On the ranked excursion heights of a Kiefer process

by

Endre CSÁKI^{1} and Yueyun HU^{2}

Abstract: Let (K(s, t), 0 ≤ s ≤ 1, t ≥ 1) be a Kiefer process, i.e. a continuous twoparameter centered Gaussian process indexed by [0, 1]×ℝ_{+} whose covariance function is given by 𝔼(K(s_{1}, t_{1})K(s_{2}, t_{2})) = (s_{1}∧s_{2}−s_{1}s_{2}) t_{1}∧t_{2}, 0 ≤ s_{1}, s_{2} ≤ 1, t_{1}, t_{2} ≥ 0. For each t > 0, the process K(·, t) is a Brownian bridge on the scale of \sqrt{t}. Let M_{1}^{∗}(t) ≥ M_{2}^{∗}(t) ≥ ...M_{j}^{∗}(t) ≥ ...0 be the ranked excursion heights of K(·, t). In this paper, we study the path properties of the process t → M_{j}^{∗}(t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M_{j}^{∗}(t) as t goes to infinity.

Keywords. Kiefer process, excursions, ranked heights.

AMS Classification 2000. 60F15, 60G55.

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

127, Budapest, H–1364, Hungary. E-mail: csaki@renyi.hu. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 029621 and T 037886

2Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR–7599), Université Paris VI, 4 Place Jussieu, F–75252 Paris cedex 05, France. E-mail: hu@ccr.jussieu.fr

1 Introduction

Let {B(t), t ≥ 0} be a standard one-dimensional Brownian motion, i.e. a continuous centered Gaussian process with covariance

𝔼(B(t_{1})B(t_{2})) = t_{1} ∧ t_{2}, t_{1}, t_{2} ≥ 0.

We consider also standard Brownian bridge {p(s), 0 ≤ s ≤ 1}, i.e. a centered Gaussian process with covariance

𝔼 (p(s_{1})p(s_{2})) = s_{1} ∧ s_{2} − s_{1}s_{2}, 0 ≤ s_{1}, s_{2} ≤ 1.

It is well-known that almost all sample paths of B consists of countable many zero-free intervals called excursions. Let (a, b) an excursion interval, i.e. B(a) = B(b) = 0 and either B(s) > 0, a < s < b called positive excursion, or B(s) < 0, a < s < b called negative excursion. The height of excursion is defined by either

H =^{def} max_{a≤s≤b} B(s)

or

H^{∗} =^{def} max_{a≤s≤b} |B(s)|.

Clearly, H > 0 holds only for positive excursions. Pitman and Yor [11] introduced the ranked heights of excursions up to time t: let

H_{1}(t) ≥ H_{2}(t) ≥ … H_{j}(t) ≥ …

and

H_{1}^{∗}(t) ≥ H_{2}^{∗}(t) ≥ … H_{j}^{∗}(t) ≥ …

be the heights of positive and all excursions respectively, of {B(s), 0 ≤ s ≤ t}, including the meander heights sup_{g_{t}≤s≤t} B(s) and sup_{g_{t}≤s≤t} |B(s)|, where g_{t} denotes the last zero of B before t. The ranked heights of excursions of p can be defined similarly.

Let furthermore {K(s, t), 0 ≤ s ≤ 1, t ≥ 0} be a Kiefer process, i.e. a continuous twoparameter centered Gaussian process indexed by [0, 1] × ℝ_{+} whose covariance function is given by

𝔼(K(s_{1}, t_{1})K(s_{2}, t_{2})) = (s_{1} ∧ s_{2} − s_{1}s_{2}) t_{1} ∧ t_{2}, 0 ≤ s_{1}, s_{2} ≤ 1, t_{1}, t_{2} ≥ 0.

Kiefer [7] introduced this process K to approximate the empirical process. See Csörgő and Révész [4] for detailed studies and related references on Kiefer process and on the invariance principle between empirical process and Kiefer process. Note that for fixed t > 0, the process s ∈ [0, 1] → \frac{K(s,t)}{\sqrt{t}} is a standard Brownian bridge. Denote by

M_{1}(t) ≥ M_{2}(t) ≥ ... ≥ M_{j}(t) ≥ ...

the ranked heights of the positive excursions of the Brownian bridge K(·, t) over the whole time interval [0, 1]. Denote by

M_{1}^{∗}(t) ≥ M_{2}^{∗}(t) ≥ ... ≥ M_{j}^{∗}(t) ≥ ...

the ranked heights of the excursions of |K(·, t)|. By scaling properties, the distributions of (\frac{M_{j}(t)}{\sqrt{t}}, j ≥ 1) and (\frac{M_{j}^{∗}(t)}{\sqrt{t}}, j ≥ 1) are the same as that of the ranked excursion heights of a standard Brownian bridge. See Pitman and Yor [12] for studies on these distribution.

We are interested in the path properties of the processes t → M_{j}(t) and t → M_{j}^{∗}(t). In particular, we aim at the asymptotic behaviors of M_{j}(t) and M_{j}^{∗}(t) as t → ∞.

Observe that M_{1}(t) = sup_{0≤s≤1} K(s, t) and M_{1}^{∗}(t) = sup_{0≤s≤1} |K(s, t)|. The following laws of the iterated logarithm are known, see respectively Csörgő and Révész ([4], pp. 81), Mogul’skii [8] and Csáki and Shi [3]:

Theorem A ([4], [8], [3]). We have

lim sup_{t→∞} \frac{M_{1}^{∗}(t)}{\sqrt{t log log t}} = \frac{1}{\sqrt{2}},  a.s.  (1.1)
lim inf_{t→∞} \sqrt{\frac{log log t}{t}}M_{1}^{∗}(t) = \frac{π}{\sqrt{8}},  a.s.  (1.2)

lim inf_{t→∞} \frac{(log t)^{χ}}{\sqrt{t}} M_{1}(t) = \{ \begin{matrix}0 & if χ ≤ \frac{1}{2} \\ ∞ & if χ > \frac{1}{2} \end{matrix}  a.s.  (1.3)

In (1.1) we may replace M_{1}^{∗}(t) by M_{1}(t).

