Asymptotic properties of ranked heights in Brownian excursions

by

Endre CS^{}ÁKI^{1} and Yueyun HU^{2}

Summary. Pitman and Yor [20,21] recently studied the distributions related to the ranked excursion heights of a Brownian bridge. In this paper, we study the asymptotic properties of the ranked heights of Brownian excursions. The heights of both high and low excursions are characterized by several integral tests and laws of the iterated logarithm. Our analysis relies on the distributions of the ranked excursion heights considered up to some random times.

KEY WORDS: Ranked heights, Brownian and Bessel excursions, integral test, law of the iterated logarithm.

1991 Mathematics Subject Classification. 60F15, 60G55.

Running title: Ranked heights in Brownian excursions

1 A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13–15, P.O.B. 127, Budapest, H–1364, Hungary. E-mail: csaki@math-inst.hu. Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 019346 and T 029621

2 Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR–7599, Université Paris VI, Tour 56, 3e étage, 4 Place Jussieu, F–75252 Paris cedex 05, France. E-mail: hu@proba.jussieu.fr

1. INTRODUCTION

Let {B(t), t ≥ 0} be a one-dimensional Brownian motion starting from 0 and consider the sequence

M_{1}(t) ≥ M_{2}(t) ≥ ... ≥ M_{n}(t) ≥ ...  (1.1)

the ranked heights of all the excursions of the reflected Brownian motion |B| considered up to time t (including the meander height sup_{g_{t}≤u≤t} |B(u)|, where g_{t} denotes the last zero of B before t), which gives a natural way to order the countable many Brownian excursions. In the literature, there is an another well-studied way to order excursions, namely by considering the ranked excursion lengths, see e.g. Csáki et al. [6] for random walk and Brownian excursion lengths, and some recent papers: Pitman and Yor [18,19] combined with their references for studies of laws and many further developments, Révész [23], Hu and Shi [14,15] and the references therein for the behaviors of longest and shortest excursion lengths. For the ranked excursion heights, the most recent references are Pitman and Yor [20,21] who have characterized the laws of ranked heights of excursions of a Brownian bridge, or more generally, of a bridge of a recurrent self-similar Markov process, considered up to some random times. Let us mention in particular Pitman and Yor [20] for the law of (M_{n}, n ≥ 1) taking at an independent exponential time (cf. Remark 2.2 below).

Here, we are interested in the almost sure asymptotic behaviors of {M_{n}(t), n ≥ 1} as t → ∞. Our analysis relies on a distribution result which will be stated in Section 2 for all recurrent Bessel processes. First of all, let us observe that M_{1}(t) = sup_{0≤s≤t} |B(s)|; hence the classical Erdős-Feller-Kolmogorov-Petrowski (EFKP) and Chung’s integral tests for the Brownian motion can be directly applied to M_{1}(t). Consequently, their respective laws of the iterated logarithm (LIL) read as follows:

lim sup_{t→∞} \frac{M_{1}(t)}{\sqrt{2t log log t}} = 1, a.s.  (1.2)
lim inf_{t→∞} \sqrt{\frac{log log t}{t}} M_{1}(t) = \frac{π}{\sqrt{8}}, a.s.  (1.3)

(see Csörgő and Révész [8], Révész [22] for detailed accounts). Let us consider M_{n}(t) for n ≥ 2. Obviously, when M_{n}(t) is very big, then M_{1}(t), ..., M_{n−1}(t) should also be very big. This simple fact prevents M_{n}(t) from reaching the same bound as M_{1}(t). On the other hand, for n ≥ 2, M_{n}(t) could reach much smaller values than M_{1}(t). The following result confirms this intuitive idea:

Theorem 1.1. Let f > 0 be a nondecreasing function. For fixed n ≥ 2, we have

ℙ(M_{n}(t) > \sqrt{t} f(t), i.o.) = \{ \begin{matrix}0 \\ 1 \end{matrix} ⟺ ∫ ^{∞} \frac{dtf(t)}{t} exp ( − \frac{(2n − 1)^{2}f^{2}(t)}{2}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} , (1.4)
ℙ(M_{n}(t) < \frac{\sqrt{t}}{f(t)}, i.o.) = \{ \begin{matrix}0 \\ 1 \end{matrix} ⟺ ∫ ^{∞} \frac{dt}{t f(t)} \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  (1.5)

where, here and in the sequel, “i.o.” means “infinitely often” as the relevant index goes to infinity. Consequently, we have

lim sup_{t→∞} \frac{M_{n}(t)}{\sqrt{2t log log t}} = \frac{1}{2n − 1},  a.s.  (1.6)
lim inf_{t→∞} \frac{(log t)^{a}}{\sqrt{t}} M_{n}(t) = \{ \begin{matrix}0 \\ ∞ \end{matrix} ⟺ \{ \begin{matrix}a ≤ 1 \\ a > 1 \end{matrix} ,  a.s.  (1.7)

