − x_{i−1}} ) > 1.

For the ith interval of the above partition consider the following three cases: (i) f_{i−1} = f_{i}, (ii) f_{i−1} ≠ f_{i}, and f_{i−1}f_{i} ⩾ 0, (iii) f_{i−1} ≠ f_{i}, and f_{i−1}f_{i} < 0. In case (i) we can simply write

\frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} = \frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} 1l_{\{g_{i}−g_{i−1}=0\}}.

In case (ii) let x′_{i} = max{x ⩽ x_{i} : f(x) = f(x_{i−1})} and x′′_{i} = min{x ⩾ x′_{i} : f(x) = f(x_{i})}. (It is possible that x′_{i} = x_{i−1} or x′′_{i} = x_{i}.) Consider the refinement of the partition by replacing (x_{i−1}, x_{i}) with (x_{i−1}, x′_{i}), (x′_{i}, x′′_{i}), (x′′_{i}, x_{i}). In the interval (x′_{i}, x′′_{i}) f(x) must strictly be between f_{i−1} and f_{i}, so f(x) ≠ 0, hence g′(x) = 0 for all x ∈ (x′_{i}, x′′_{i}), thus g(x′_{i}) = g(x′′_{i}). Using the elementary inequality

\frac{(a + b)^{2}}{c + d} ⩽ \frac{a^{2}}{c} + \frac{b^{2}}{d} ,

we may write

\frac{(f_{i} − f_{i−1})^{2}}{x_{i} − x_{i−1}} ⩽ \frac{(f(x′_{i}) − f_{i−1})^{2}}{x′_{i} − x_{i−1}} + \frac{(f(x′′_{i}) − f(x′_{i}))^{2}}{x′′_{i} − x′_{i}} + \frac{(f_{i} − f(x′′_{i}))^{2}}{x_{i} − x′′_{i}}
= \frac{(f(x′_{i}) − f_{i−1})^{2}}{x′_{i} − x_{i−1}} 1l_{\{g(x′_{i})−g_{i−1}=0\}} + \frac{(f(x′′_{i}) − f(x′_{i}))^{2}}{x′′_{i} − x′_{i}} 1l_{\{g(x′′_{i})−g(x′_{i})=0\}}
(3.3)  + \frac{(f_{i} − f(x′′_{i}))^{2}}{x_{i} − x′′_{i}} 1l_{\{g_{i}−g(x′′_{i})=0\}}.

In case (iii) let x′_{i} = min{x ⩾ x_{i−1} : f(x) = 0} and x′′_{i} = max{x ⩽ x_{i} : f(x) = 0}. Consider again the refinement of the partition by replacing (x_{i−1}, x_{i}) with (x_{i−1}, x′_{i}), (x′_{i}, x′′_{i}), (x′′_{i}, x_{i}). In the first and the last of these three intervals f(x) ≠ 0, hence g′(x) = 0, thus g(x′_{i}) = g_{i−1} and g(x′′_{i}) = g_{i}. On the other hand, f(x′_{i}) = f(x′′_{i}) = 0. So we again have (3.3).

By repeating this argument, we get finally a partition for which (3.2) holds. This completes the proof of the Lemma. □

Returning to the main course of the proof, choose ε > 0 such that

Λ_{k} − 20ε ∑_{i=1}^{k} \frac{1}{x_{i} − x_{i−1}} > 1,

(f_{i−1} − ε, f_{i−1} + ε) and (f_{i} − ε, f_{i} + ε) are disjoint if f_{i} ≠ f_{i−1},

|g_{i} − g_{i−1}| > 6ε if g_{i} ≠ g_{i−1}.

Here f_{i} = f(x_{i}) and g_{i} = g(x_{i}), i = 1, …, k. We may also assume that |f_{i} − f_{i−1}| ⩽ 1 and |g_{i} − g_{i−1}| ⩽ 1, i = 1, …, k, otherwise (f, g) cannot be a limit point by the usual law of the iterated logarithm.