The almost sure behavior of H_{j}^{∗}(t) was studied in Csáki and Hu [2]:

Theorem B ([2]). We have

lim sup_{t→∞} \frac{H_{j}^{∗}(t)}{\sqrt{t log log t}} = \frac{\sqrt{2}}{2j − 1},  a.s. j ≥ 1  (1.4)
lim inf_{t→∞} \frac{(log t)^{χ}}{\sqrt{t}} H_{j}^{∗}(t) = \{ \begin{matrix}0 & if χ ≤ 1 \\ ∞ & if χ > 1 \end{matrix}  a.s. j ≥ 2.  (1.5)

A natural question is to ask what happens with (M_{j}^{∗}(t), t ≥ 0) for j ≥ 2. As a process indexed by t, the j-highest heights M_{j}^{∗}(t) may share some unusual properties different from M_{1}^{∗}(t). For instance, t → M_{j}^{∗}(t) is not continuous for j ≥ 2 in contrast with the continuity of M_{1}^{∗}(·).

Theorem 1.1 Fix j ≥ 1. We have

lim sup_{t→∞} \frac{M_{j}^{∗}(t)}{\sqrt{t log log t}} = \frac{1}{j\sqrt{2}}, a.s.

The same result remains true when M_{j}^{∗}(t) is replaced by M_{j}(t).

It is also of interest to find the liminf behavior of M_{j}(·):

Theorem 1.2 Fix j ≥ 2. We have

lim inf_{t→∞} \frac{(log t)^{χ}}{\sqrt{t}} M_{j}^{∗}(t) = \{ \begin{matrix}0 & if χ ≤ \frac{1}{2} \\ ∞ & if χ > \frac{1}{2} \end{matrix} a.s.

The same result remains true when we replace M_{j}^{∗}(t) by M_{j}(t).

Comparing (1.2) with Theorem 1.2, we can see that the liminf behaviors of M_{1}^{∗} and M_{j}^{∗} (j > 1) are completely different.

The proof of Theorem 1.1 is based on an estimate on the downcrossings of a Brownian bridge, this estimate will be given in Section 2. To show Theorem 1.2, a usual way would be to estimate ℙ( inf_{1≤t≤2} M_{j}(t) < ϵ) as ϵ goes to 0. This problem remains open to our best knowledge. To overcome this difficulty, we shall adopt the method of Csáki and Shi [3], which consists of reducing the problem for the Kiefer process to that for an Ornstein-Uhlenbeck process. Section 2 also contains several preliminary results to complete the proofs of Theorems 1.1 and 1.2, which will be presented respectively in Sections 3 and 4.

Throughout this paper, (C_{k}, 1 ≤ k ≤ 6) denote some positive constants whose exact values are unimportant.

2 Downcrossings

Consider a continuous function f : I = [a, b] → ℝ with a, b ∈ ℝ. For two real numbers x < y, we define inductively

α_{1} = α_{1}(y) =^{def} inf{v ≥ a : f(v) ≥ y},  (2.1)
β_{k} = β_{k}(x) =^{def} inf{v ≥ α_{k} : f(v) ≤ x},  k ≥ 1,  (2.2)
α_{k} = α_{k}(y) =^{def} inf{v ≥ β_{k−1} : f(v) ≥ y},  k ≥ 2,  (2.3)

with the convention inf ∅ = ∞. Define the number of downcrossings of (x, y) by f during the time interval I as

D_{f}(x, y; I) = sup{k : α_{k}(y) ≤ b}.  (2.4)

We adopt the above definition of downcrossings, which is slightly different from the usual one, to keep the following equivalence:

sup_{v∈I} f(v) ≥ y  ⟺  D_{f}(x, y; I) ≥ 1.

Remark that the condition {D_{f}(x, y; I) ≥ 1} does not depend on x. In the following two subsections, we shall discuss respectively the numbers of downcrossings by a standard Brownian motion, a Brownian bridge and by an Ornstein-Uhlenbeck process.

2.1 Brownian motion and Brownian bridge

Let {B(s), s ≥ 0} be a standard Brownian motion and let {p(s), 0 ≤ s ≤ 1} be a standard Brownian bridge from 0 to 0. First, we present a preliminary result based on the reflection principle.

Lemma 2.1 Fix j ≥ 1 and max(x, 0) < y. We have

ℙ(D_{B}(x, y; [0, 1]) ≥ j, B(1) ∈ dz) = \{ \begin{matrix}φ(2jy − 2(j − 1)x − z)dz & if z ≤ y, \\ φ(2(j − 1)y − 2(j − 1)x + z)dz & if z > y, \end{matrix} (2.5)

where φ is the standard normal density function.

Proof: We use the reflection principle formulated by (cf., e.g. [5])

Fact 2.2 Let {B(s), s ≥ 0} be a standard Brownian motion and let τ be a stopping time for B. Then

B^{(τ)}(s) =^{def} \{ \begin{matrix}B(s) & if 0 ≤ s ≤ τ \\ 2B(τ) − B(s) & if τ ≤ s \end{matrix}

is also a standard Brownian motion.

Let us make use of the stopping times α_{k} = α_{k}(y) and β_{k} = β_{k}(x) introduced in (2.1)– (2.3), corresponding to f(t) = B(t), I = [0, 1].

Our Lemma 2.1 is well-known for j = 1.