In the case of n = 1, (1.4) is just the famous EFKP’s test. Now, we turn to the problem of low excursion heights, i.e. of the asymptotic behaviors of M_{[n(t)]}(t) as t → ∞, with n(t) depending on t and [n(t)] meaning the integer part of n(t). We have

Theorem 1.2. Let n(t) ↑ ∞ be a nondecreasing function such that n(t)/\sqrt{t log log t} is nonincreasing. Assume that lim_{t→∞} n(t)/ log log t = c with c ∈ [0, ∞]. We have

lim sup_{t→∞} \frac{n(t) M_{[n(t)]}(t)}{\sqrt{t log log t}} = r_{c}, a.s.  (1.8)

where r_{0} = 1/\sqrt{2}, r_{∞} = \sqrt{2}, and for 0 < c < ∞ the constant r_{c} is the unique r > 0 such that μ(c r^{−2}) = r^{−2}, where μ(x) = x(log cosh λ_{0} + λ_{0} − λ_{0}^{2}x/2), and λ_{0} is the unique positive solution λ = λ_{0}(x) of λx = 1 + tanh λ.

The rest of this paper is organized as follows: in Section 2 we give the distributions of the ranked heights of excursions of recurrent Bessel processes taken at some random times, which consist of the core of the proofs of Theorem 1.1 and 1.2. In Section 3, we prove Theorem 1.1, whereas Section 4 is devoted to the study of low excursions, and to the proof of Theorem 1.2.

Before closing this introduction, we would like to point out that Theorems 1.1 and 1.2 admit their natural generalizations to excursions of all the recurrent Bessel processes by using Corollary 2.1 and Lemma 2.2 below, and to excursions of a simple random walk on ℤ by using the Skorokhod embedding. Let us also mention the difference of the lower functions for M_{1}(t) and for M_{2}(t) (cf. (1.3) and (1.5)), one way to get better understanding of this difference is to consider the joint lower functions of (M_{1}(t), M_{2}(t)). Some further studies of this kind will be presented in [7]. We refer to [5] for the characterizations of the joint lower functions of (sup_{0≤s≤t} B(s), − inf_{0≤s≤t} B(s)) (and random walk case), where Chung-type and Hirsch-type tests are unified.

Throughout this paper, we use the notation f(x) ∼ g(x) as x → x_{0} ∈ [0, ∞] (resp: f(x) ≍ g(x) as x ∈ I ⊂ ℝ_{+}) meaning lim_{x→x_{0}} f(x)/g(x) = 1 (resp: 0 < C_{1} ≤ f(x)/g(x) ≤ C_{2} < ∞, for all x ∈ I), where, here and in the sequel, C_{1}, C_{2}, ..., C_{13} denote some universal positive constants.

Acknowledgements: We are very grateful to Professor Marc Yor for helpful discussions and for references, who also kindly pointed out another proof of (2.3) below by using a martingale argument. We are also indebted to the referee for useful remarks. The cooperation between the authors was supported by the joint French-Hungarian Intergovernmental Grant “Balaton” (grant no. F25/97).

2. DISTRIBUTION OF BESSEL EXCURSION HEIGHTS

In this Section, we consider a recurrent Bessel process {R(t), t ≥ 0} starting from 0, of dimension 0 < d < 2. See Revuz and Yor [24, Chap. XI] for detailed accounts on Bessel processes. We only mention that in the case of d = 1, R is in fact a reflected Brownian motion. Denote by

M_{1}^{(d)}(t) ≥ M_{2}^{(d)}(t) ≥ ... ≥ M_{n}^{(d)}(t) ≥ ...  (2.1)

the sequence of ranked excursion heights of R over [0, t] (including the meander height sup_{g_{R}(t)≤s≤t} R(s) with g_{R}(t) the last zero of R before t). Let

H_{n}^{(d)}(r) =^{def} inf{t > 0 : M_{n}^{(d)}(t) > r},  r > 0.  (2.2)

We aim at characterizing the law of the sequence {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≥ 1; H_{n}^{(d)}(1)}. Note that by definition M_{n}^{(d)}(H_{n}^{(d)}(1)) = 1, a.s. Write ν ≡ (2 − d)/2 ∈ (0, 1) for the sake of notational simplification.