Define the events

A_{t}^{(i)} = {f_{i} − ε ⩽ U_{t}(x_{i}) ⩽ f_{i} + ε, g_{i} − g_{i−1} − 2ε ⩽ Z_{t}(x_{i}) − Z_{t}(x_{i−1}) ⩽ g_{i} − g_{i−1} + 2ε}
= {a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i}}

with s_{i} = x_{i}t and

a_{i} = (f_{i} − ε)(2t log log t)^{1/2}, b_{i} = (f_{i} + ε)(2t log log t)^{1/2},
α_{i} = (g_{i} − g_{i−1} − 2ε)2(2t log log t)^{1/2}, β_{i} = (g_{i} − g_{i−1} + 2ε)2(2t log log t)^{1/2}.

It follows from Lemma 2.1 putting λ = (2t log log t)^{1/2} there

ℙ(A_{t}^{(i)} | W (s_{i−1}) = z_{i−1}) ⩽ \sqrt{\frac{2 log log t}{x_{i} − x_{i−1}}} exp (−\frac{(f_{i} − f_{i−1})^{2} − 8ε}{x_{i} − x_{i−1}} log log t)

and if g_{i} ≠ g_{i−1}, then

ℙ(A_{t}^{(i)} | W (s_{i−1}) = z_{i−1}) ⩽ c_{15} exp (−\frac{(g_{i} − g_{i−1})^{2} − 20ε}{(8 + δ)(x_{i} − x_{i−1})}8 log log t)

with some c_{15} > 0. So for large enough t we have

ℙ(A_{t}^{(i)} | W (s_{i−1}) = z_{i−1}) ⩽ c_{16} \sqrt{\frac{log log t}{x_{i} − x_{i−1}}} [ exp (−\frac{(f_{i} − f_{i−1})^{2} − 8ε}{x_{i} − x_{i−1}} log log t) 1l_{\{g_{i}=g_{i−1}\}}
+ exp (−\frac{(g_{i} − g_{i−1})^{2} − 20ε}{(8 + δ)(x_{i} − x_{i−1})}8 log log t) 1l_{\{g_{i}≠g_{i−1}\}}].

It follows that for all large t and some constants c_{17} > 0 and δ̃ > 0,

ℙ(∩_{i=1}^{k}A_{t}^{(i)}) ⩽ c_{17} (log log t)^{3k/2} exp (− (Λ_{k} − 20ε ∑_{i=1}^{k} \frac{1}{x_{i} − x_{i−1}} ) log log t)
⩽ exp(−(1 + δ̃) log log t).

Let t = t_{n} = exp(n/(log n)). Then ∑_{n} ℙ(A_{t_{n}}) < ∞. By the Borel–Cantelli lemma,

(3.4)  lim inf_{n→∞} d ((U_{t_{n}}, Z_{t_{n}}) , (f, g)) ⩾ ε a.s.

On the other hand, we infer from the increment results in Section 2.3 that

(3.5)  lim_{n→∞} sup_{t∈[t_{n},t_{n+1}]} sup_{x∈[0,1]} |U_{t}(x) − U_{t_{n}}(x)| = 0 a.s.,
(3.6)  lim_{n→∞} sup_{t∈[t_{n},t_{n+1}]} sup_{x∈[0,1]} |Z_{t}(x) − Z_{t_{n}}(x)| = 0 a.s.,

Combining (3.5)–(3.6) with (3.4) gives that

lim inf_{t→∞} d ((U_{t}, Z_{t}) , (f, g)) ⩾ ε a.s.

for some ε > 0.

Thus we proved that if (f, g) ∉ S̃_{J}^{(2)}, then it is not a limit point with probability one, i.e. (f, g) has an open ball neighboorhood of radius ε not containing (U_{t}, Z_{t}) for large enough t. However the exceptional ω-set of probability zero may depend on (f, g). Now we prove that the totality of these exceptional ω-sets is still of probability zero. Denote the complement of S̃_{J}^{(2)} by D and for each (f, g) ∈ D consider the open balls defined above. Their union covers D and being C^{(2)} separable, we can select a countable subcover (cf. e.g. [2], p. 217). The union of exceptional ω-sets belonging to this countable subcover is still of probability zero. We call the complement of this last set of probability zero as our universal ω-set. Each (f, g) ∈ D has a neighborhood which is completely contained in one of the elements of the countable subcover, hence on the universal set this neighborhood for large enough t does not contain (U_{t}, Z_{t}), i.e. (f, g) is not a limit point. This completes the proof of (1). □

Proof of (2): Assume that (f, g) ∈ S̃_{J}^{(2)} with strict inequality in the integral criterion, i.e.