We illustrate the proof in the simple case j = 2, using the reflection principle subsequently for our stopping times. Let {B(s), 0 ≤ s ≤ 1} be a Brownian motion such that α_{2} < 1 and B(1) = z ≤ y. Then by Fact 2.2, B_{1}(s) =^{def} B^{(α_{1})}(s), 0 ≤ s ≤ 1 is a Brownian motion with B_{1}(1) = 2y − z, β_{1} is its first hitting time of 2y − x and α_{2} is its first hitting time of y after β_{1}. In the next step consider B_{2}(s) =^{def} B_{1}^{(β_{1})}(s), 0 ≤ s ≤ 1. Then B_{2}(1) = 2y − 2x + z, and α_{2} is its first hitting time of 3y − 2x. Finally, consider B_{3}(s) = B_{2}^{(α_{2})}(s), 0 ≤ s ≤ 1 for which we have B_{3}(1) = 4y − 2x − z. By reversing this procedure, starting from a Brownian motion with endpoints 4y − 2x − z at s = 1, we get a Brownian motion with α_{1} < 1 and B(1) = z. This proves the first equality of (2.5) in the case j = 2. The procedure is similar for z > y, except that we stop with B_{2}, so the last reflection (at α_{2}) is not performed. Using this idea in obvious manner for the general case j > 2, yields our lemma.

Since a Brownian bridge {p(s), 0 ≤ s ≤ 1} is a Brownian motion conditioned to B(1) = 0, we have the following

Corollary 2.3 For j ≥ 1 and max(x, 0) < y, we have

ℙ(D_{p}(x, y; [0, 1]) ≥ j) = exp ( − 2(jy − (j − 1)x)^{2}).

Proof: Putting z = 0 in (2.5) we get

ℙ(D_{p}(x, y; [0, 1]) ≥ j) = \frac{φ(2jy − 2(j − 1)x)}{φ(0)} = exp ( − 2(jy − (j − 1)x)^{2}).

Taking x = 0, we recover Pitman and Yor [12]’s formula for the distribution of M_{j}(1):

ℙ(M_{j}(1) > y) = ℙ(D_{p}(0, y; [0, 1]) ≥ j) = exp ( − 2j^{2}y^{2}).  (2.6)

Another corollary can be obtained by taking x = 0 and integrating with respect to z:

Corollary 2.4 For j ≥ 1, y > 0 we have

ℙ(H_{j}(1) > y) = 2 (1 − Φ((2j − 1)y)) ,

where Φ is the standard normal distribution function, and H_{j}(1) denotes the height of the j-th highest positive Brownian excursion up to time 1.

Now we present an estimate on sup_{0≤t≤T} M_{j}^{∗}(t).

Proposition 2.5 Fix j ≥ 2. There exists some constant C_{1} > 1 such that for all u > 0 and λ ≥ \sqrt{u}, we have

ℙ( sup_{0≤t≤u} M_{j}^{∗}(t) > λ) ≤ C_{1} exp ( − 2 (\frac{jλ}{\sqrt{u}} − \frac{2j − 1}{2})^{2} )

In the proof of Proposition 2.5, we need the following lemma:

Lemma 2.6 For 0 ≤ x < y, j ≥ 1, we have

ℙ(D_{|p|}(x, y; [0, 1]) ≥ j) ≤ 2^{j}ℙ(D_{p}(x, y; [0, 1]) ≥ j).

Proof of Lemma 2.6: Again, we present the proof for j = 2. Upcrossings from x to y by |p| are either upcrossings by p from x to y or downcrossings by p from −x to −y. Define the following events:

A^{++} =^{def} {There are at least two upcrossings by p from x to y}
A^{+−} =^{def} {There is at least one downcrossing by p from −x to −y

after an upcrossing by p from x to y}

A^{−+} =^{def} {There is at least one upcrossing by p from x to y

after a downcrossing by p from −x to −y}

A^{−−} =^{def} {There are at least two downcrossings by p from −x to −y}.

Obviously

ℙ(D_{|p|}(x, y; [0, 1]) ≥ 2) ≤ ℙ (A^{++}) + ℙ (A^{+−}) + ℙ (A^{−+}) + ℙ (A^{−−})

and by symmetry, ℙ(A^{++}) = ℙ(A^{−−}), ℙ(A^{+−}) = ℙ(A^{−+}). Moreover, ℙ(A^{+−}) ≤ ℙ(A^{++}), since by Corollary 2.3 we have

ℙ (A^{++}) = exp (−2(2y − x)^{2})

and an argument, similar to the proof of Lemma 2.1 shows that

ℙ (A^{+−}) = exp (−8y^{2}) .

Hence,

ℙ(D_{|p|}(x, y; [0, 1]) ≥ 2) ≤ 4ℙ (A^{++}) = 2^{2} exp (−2(2y − x)^{2}) ,

proving Lemma 2.6 for j = 2. Extension of the above argument in an obvious manner for j > 2, proves our Lemma 2.6.

Now we proceed with the proof of Proposition 2.5. Proof of Proposition 2.5: For t > 0, we define σ_{0}^{(t)}(0) = 0 and for i ≥ 1,

τ_{i}^{(t)}(x) =^{def} inf{s ≥ σ_{i−1}^{(t)}(0) : |K(s, t)| = x},
σ_{i}^{(t)}(0) =^{def} inf{s ≥ τ_{i}^{(t)}(x) : K(s, t) = 0},

(write τ_{i}^{(t)}(x) = 1 if such s does not exist). Therefore,

ℙ( sup_{0≤t≤u} M_{j}^{∗}(t) > λ) = ℙ(∃t ∈ [0, u] : τ_{j}^{(t)}(λ) < 1) = ℙ(Θ ≤ u),

where we define Θ =^{def} inf{t ≥ 0 : M_{j}^{∗}(t) > λ}. Let F_{t} =^{def} σ{K(s, u), 0 ≤ s ≤ 1, 0 ≤ u ≤ t}, the sigma-algebra generated by {K(s, u), 0 ≤ s ≤ 1, 0 ≤ u ≤ t}. Then Θ is a stopping time with respect to (F_{t}). Notice that the process t → (K(·, Θ + t) − K(·, Θ)) is independent of F_{Θ} and has the same law as (K(·, t), t ≥ 0). Using the self similarity: K(·, v + Θ) − K(·, Θ) =^{law} \sqrt{v}K(·, 1) for any fixed v > 0, we get

ℙ( sup_{0≤s≤1} |K(s, u) − K(s, Θ)| < \frac{\sqrt{u}}{2} | Θ ≤ u) ≥ ℙ( sup_{0≤s≤1} |K(s, 1)| < \frac{1}{2}) =^{def} \frac{2^{j}}{C_{1}} > 0.