Proposition 2.1. The two sequences {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≤ n} and {M_{j}^{(d)}(H_{n}^{(d)}(1)), j ≥ n + 1} are independent. Furthermore, we have (i) For n ≥ 2,

\{0 ≤ \frac{1}{M_{1}^{(d)}(H_{n}^{(d)}(1))} ≤ ... ≤ \frac{1}{M_{n−1}^{(d)}(H_{n}^{(d)}(1))} ≤ 1\} =^{law} \{0 ≤ ξ_{1} ≤ … ≤ ξ_{n−1} ≤ 1\} ,

(2.3)

where {ξ_{1}, …, ξ_{n−1}} is a rearranged nondecreasing sequence of n − 1 i.i.d. variables taking values in [0, 1] with common distribution 2νx^{2ν−1}1l_{(0≤x≤1)}dx;

(ii) The law of {M_{j}^{(d)}(H_{n}^{(d)}(1)), j ≥ n + 1} is characterized as follows: for every measurable function f ≥ 0, we have

𝔼 exp ( − ∑_{j≥n+1} f(M_{j}^{(d)}(H_{n}^{(d)}(1)))) = (1 + 2ν ∫_{0}^{1} \frac{dx}{x^{2ν+1}} (1 − e^{−f(x)}))^{−n}; (2.4)

(iii) The law of H_{n}^{(d)}(1) conditioned on {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≥ 1} is determined as follows: for every λ > 0, we have

𝔼(e^{− \frac{λ^{2}}{2} H_{n}^{(d)}(1)} | \{M_{i}^{(d)}(H_{n}^{(d)}(1)) = x_{i} > 0, i ≠ n\}) = ℓ_{ν}(λ) ∏_{i≠n} ℓ_{ν}^{2}(λ x_{i}), (2.5)

where here and in the sequel, ℓ_{μ}(x) =^{def} \frac{x^{μ}}{2^{μ}Γ(1+μ) I_{μ}(x)} for all μ > −1 and x > 0, with I_{μ} the modified Bessel function with index μ.

Remark 2.1. By BES_{r}^{μ} we mean a Bessel process of dimension 2(1 + μ), starting from r ≥ 0 (hence R =^{law} BES_{0}^{−ν}). Write T_{r→a}^{(μ)} its first hitting time at a ≥ 0 (if it is finite) by the BES_{r}^{μ}. According to Kent [16], we have

𝔼 exp ( − \frac{x^{2}}{2} T_{0→1}^{(μ)}) = ℓ_{μ}(x),  x > 0, μ > −1,  (2.6)
𝔼 exp ( − \frac{x^{2}}{2} T_{1→0}^{(μ)}) = \frac{2^{1+μ}}{Γ(−μ)} x^{−μ} K_{μ}(x),  x > 0, μ < 0,  (2.7)

where K_{μ} denotes the modified Bessel function with index μ (cf. [1] for the modified Bessel functions I_{μ} and K_{μ}). Consequently, (2.5) is in fact a decomposition of H_{n}^{(d)}(1) conditioning on {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≥ 1}, as the sum of independent hitting times related to BES_{0}^{ν}.

Proof of Proposition 2.1. The proof relies on Itô’s excursion theory for the recurrent Bessel process R, cf. also Bertoin [2]. Here, we begin by taking one choice of local times {L(t, x), t ≥ 0, x ≥ 0} of the Bessel process R, determined by the following density formula: for every measurable function f : ℝ_{+} → ℝ_{+}, we have

∫_{0}^{t} f(R(s)) ds = \frac{1}{ν} ∫_{ℝ_{+}} f(x)L(t, x)x^{1−2ν} dx,

(notice that in the case d = 1, ν = 1/2, L(t, 0) is equal to the local time at 0 of a Brownian motion, which is the half of that of a reflected Brownian motion).

Denote by (e(t), t ≥ 0) the excursion process of R associated with {L(t, x), t ≥ 0, x ≥ 0}, and n the Itô measure. Denote by h(ϵ) ≡ sup{ϵ(u), u ≥ 0}, the height for a generic excursion ϵ and V (ϵ) ≡ inf{t > 0 : ϵ(t) = 0} its lifetime. According to Biane and Yor [3, pp. 43–45, formula (3h)] together with our choice of local times, we have (n equals in fact their n̂_{ν}, notice that in formula (3h) the constant 4 should be 2)

n(h ≥ x) = x^{−2ν},  x > 0.  (2.8)

Let α_{n} =^{def} L(H_{n}^{(d)}(1), 0) be the local time at 0 up to time H_{n}^{(d)}(1). Observe that in the excursion time scale, α_{n} is the first time when there are exactly n excursions whose heights are larger than 1. Remark that {M_{1}^{(d)}(H_{n}^{(d)}(1)) ≥ ... ≥ M_{n−1}^{(d)}(H_{n}^{(d)}(1)) ≥ 1} is the rearranged nonincreasing sequence of {h(e(α_{1})), ..., h(e(α_{n−1}))}, and the latter n − 1 variables are measurable with respect to the excursion process {e(t) 1l_{(h(e(t))≥1)}, t ≥ 0}, which is independent of {e(t) 1l_{(h(e(t))<1)}, t ≥ 0}. Hence the two sequences {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≤ n−1} and {M_{j}^{(d)}(H_{n}^{(d)}(1)), j ≥ n+1} are independent, since the latter is measurable with respect to {e(t) 1l_{(h(e(t))<1)}, t ≥ 0} (here, we suppose that n ≥ 2, for there is nothing to do for showing the independence if n = 1).