∫_{0}^{1}((f′(x))^{2} + (g′(x))^{2}) dx < 1.

For given ε_{1} > 0, choose a partition x_{0} = 0 < x_{1} … < x_{k} = 1 of the interval [0, 1] such that

sup_{1⩽i⩽k}(x_{i} − x_{i−1}) ⩽ ε_{1}^{2},
sup_{1⩽i⩽k} sup_{x∈[x_{i−1},x_{i}]} |f(x) − f_{i}| ⩽ ε_{1},
sup_{1⩽i⩽k} sup_{x∈[x_{i−1},x_{i}]} |g(x) − g_{i}| ⩽ ε_{1},

where f_{i} = f(x_{i}), g_{i} = g(x_{i}). We may assume that if g_{i−1} ≠ g_{i}, then f_{i−1} = f_{i} = 0. Otherwise if it happens that g_{i−1} ≠ g_{i} but either f_{i−1} ≠ 0 or f_{i} ≠ 0 (or both), then we can choose x′ = min{x : x > x_{i−1}, f(x) = 0}, x′′ = max{x : x < x_{i}, f(x) = 0}. We must have g(x′) = g_{i−1} and g(x′′) = g_{i} so by refining the original partition by inserting new points x′, x′′, the new partition satisfies the above assumption. Since

\frac{(f(x_{i}) − f(x_{i−1})^{2}}{x_{i−1} − x_{i}} ⩽ ∫_{x_{i−1}}^{x_{i}}(f′(x))^{2} dx, \frac{(g(x_{i}) − g(x_{i−1}))^{2}}{x_{i−1} − x_{i}} ⩽ ∫_{x_{i−1}}^{x_{i}}(g′(x))^{2} dx,

(cf. for example [10], p. 52), we have

Λ̄_{k} := ∑_{i=1}^{k} \frac{(f_{i} − f_{i−1})^{2} + (g_{i} − g_{i−1})^{2}}{x_{i} − x_{i−1}} < 1. (3.7)

Now choose 0 < δ < 1 such that Λ̄_{k} < (1 − δ)^{2} and then choose ε > 0 such that

Γ := \frac{Λ̄_{k}}{(1 − δ)^{2}} + (\frac{20ε}{(1 − δ)^{2}} + \frac{2ε}{δ^{2}}) ∑_{i=1}^{k} \frac{1}{x_{i} − x_{i−1}} < 1 (3.8)

and 5ε < |f_{i}|, i = 1, 2, …, k.

Introduce the notations λ = (2t log log t)^{1/2}, s_{i} = tx_{i},

a_{i} = (f_{i} − ε)λ, b_{i} = (f_{i} + ε)λ,

α_{i} = 2(g_{i} − g_{i−1} − ε)λ, β_{i} = 2(g_{i} − g_{i−1} + ε)λ.

By using the strong Markov property of the Wiener process, it is readily seen that

ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i}, i = 1, 2, …, k)

⩾ ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i}, i = 1, 2, …, k − 1)×

× inf_{a_{k−1}⩽z_{k−1}⩽b_{k−1}} ℙ(a_{k} ⩽ W (s_{k}) ⩽ b_{k}, α_{k} ⩽ Y (s_{k}) − Y (s_{k−1}) ⩽ β_{k} | W (s_{k−1}) = z_{k−1}).

Iterating this argument we can see that

ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i}, i = 1, 2, …, k)

(3.9) ⩾ ∏_{i=1}^{k} inf_{a_{i−1}⩽z_{i−1}⩽b_{i−1}} ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i} | W (s_{i−1}) = z_{i−1}).

Next we show that for i = 1, 2, …, k we have

ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i} | W (s_{i−1}) = z_{i−1}) ⩾ \frac{c_{18}(δ)}{(log log t)^{1/2}} ×

(3.10) × exp (− (\frac{(f_{i} − f_{i−1})^{2} + (g_{i} − g_{i−1})^{2} + 20ε}{(1 − δ)^{2}(x_{i} − x_{i−1})} + \frac{2ε}{δ^{2}(x_{i} − x_{i−1})}) log log t)

with some c_{18}(δ) > 0. To see (3.10) we apply Lemmas 2.4–2.7 with s = s_{i} − s_{i−1} = t(x_{i} − x_{i−1}), λ = (2t log log t)^{1/2} and t large enough and use the inequalities |f_{i} − f_{i−1}| ⩽ 1, |g_{i} − g_{i−1}| ⩽ 1, ε < 1.