Denote by

E_{1} =^{def} \{ sup_{0≤s≤1} |K(s, u) − K(s, Θ)| < \frac{\sqrt{u}}{2}\} ∩ \{Θ ≤ u\},

we have shown that

ℙ(Θ ≤ u) ≤ 2^{−j}C_{1} ℙ(E_{1}).

On E_{1}, we can decompose K(s, u) = K(s, Θ) + K̂(s) with sup_{0≤s≤1} |K̂(s)| ≤ \frac{\sqrt{u}}{2}. Since |K(τ_{i}^{(Θ)}(λ), Θ)| = λ and K(σ_{i}^{(Θ)}(0), Θ) = 0 for 1 ≤ i ≤ j, it follows that for such random times 0 < s_{1} =^{def} τ_{1}^{(Θ)}(λ) < v_{1} =^{def} σ_{1}^{(Θ)}(0) < ... < s_{j} =^{def} τ_{j}^{(Θ)}(λ) < 1, we have respectively,

|K(s_{1}, u)| ≥ λ − \frac{\sqrt{u}}{2}, |K(v_{1}, u)| ≤ \frac{\sqrt{u}}{2}, ..., |K(s_{j}, u)| ≥ λ − \frac{\sqrt{u}}{2}.

Namely, we have

E_{1} ⊂ \{D_{|K(·,u)|} (\frac{\sqrt{u}}{2}, λ − \frac{\sqrt{u}}{2}; [0, 1]) ≥ j\}.

It follows from scaling, Corollary 2.3 and Lemma 2.6 that

ℙ(E_{1}) ≤ ℙ(D_{|K(·,u)|} (\frac{\sqrt{u}}{2}, λ − \frac{\sqrt{u}}{2}; [0, 1]) ≥ j)

= ℙ (D_{|p|} (\frac{1}{2}, \frac{λ}{\sqrt{u}} − \frac{1}{2}; [0, 1]) ≥ j) ≤ 2^{j} exp ( − 2\frac{(jλ − (2j − 1)\frac{\sqrt{u}}{2})^{2}}{u} ),

proving the result.

2.2 Ornstein-Uhlenbeck process

Let us consider a stationary Ornstein-Uhlenbeck process (U(t), t ≥ 0) with parameter \frac{1}{2},

which is a stationary centered Gaussian process with covariance 𝔼(U(t)U(s)) = e^{− \frac{|t−s|}{2}} . We mention a paper by Pitman and Yor [10] for the study of distributions of excursion lengths of U.

Recall some known facts on the hitting times of U. Fix −∞ ≤ z_{1} < z_{2} ≤ ∞ and define

σ(z_{1}, z_{2}) = inf{s ≥ 0 : U(s) ∉ [z_{1}, z_{2}]}

to be the first exit time from the interval [z_{1}, z_{2}]. Consider the Sturm-Liouville equation:

\frac{1}{2}ϕ′′(x) − \frac{x}{2}ϕ′(x) = −λϕ(x),  x ∈ (z_{1}, z_{2});  ϕ(z_{i}) = 0 if |z_{i}| < ∞, i = 1, 2.

Fact 2.7 ([14], [6], [9]) Assume that min(|z_{1}|, |z_{2}|) < ∞. There is a sequence of simple eigenvalues 0 < λ_{1}(z_{1}, z_{2}) < ... < λ_{n}(z_{1}, z_{2}) < ... whose corresponding eigenfunctions ψ_{1}(z_{1}, z_{2}; x), ..., ψ_{n}(z_{1}, z_{2}; x), ... form a complete orthonormal system with respect to m(dx) = e^{−x^{2}/2}dx. The function (z_{1}, z_{2}) → λ_{1}(z_{1}, z_{2}) is strictly positive and jointly continuous on Ξ = {(z_{1}, z_{2}) ∈ [−∞, ∞]^{2} : z_{1} < z_{2}, min(|z_{1}|, |z_{2}|) < ∞}, strictly increasing in z_{1} ∈ (−∞, z_{2}] for z_{2} ≤ ∞ and strictly decreasing in z_{2} ∈ [z_{1}, ∞) for z_{1} ≥ −∞:

λ_{1}(−∞, 0) = λ_{1}(0, ∞) = \frac{1}{2},  lim_{(z_{1},z_{2})→0} λ_{1}(z_{1}, z_{2}) = ∞,  lim_{(z_{1},z_{2})→(−∞,∞)} λ_{1}(z_{1}, z_{2}) = 0.

Fact 2.8 ([14], [6], [9], [1], [3]) Assume that min(|z_{1}|, |z_{2}|) < ∞. There exists some constant C_{2} > 0 such that uniformly on x ∈ ℝ,

ℙ(σ(z_{1}, z_{2}) > t | U(0) = x) = e^{−λ_{1}(z_{1},z_{2}) t}(θ(z_{1}, z_{2})ψ_{1}(z_{1}, z_{2}; x) + r(t, x)),

where θ(z_{1}, z_{2}) = ∫_{z_{1}}^{z_{2}} ψ_{1}(z_{1}, z_{2}; x)m(dx) and

|r(t, x)| ≤ C_{2} exp (\frac{x^{2}}{9} − \frac{t}{2}).

When z_{1} = −z_{2} = −z with z > 0, we get

lim_{t→∞} \frac{1}{t} log ℙ( sup_{0≤s≤t} |U(s)| < z) = −λ_{1}(−z, z).  (2.7)

Moreover, lim_{z→∞} λ_{1}(−z, z) = 0.

We shall need the probability that the process U downcrosses a given interval (z_{1}, z_{2}) only a few times during [−t, t]. This is stated in the following lemma:

Lemma 2.9 Fix −∞ < z_{1} < z_{2} < ∞ and k ≥ 1; We have

lim_{t→∞} \frac{1}{t} log ℙ(D_{U}(z_{1}, z_{2}; [−t, t]) ≤ k) = lim_{t→∞} \frac{1}{t} log ℙ(D_{U}(z_{1}, z_{2}; [0, 2t]) ≤ k) = −2μ(z_{1}, z_{2}),

where μ(z_{1}, z_{2}) =^{def} min(λ_{1}(−∞, z_{2}), λ_{1}(z_{1}, ∞)) > 0. Moreover, we have

lim_{z_{1},z_{2}→0} μ(z_{1}, z_{2}) = \frac{1}{2}.

Note that the constant k, arbitrary but fixed, does not influence the rate of exponential decay of the two probability terms in the above lemma.

Proof: The above first equality is due to the stationarity of the Ornstein-Uhlenbeck process. Using again the stopping times α_{j} and β_{j} defined in (2.1)–(2.3) associated with a = 0, b = 2t, x = z_{1}, y = z_{2}, I = [0, 2t] and f(v) = U(v), we have ℙ(D_{U}(z_{1}, z_{2}; [0, 2t]) ≤ k) = ℙ(α_{k+1} >

2t).

Remark that α_{1} = inf{s ≥ 0 : U(s) ≥ z_{2}} = σ(−∞, z_{2}). The strong Markov property implies that the random variables of the family {α_{1}, β_{j} − α_{j}, α_{j+1} − β_{j}, j ≥ 1} are mutually independent. Furthermore, β_{1} − α_{1} = σ(z_{1}, ∞) ∘ θ_{α_{1}} where θ is the usual shift operator. And for j ≥ 2, β_{j} − α_{j} (resp: α_{j} − β_{j−1}) has the same law as T_{z_{2}→z_{1}} (resp: T_{z_{1}→z_{2}}), where T_{x→y} denotes the hitting time of y by an Ornstein-Uhlenbeck process starting from x. Based on Fact 2.8, simple convolution computation yields that

lim_{t→∞} \frac{1}{t} log ℙ(α_{k+1} > t) = −μ(z_{1}, z_{2}),

and the desired conclusion follows.

2.3 A technical lemma

Recall that {p(s), 0 ≤ s ≤ 1} denotes a standard Brownian bridge. Let 0 ≤ y < z/4 and consider the event

G_{y,z} = {∃ 0 < a_{1} < c_{1} < b_{1} < a_{2} < c_{2} < b_{2} < 1 :
|p(a_{i})| ≤ y, |p(b_{i})| ≤ y, |p(c_{i})| ≥ z, i = 1, 2}  (2.8)

Remark that G_{y,z} ⊃ G_{0,z} and that G_{0,z} is in fact the event that the height of the second highest excursion of |p(·)| is larger than z. We shall need to bound ℙ(G_{y,z}) in the proof of the upper bound of Theorem 1.2.

Lemma 2.10 There exists an absolute constant C_{3} > 0 such that for all 0 ≤ y < \frac{z}{4} and 0 < z < \frac{1}{2},

ℙ(G_{y,z}) ≤ 1 − C_{3} z^{2}.

We note that this estimate is nearly sharp, since we can also obtain a lower bound from (2.6) as follows:

ℙ(G_{y,z}) ≥ ℙ(G_{0,z}) = ℙ(M_{2}^{∗}(1) ≥ z) ≥ ℙ(M_{2}(1) ≥ z) ≥ 1 − C_{4}z^{2}, 0 < z < \frac{1}{2}.

Proof of Lemma 2.10: Fix (y, z) such that 0 ≤ y < \frac{z}{4} and 0 < z < \frac{1}{2}. Define two stopping times for any continuous process X(·):

T_{z}^{∗}(X) =^{def} inf{t ≥ 0 : |X(t)| ≥ z}
Υ(X) =^{def} inf{t > T_{z}^{∗}(X) : |X(t)| ≤ y},

with inf ∅ = ∞. Observe that

G_{y,z}^{c} ⊃ \{T_{z}^{∗}(p) < z^{2}; 1 − 2z^{2} ≤ Υ(p) ≤ 1 − z^{2}; sup_{Υ(p)≤t≤1} |p(t)| ≤ \frac{z}{2}\}.

Applying the strong Markov property at Υ(p), we deduce from the symmetry that

ℙ(G_{y,z}^{c}) ≥ 𝔼(1_{(T_{z}^{∗}(p)<z^{2}; 1−2z^{2}≤Υ(p)≤1−z^{2})} f(y, 1 − Υ(p); z)),

because p(Υ(p)) = ±y on the event {Υ(p) < 1}, and where the function f is given by

f(y, s; z) =^{def} ℙ(The Brownian bridge from y to 0 of length s always lives in [−\frac{z}{2}, \frac{z}{2}] ).

It follows from the scaling property that for all z^{2} ≤ s ≤ 2z^{2},

f(y, s; z) = f (\frac{y}{\sqrt{s}}, 1; \frac{z}{\sqrt{s}}) ≥ inf_{0≤a≤ \frac{1}{4}} f (a, 1, \frac{1}{\sqrt{2}}) =^{def} C_{5} > 0,

because a = \frac{y}{\sqrt{s}} ≤ \frac{1}{4} and \frac{z}{\sqrt{s}} ≥ \frac{1}{\sqrt{2}}. Hence we have shown that

ℙ(G_{y,z}^{c}) ≥ C_{5} ℙ(T_{z}^{∗}(p) < z^{2}; 1 − 2z^{2} ≤ Υ(p) ≤ 1 − z^{2}).

Recall the following absolute continuity between the law of a standard Brownian bridge and that of a standard Brownian motion: Denote by ℙ_{0,0} the law of p(·) and by ℙ_{0} that of B(·), on the canonical space (C([0, 1] → ℝ), (X(t), 0 ≤ t ≤ 1), (X_{t})_{0≤t≤1}), we have

dℙ_{0,0} |_{X_{t}} = \frac{1}{\sqrt{1 − t}} exp ( − \frac{X^{2}(t)}{2(1 − t)}) dℙ_{0} |_{X_{t}}, t < 1.