(1) In case f_{i} = f_{i−1} = 0, apply Lemma 2.4 with α = (g_{i} − g_{i−1} − ε)λ, |z| ⩽ ελ and observe that by (2.10), \overline{Φ} gives a constant×(log log t)^{−1/2} factor in front of the exponent.

(2) In case g_{i} = g_{i−1}, f_{i}f_{i−1} > 0, apply Lemma 2.5 with a = (f_{i}−ε)λ and use |z−f_{i−1}λ| ⩽ ελ.

(3) In case g_{i} = g_{i−1}, f_{i}f_{i−1} < 0, apply Lemma 2.6 with a = (f_{i}−ε)λ and use |z−f_{i−1}λ| ⩽ ελ.

(4) In case g_{i} = g_{i−1}, f_{i} = 0, f_{i−1} ≠ 0, apply Lemma 2.4 with α = −2ελ, use that |z − f_{i−1}λ| ⩽ ελ and replace δ by 1 − δ.

(5) In case g_{i} = g_{i−1}, f_{i} = 0, f_{i−1} ≠ 0, apply Lemma 2.7 with a = (f_{i} − ε)λ, |z| ⩽ ελ. Assembling all these estimations, (3.10) follows. This combined with (3.9) gives

ℙ(a_{i} ⩽ W (s_{i}) ⩽ b_{i}, α_{i} ⩽ Y (s_{i}) − Y (s_{i−1}) ⩽ β_{i}, i = 1, 2, …, k)

(3.11) ⩾ \frac{(c_{18}(δ))^{k}}{(log log t)^{k/2}} exp(−Γ log log t),

where Γ < 1 is given by (3.8).

Now let t_{i} = exp(7i log i), i = 1, 2, … and define

η_{0} = 0, T_{i} = η_{i−1} + t_{i}, η_{i} = inf{t : t > T_{i}, W (t) = 0}, i = 1, 2, …

It was shown in [6] that we have almost surely for all large enough n,

t_{n} ⩽ T_{n} ⩽ t_{n} (1 + \frac{1}{n}) .

Define

Ŵ^{(n)}(t) = W (t + η_{n−1}), t ⩾ 0,
Ŷ^{(n)}(t) = Y (t + η_{n−1}) − Y (η_{n−1}), t ⩾ 0,
Û^{(n)}(x) = \frac{Ŵ^{(n)}(xt_{n})}{\sqrt{2t_{n} log log t_{n}}} , x ∈ [0, 1],
Ẑ^{(n)}(x) = \frac{Ŷ^{(n)}(xt_{n})}{\sqrt{2t_{n} log log t_{n}}} , x ∈ [0, 1].

Now let x_{0} = 0 < x_{1} < … < x_{k} = 1 be a partition as before and consider the events Ê_{n} = ∩_{i=1}^{k}Ê_{n}^{(i)} with

Ê_{n}^{(i)} = {â_{i} ⩽ Ŵ^{(n)}(ŝ_{i}) ⩽ b̂_{i}, α̂_{i} ⩽ Ŷ^{(n)}(ŝ_{i}) − Ŷ^{(n)}(ŝ_{i−1}) ⩽ β̂_{i}},

ŝ_{i} = x_{i}t_{n},

â_{i} = (f_{i} − ε)(2t_{n} log log t_{n})^{1/2}, b̂_{i} = (f_{i} + ε)(2t_{n} log log t_{n})^{1/2},
α̂_{i} = (g_{i} − g_{i−1} − ε)^{+}(2t_{n} log log t_{n})^{1/2}, β̂_{i} = (g_{i} − g_{i−1} + ε)(2t_{n} log log t_{n})^{1/2}.

It follows from (3.11) that ∑_{n} ℙ(Ê_{n}) = ∞ and since Ê_{n} are independent, we have by the Borel–Cantelli lemma ℙ(Ê_{n} i.o.) = 1. Since ε > 0 is arbitrary, this implies

lim inf_{n→∞} sup_{1⩽i⩽k} |Û^{(n)}(x_{i}) − f(x_{i})| = 0 a.s.
lim inf_{n→∞} sup_{1⩽i⩽k} |Ẑ^{(n)}(x_{i}) − g(x_{i})| = 0 a.s.