Applying the above formula to the stopping time Υ(X), we have

ℙ(T_{z}^{∗}(p) < z^{2}; 1 − 2z^{2} ≤ Υ(p) ≤ 1 − z^{2})
= 𝔼_{0}(1_{(T_{z}^{∗}(X)<z^{2};1−2z^{2}≤Υ(X)≤1−z^{2})} \frac{1}{\sqrt{1 − Υ(X)}} exp ( − \frac{y^{2}}{2(1 − Υ(X))}))
≥ \frac{e^{−1/32}}{\sqrt{2}} \frac{1}{z} ℙ_{0}(T_{z}^{∗}(X) < z^{2}; 1 − 2z^{2} ≤ Υ(X) ≤ 1 − z^{2})
≥ \frac{e^{−1/32}}{\sqrt{2}} \frac{1}{z} ℙ_{0}(T_{z}^{∗}(X) < \frac{z^{2}}{2} ) ℙ_{z}(1 − 2z^{2} ≤ T_{y}(X) ≤ 1 − \frac{3z^{2}}{2} ),

where ℙ_{z} means that the Brownian motion X(·) starts from z and T_{y}(X) denotes the first hitting time at y of X. Thanks to the scaling property, the first probability in the above inequality ℙ_{0}(T_{z}^{∗}(X) < \frac{z^{2}}{2} ) is bounded below by some numerical constant. Using the well-

known distribution of the Brownian hitting time: ℙ_{z}(T_{y}(X) ∈ dt) = \frac{z−y}{\sqrt{2πt^{3}}} e^{−(z−y)^{2}/(2t)}dt, we obtain that

ℙ_{z}(1 − 2z^{2} ≤ T_{y}(X) ≤ 1 − \frac{3z^{2}}{2} ) ≥ C_{6} z^{3}.

Assembling these estimates, we get

ℙ(G_{y,z}^{c}) ≥ C_{3}z^{2},

for some universal constant C_{3} > 0, as desired.

3 Proof of Theorem 1.1

We begin with the proof of the upper bound:

lim sup_{t→∞} \frac{M_{j}^{∗}(t)}{\sqrt{t log log t}} ≤ \frac{1}{j\sqrt{2}},  a.s.  (3.1)

This follows from Proposition 2.5: Fix an arbitrary constant a > \frac{1}{j\sqrt{2}}. Let n ≥ 3 and t_{n} = e^{n/ log n}. We have from Proposition 2.5 that

ℙ( sup_{0≤t≤t_{n+1}} M_{j}^{∗}(t) > a\sqrt{t_{n} log log t_{n}}) ≤ C_{1} exp ( − (2j^{2}a^{2} + o(1)) log log t_{n}),

whose sum over n converges; this in view of a simple application of Borel-Cantelli lemma yields (3.1).

Now, fix an arbitrary constant a < \frac{1}{j\sqrt{2}}. It suffices to prove that

lim sup_{t→∞} \frac{M_{j}(t)}{\sqrt{t log log t}} ≥ a,  a.s.  (3.2)

To this end, let t_{n} = n^{n} and λ_{n} = a\sqrt{t_{n} log log t_{n}}, we consider the event

E_{n} =^{def} {M_{j}(t_{n}) > λ_{n}},

which is F_{t_{n}} =^{def} σ{K(s, u), 0 ≤ s ≤ 1, 0 ≤ u ≤ t_{n}}-measurable. If we can show that

∑_{n} ℙ(E_{n} | F_{t_{n−1}}) = ∞,  a.s.  (3.3)

then according to Lévy’s version of Borel-Cantelli lemma (cf. [13]), we get ℙ(E_{n}, i.o. ) = 1 hence (3.2).

Consider the process K̃(s, u) =^{def} K(s, u + t_{n−1}) − K(s, t_{n−1}) for 0 ≤ s ≤ 1 and u ≥ 0. The independent increment property says that K̃(·, ·) is independent of F_{t_{n−1}} and has the same law as K(·, ·). Fix a small ϵ > 0 such that 2j^{2}a^{2}(1 + 2ϵ) ≤ (1 − 2ϵ).

Recall the notation D_{K̃(·,t_{n}−t_{n−1})} in Section 2 for the downcrossings by the process K̃(·, t_{n}− t_{n−1}). Observe that

{D_{K̃(·,t_{n}−t_{n−1})}(−ϵλ_{n}, (1 + ϵ)λ_{n}; [0, 1]) ≥ j} ∩ {M̃_{1}^{∗}(t_{n−1}) < ϵλ_{n}} ⊂ E_{n},

where M̃_{1}^{∗}(t_{n−1}) =^{def} sup_{0≤s≤1,0≤u≤t_{n−1}} |K̃(s, u)|. Therefore, we apply Corollary 2.3 and obtain that for all large n,

ℙ(E_{n} | F_{t_{n−1}}) ≥ 1_{(M̃_{1}^{∗}(t_{n−1})<ϵλ_{n})} ℙ(D_{K̃(·,t_{n}−t_{n−1})}(−ϵλ_{n}, (1 + ϵ)λ_{n}; [0, 1]) ≥ j)
= 1_{(M̃_{1}^{∗}(t_{n−1})<ϵλ_{n})} ℙ(D_{p} (−ϵ\frac{λ_{n}}{\sqrt{t_{n} − t_{n−1}}} , (1 + ϵ)\frac{λ_{n}}{\sqrt{t_{n} − t_{n−1}}} ; [0, 1]) ≥ j)
≥ 1_{(M̃_{1}^{∗}(t_{n−1})<ϵλ_{n})} exp ( − 2j^{2}a^{2}(1 + 2ϵ) log log t_{n})
≥ 1_{(M̃_{1}^{∗}(t_{n−1})<ϵλ_{n})} n^{−(1−ϵ)},  (3.4)

where the above equality is due to the self-similarity: K̃(·, v) =^{law} \sqrt{v} p(·) for any fixed v > 0, and p(·) is a standard Brownian bridge. Now, applying (1.1), we obtain that almost surely, M̃_{1}^{∗}(t_{n−1}) < ϵλ_{n} for all large n. This together with (3.4) implies (3.3), completing the proof of Theorem 1.1.

4 Proof of Theorem 1.2

4.1 Upper bound

It suffices to show that

lim inf_{t→∞} \begin{matrix}√ & \overline{log t} \\ √ & \overline{t} \end{matrix} M_{2}^{∗}(t) = 0, a.s.

According to Lévy’s version of Borel-Cantelli’s lemma (cf. [13]), the above result follows if we can prove that for any constant ϵ > 0 and for some sequence (t_{n} ↑ ∞),

∑_{n} ℙ(M_{2}^{∗}(t_{n}) < ϵ\sqrt{\frac{t_{n}}{log t_{n}}} | F_{t_{n−1}}) = ∞,  a.s.  (4.1)

where F_{t} = σ{K(s, u), 0 ≤ s ≤ 1, 0 ≤ u ≤ t}. Let us consider t_{n} = n^{3n}. By means of (1.1), we have almost surely for all large n,

sup_{0≤s≤1} |K(s, t_{n−1})| ≤ \sqrt{t_{n−1} log n} =^{def} λ_{n}.  (4.2)

Consider large n. Observe that λ_{n} ≤ \frac{1}{4}ϵ\sqrt{\frac{t_{n}}{log t_{n}}} =^{def} \frac{x_{n}}{4} . By the independent increment property,

K(·, t_{n}) = K(·, t_{n−1}) + K̃(·, t_{n} − t_{n−1}),

with K̃ a Kiefer process independent of F_{t_{n−1}}. The key observation is that

{M_{2}^{∗}(t_{n}) ≥ x_{n}} ∩ { sup_{0≤s≤1} |K(s, t_{n−1})| ≤ λ_{n}}

⊂ {∃0 < a_{1} < c_{1} < b_{1} < a_{2} < c_{2} < b_{2} < 1 : |K̃(a_{i}, t_{n} − t_{n−1})| ≤ λ_{n},

|K̃(b_{i}, t_{n} − t_{n−1})| ≤ λ_{n}, |K̃(c_{i}, t_{n} − t_{n−1})| ≥ x_{n} − λ_{n}, i = 1, 2} =^{def} F̃_{n},

which implies that

F̃_{n}^{c} ∩ { sup_{0≤s≤1} |K(s, t_{n−1})| ≤ λ_{n}} ⊂ {M_{2}^{∗}(t_{n}) < x_{n}}.

It follows from the independence of F̃_{n}^{c} and F_{t_{n−1}} that

ℙ(M_{2}^{∗}(t_{n}) < x_{n} | F_{t_{n−1}}) ≥ 1_{(sup_{0≤s≤1} |K(s,t_{n−1})|≤λ_{n})} ℙ(F̃_{n}^{c})
= 1_{(sup_{0≤s≤1} |K(s,t_{n−1})|≤λ_{n})} ℙ(G_{y,z}^{c})
≥ C_{3}1_{(sup_{0≤s≤1} |K(s,t_{n−1})|≤λ_{n})} z^{2}
≥ C_{3} \frac{ϵ^{2}}{4} 1_{(sup_{0≤s≤1} |K(s,t_{n−1})|≤λ_{n})} \frac{1}{n log n},

where the above equality is due to scaling with y = \frac{λ_{n}}{\sqrt{t_{n}−t_{n−1}}} , z = \frac{x_{n}−λ_{n}}{\sqrt{t_{n}−t_{n−1}}} , G_{y,z}^{c} is the complement event of G_{y,z} which was defined in (2.8), and the second inequality follows from Lemma 2.10. The above lower bound together with (4.2) implies (4.1).

4.2 Lower bound

Fix j ≥ 2 and χ > \frac{1}{2}. We want to show that almost surely for all large t:

M_{j}(t) > \sqrt{t} (log t)^{−χ}.

Consider the two-parameter Ornstein-Uhlenbeck process (U(v, t), v ∈ ℝ, t ≥ 0) defined by

U( log (\frac{s}{1 − s}) , t) = \frac{K(s, t)}{\sqrt{s(1 − s)}}, 0 < s < 1, t ≥ 0.

Namely, {U(v, t), v ∈ ℝ, t ≥ 0} is a centered Gaussian process with covariance

𝔼(U(v_{1}, t_{1})U(v_{2}, t_{2})) = e^{− \frac{|v_{1}−v_{2}|}{2}} t_{1} ∧ t_{2}, v_{1}, v_{2} ∈ ℝ, t_{1}, t_{2} ≥ 0.

Let 0 < δ < 1 be small. First,if there exist some (random) times δ ≤ u_{1} < v_{1} < ... < u_{j−1} < v_{j−1} < u_{j} ≤ 1−δ such that U( log(\frac{u_{i}}{1−u_{i}}), t) ≥ x for i = 1, ..., j and U( log(\frac{v_{i}}{1−v_{i}}), t) =

0 for i = 1, ..., j − 1, then K(u_{i}, t) ≥ x\sqrt{δ(1 − δ)} and K(v_{i}, t) = 0. This implies in particular that M_{j}(t) ≥ x\sqrt{δ(1 − δ)}.

Recall (2.4). If we denote by D_{U(·,t)}(x, y; [− log(\frac{1−δ}{δ}), log(\frac{1−δ}{δ})]) the number of downcrossings of (x, y) by U(·, t) during the time interval [− log(\frac{1−δ}{δ}), log(\frac{1−δ}{δ})], then

\{D_{U(·,t)} (0, x; [− log (\frac{1 − δ}{δ} ) , log (\frac{1 − δ}{δ} )]) ≥ j\} ⊂ \{M_{j}(t) ≥ x\sqrt{δ(1 − δ)}\}.