Again, from the increment results in Subsection 2.2 it follows that

lim sup_{n→∞} sup_{1⩽i⩽k} sup_{x∈[x_{i−1},x_{i})} |Û^{(n)}(x_{i−1}) − Û^{(n)}(x)| ⩽ ε_{1} a.s.
lim sup_{n→∞} sup_{1⩽i⩽k} sup_{x∈[x_{i−1},x_{i})} |Ẑ^{(n)}(x_{i−1}) − Ẑ^{(n)}(x)| ⩽ ε_{1} a.s.

Since ε_{1} > 0 is arbitrary, these yield

lim inf_{n→∞} d ((Û^{(n)}, Ẑ^{(n)}), (f, g)) = 0 a.s.

On the other hand, the increment results in Subsection 2.2 once again yields that, as n → ∞, d((Û^{(n)}, Ẑ^{(n)}), (U_{T_{n}}, Z_{T_{n}})) converges to 0 almost surely. Therefore,

lim inf_{n→∞} d ((U_{T_{n}}, Z_{T_{n}}), (f, g)) = 0 a.s.,

Hence, (f, g) is a limit point of (U_{t}, Z_{t}) with probability 1.

To complete the proof of Theorem 1.1, we have to show that there exists an ω-set of probability one for which every (f, g) ∈ S̃_{J}^{(2)} is a limit point.

First we show that there exists a countable dense subset K ⊂ S̃_{J}^{(2)}. For any (f, g) ∈ S̃_{J}^{(2)} and ε > 0, as before, choose a partition x_{0} = 0 < x_{1} < … < x_{k−1} < x_{k} = 1 such that

sup_{x_{i−1}⩽x⩽x_{i}} |f(x) − f(x_{i})| ⩽ ε,  sup_{x_{i−1}⩽x⩽x_{i}} |g(x) − g(x_{i})| ⩽ ε

and g(x_{i−1}) ≠ g(x_{i}) implies f(x_{i−1}) = f(x_{i}) = 0. Define (f̃, g̃) ∈ S_{J}^{(2)} such that f̃(x_{i}) = f(x_{i}), g̃(x_{i}) = g(x_{i}), i = 1, 2, …, k and let f̃ and g̃ be linear in between. Then

d((f, g), (f̃, g̃)) < 2\sqrt{2}ε,

meaning that the set of pairs (f, g), where both f and g are piecewise linear (with the same cut-off points), is dense. It can be seen that one can choose a countable dense subset K = {(f_{n}, g_{n})}_{n=1}^{∞} (for example by taking all x_{i}, f_{n}(x_{i}), g_{n}(x_{i}) rational) such that

∫_{0}^{1}(f′_{n}(x))^{2} + (g′_{n}(x))^{2} dx < 1.

It follows that there exists an ω-set of probability one such that all (f_{n}, g_{n}) ∈ K are limit points. Next we show that for this ω-set every (f, g) ∈ S̃_{J}^{(2)} is a limit point. Since K is dense, for each n we find (f_{n}, g_{n}) ∈ K such that

d((f, g), (f_{n}, g_{n})) < \frac{1}{n}

and since (f_{n}, g_{n}) is a limit point, we can find t_{n} such that d((f_{n}, g_{n}), (U_{t_{n}}, Z_{t_{n}})) < \frac{1}{n}. Hence d((f, g), (U_{t_{n}}, Z_{t_{n}})) < \frac{2}{n}. Consequently,

lim_{n→∞}(U_{t_{n}}, Z_{t_{n}}) = (f, g),

i.e., (f, g) is a limit point.

This completes the proof of Theorem 1.1. □

4 Proof of Corollaries

The proof of Corollary 1.2 is obvious. To show Corollary 1.3 we need the following lemma.

Lemma 4.1 If f and g are absolutely continuous functions and f(x)g′(x) = 0 a.e., then

∫_{0}^{1}(f′(x))^{2} 1l_{{g′(x)≠0}} dx = 0. (4.1)

Proof: Let

A = {x ∈ [0, 1] : f(x) = 0, f′(x) ≠ 0}.