Fix a small constant c = c(χ) > 0 whose value will be determined later. Define n_{k} = exp(\frac{k}{log k}) and let δ_{k} = (log n_{k})^{−2χ}, I_{k} = [− log(\frac{1−δ_{k}}{δ_{k}}), log(\frac{1−δ_{k}}{δ_{k}})], x_{k} = c\sqrt{n_{k+1}} for k ≥ 3. Consider the event

F_{k} =^{def} {∃t ∈ [n_{k}, n_{k+1}) : D_{U(·,t)}(0, x_{k}; I_{k}) ≤ j − 1}.

If we can show that

∑_{k} ℙ(F_{k}) < ∞,  (4.3)

then the Borel-Cantelli lemma implies that almost surely for all large k, F_{k}^{c} realizes; hence for all large t, we have that n_{k} ≤ t < n_{k+1}, and D_{U(·,t)}(0, x_{k}; I_{k}) ≥ j, which implies that M_{j}(t) ≥ x_{k}\sqrt{δ_{k}(1 − δ_{k})} ≥ \frac{c}{2} \sqrt{t} (log t)^{−χ}. This yields the convergence part of Theorem 1.2, since χ > \frac{1}{2} is arbitrary.

To estimate ℙ(F_{k}), we consider the following stopping time ζ with respect to F_{t}^{U} = σ{U(x, s), x ∈ ℝ, s ≤ t}:

ζ = inf{t ≥ n_{k} : D_{U(·,t)}(0, x_{k}; I_{k}) ≤ j − 1}.

We want to estimate ℙ(F_{k}) = ℙ(ζ < n_{k+1}). Define Ũ(v, t) =^{def} U(v, t + ζ) − U(v, ζ) for

v ∈ ℝ and t ≥ 0. The independent increments property says that Ũ is independent of F_{ζ}^{U} and has the same law as U. On {ζ < n_{k+1}}, we have D_{U(·,ζ)}(0, x_{k}; I_{k}) ≤ j − 1; Fix a small constant ϵ > 0. Consider the event

G_{k} =^{def} \{ sup_{δ_{k}≤s≤1−δ_{k}} |Ũ( log (\frac{1 − s}{s}) , n_{k+1} − ζ)| < ϵx_{k}; ζ < n_{k+1}\} ⊂ F_{k}.

Using the scaling property: Ũ(·, t) =^{law} \sqrt{t}Ũ(·, 1) for any fixed t > 0, we obtain:

ℙ(G_{k}) = ∫_{[n_{k},n_{k+1})} ℙ(ζ ∈ dv) ℙ( sup_{δ_{k}≤s≤1−δ_{k}} |Ũ( log (\frac{1 − s}{s}) , n_{k+1} − v)| < ϵx_{k})
= ∫_{[n_{k},n_{k+1})} ℙ(ζ ∈ dv) ℙ( sup_{δ_{k}≤s≤1−δ_{k}} |Ũ( log (\frac{1 − s}{s}) , 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1} − v}} )
≥ ℙ(ζ < n_{k+1}) ℙ( sup_{δ_{k}≤s≤1−δ_{k}} |Ũ( log (\frac{1 − s}{s}) , 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1} − n_{k}}} ). (4.4)

Observe that on G_{k}, the number of downcrossings of (−ϵx_{k}, (1 + ϵ)x_{k}) by U(·, n_{k+1}) during I_{k} = [− log(\frac{1−δ_{k}}{δ_{k}}), log(\frac{1−δ_{k}}{δ_{k}})] can not be larger or equal to j; otherwise, we would get D_{U(·,ζ)}(0, x_{k}; I_{k}) ≥ j. In view of this remark, we get

ℙ(F_{k}) ≤ \frac{ℙ(D_{U(·,n_{k+1})}(−ϵx_{k}, (1 + ϵ)x_{k}; I_{k}) ≤ j − 1)}{ℙ( sup_{δ_{k}≤s≤1−δ_{k}} |Ũ( log(\frac{1−s}{s}), 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1}−n_{k}}} )}

= \frac{ℙ(D_{U(·,1)}(−ϵc, (1 + ϵ)c; I_{k}) ≤ j − 1)}{ℙ( sup_{δ_{k}≤s≤1−δ_{k}} |U( log(\frac{1−s}{s}), 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1}−n_{k}}} )},  (4.5)

by using the scaling property. We shall bound below the denominator and bound above the numerator in (4.5): the denominator equals

ℙ( sup_{− log((1−δ_{k})/δ_{k})≤v≤log((1−δ_{k})/δ_{k})} |U(v, 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1} − n_{k}}} )

= ℙ( sup_{0≤v≤2 log((1−δ_{k})/δ_{k})} |U(v, 1)| < \frac{ϵx_{k}}{\sqrt{n_{k+1} − n_{k}}} )
≥ δ_{k}^{o(1)},  k → ∞,  (4.6)

where the above equality follows from the stationarity and the above inequality follows from (2.7) in Fact 2.8 with z = \frac{ϵx_{k}}{\sqrt{n_{k+1}−n_{k}}} → ∞. On the other hand, we have from Lemma 2.9 that

ℙ(D_{U(·,1)}(−ϵc, (1 + ϵ)c; I_{k}) ≤ j − 1) ≤ δ_{k}^{(2μ(−ϵc,(1+ϵ)c)+o(1))}

= (\frac{k}{log k} )^{−(4χ μ(−ϵc,(1+ϵ)c)+o(1))} .  (4.7)

Recall that χ > \frac{1}{2}. Since μ(−ϵc, (1 + ϵ)c) → \frac{1}{2} as c → 0, we can choose a sufficiently small constant c = c(χ) > 0 such that 4χ μ(−ϵc, (1 + ϵ)c) > 1. Putting (4.6) and (4.7) into (4.5), we obtain some constant a > 1 such that for all large k,

ℙ(F_{k}) ≤ k^{−a}

proving (4.3), as desired.

Acknowledgements

The authors are grateful to an anonymous referee for her/his suggestions. Cooperation between the authors was supported by the joint French–Hungarian Intergovernmental Grant ”Balaton” (grant no. F-39/00).

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