For each x ∈ A, there exists δ_{x} > 0 such that f(y) ≠ 0 for all y ∈ (x − δ_{x}, x + δ_{x})\{x}. The intervals {(x − δ_{x}, x + δ_{x})}_{x∈A} being disjoint and thus containing each a different rational number, they are at most countably many. This means A is a countable set. Now (4.1) follows immediately.

This proof, more elegant than our original one, was kindly communicated to us by Omer Adelman. □

Now we prove Corollary 1.3. It follows from Lemma 4.1 that if (f, g) ∈ S̃_{J}^{(2)}, then

∫_{0}^{1}(f′(x) + g′(x))^{2} dx ⩽ 1, ∫_{0}^{1}(f′(x) − g′(x))^{2} dx ⩽ 1,

from which (cf. [15])

|f(1) + g(1)| ⩽ 1, |f(1) − g(1)| ⩽ 1

showing that a limit point cannot be outside the set given in the Corollary.

To show that every point is a limit point, define

f(u) = \frac{x(u − 1 + |x|)}{|x|}1l_{\{1−|x|≤u≤1\}}, g(u) = \frac{yu}{|y|}1l_{\{0≤u≤|y|\}} + y1l_{\{|y|≤u≤1\}}.

It is easy to see that (f, g) ∈ S̃_{J}^{(2)} and f(1) = x, g(1) = y. So (x, y) is a limit point. □

5 Further consequences: additive functionals

Consider the additive functional

A(t) = ∫_{0}^{t} ψ(W (s)) ds = ∫_{ℝ} ψ(x)L(t, x) dx,

where ψ is an integrable function such that \overline{ψ} := ∫_{ℝ} ψ(x) dx ≠ 0. Then by the ratio ergodic theorem (cf. [12], p. 228)

lim_{t→∞} \frac{A(t)}{\overline{ψ} L(t)} = 1 a.s.

Hence, introducing

Ṽ_{t}(x) := \frac{A(xt)}{\overline{ψ} \sqrt{2t log log t}},

Theorem C implies Corollary 5.1 With probability one, the set {(U_{t}, Ṽ_{t})}_{t⩾1} is relatively compact in C^{(2)}, with limit set equal to

S_{J}^{(2)} := {(f, g) : f ∈ S, g ∈ S_{M}, ∫_{0}^{1}(f′(x))^{2} + (g′(x))^{2}) dx ≤ 1, f(x)g′(x) = 0 a.e.} .

On the other hand, there are additive functionals which can be approximated by the principal value Y (t). Let ψ be a function as above and consider its Hilbert transform:

H(ψ)(x) = \frac{1}{π} p.v. ∫_{ℝ} \frac{ψ(y)}{x − y} dy,

where p.v. indicates that the integral should be considered as a principal value. It was shown in [8] that if ψ is a Borel function on ℝ such that

∫_{ℝ} x^{κ}|ψ(x)| dx < ∞,

for some κ > 0, then for all sufficiently small ε > 0, when t → ∞,

B(t) := ∫_{0}^{t}(Hψ)(W (s)) ds = \frac{\overline{ψ}}{π} Y (t) + o(t^{1/2−ε}), a.s.

Introducing the notation

Z̃_{t}(x) = \frac{πB(xt)}{\overline{ψ} \sqrt{8t log log t}},

we have

Corollary 5.2 With probability one, the set {(U_{t}, Z̃_{t})}_{t⩾1} is relatively compact in C^{(2)}, with limit set equal to

S̃_{J}^{(2)} = {(f, g) : f ∈ S, g ∈ S, ∫_{0}^{1}(f′(x))^{2} + (g′(x))^{2}) dx ≤ 1, f(x)g′(x) = 0 a.e.} .

Acknowledgements

We are grateful to Omer Adelman for helpful discussions and a referee for useful remarks.

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Endre Csáki  Antónia Földes
Alfréd Rényi Institute of Mathematics  Department of Mathematics
Hungarian Academy of Sciences  City University of New York
P.O.B. 127  2800 Victory Blvd.
H-1364 Budapest  Staten Island, New York 10314
Hungary  U.S.A.

 csaki@renyi.hu                          afoldes@gc.cuny.edu

Zhan Shi Laboratoire de Probabilités UMR 7599 Université Paris VI 4 Place Jussieu F-75252 Paris Cedex 05 France

 zhan@proba.jussieu.fr