Using successively the strong Markov property at the stopping times α_{n−1}, ..., α_{1} for the excursion process e, we see that {h(e(α_{1})), ..., h(e(α_{n−1}))} are n − 1 i.i.d. variables, with common law: For x > 1,

ℙ(h(e(α_{1})) < x) = ℙ( inf{t > 0 : h(e(t)) ∈ [1, x)} < inf{t > 0 : h(e(t)) ∈ [x, ∞)})
= ∫_{0}^{∞} ds(1 − x^{−2ν})e^{−s (1−x^{−2ν})} e^{−s x^{−2ν}}
= 1 − x^{−2ν},  x > 1,  (2.9)

since by using (2.8), the two first entrance time variables in the RHS of the first equality are independent, both exponentially distributed with respective parameters 1 − x^{−2ν} and x^{−2ν}. Hence (2.3) follows.

To get the law of {M_{i}^{(d)}(H_{n}^{(d)}(1)), i ≥ n + 1}, consider {(e(s), α_{j−1} < s < α_{j})}_{1≤j≤n} (with convention α_{0} ≡ 0). Again the strong Markov property says that these n excursion processes are independent and identically distributed, with common law that of the excursion process (ê(s), 0 ≤ s < α_{1}), where ê is defined as the restriction of e on A =^{def} {ϵ : h(ϵ) < 1}. Notice that ê has characteristic measure 1l_{A}n and is independent of α_{1}, whereas α_{1} is exponentially distributed with parameter n(A^{c}) = 1. It turns out from (2.8) that

𝔼 exp ( − ∑_{i≥n+1} f(M_{i}^{(d)}(H_{n}^{(d)}(1)))) = 𝔼 exp ( − ∑_{j=1}^{n} ∑_{α_{j−1}<s<α_{j}} f(h(e(s))))

= (𝔼 exp ( − ∑_{s<α_{1}} f(h(ê(s)))))^{n}

= ( ∫_{0}^{∞} dt e^{−t} 𝔼[ exp ( − ∑_{s<t} f(h(ê(s)))) ])^{n}

= ( ∫_{0}^{∞} dte^{−t} exp ( − t ∫_{A} n(dϵ)(1 − e^{−f(h(ϵ))}))^{n}

= (1 + 2ν ∫_{0}^{1} \frac{dx}{x^{2ν+1}} (1 − e^{−f(x)}))^{−n},  (2.10)

yielding (2.4). It remains to show (2.5). By using Williams’ [26] path–decomposition of R at g_{R}(t), with g_{R}(t) =^{def} sup{s ≤ t : R(s) = 0} being the last zero of R before t, we have that H_{n}^{(d)}(1) − g_{R}(H_{n}^{(d)}(1)) is independent of σ{R(s ∧ g_{R}(H_{n}^{(d)}(1))), s ≥ 0}, and H_{n}^{(d)}(1) − g_{R}(H_{n}^{(d)}(1)) =^{law} T_{0→1}^{(ν)} (in fact, Williams’ original path–decomposition deals with Brownian motion, but the corresponding version for recurrent Bessel processes follows from a time-change argument, cf. Biane and Yor [3, Lemma 3.1]). Now, remark that

g_{R}(H_{n}^{(d)}(1)) = ∑_{s<α_{n};h(e(s))>0} V (e(s)) = ∑_{s<α_{n};h(e(s))>0} (\frac{V (e(s))}{h(e(s))^{2}} ) h(e(s))^{2},
{h(e(s)) : h(e(s)) > 0, s < α_{n}} = {M_{i}^{(d)}(H_{n}^{(d)}(1)) : i ≠ n}.

According to Biane and Yor [3, pp.43], under the Itô measure n and conditioning on {h(e(s)) : h(e(s)) > 0, s < α_{n}}, the variables {V (e(s))/h(e(s))^{2} : h(e(s)) > 0, s < α_{n}} are i.i.d., with common law as the sum of two independent copies of T_{0→1}^{(ν)}. Hence, (2.5) follows from (2.6). ⊔ ⊓

The following formula shows a decomposition of the law of H_{n}^{(d)}(1) conditioning on {M_{i}^{(d)}(H_{n}^{(d)}(1)), 1 ≤ i ≤ n − 1}, as a sum of independent Bessel hitting times, and of i.i.d. variables whose Laplace transform corresponds to ℓ_{−ν}(λ)/ℓ_{ν}(λ). See also Pitman and Yor [20] for some closely related formulas in Brownian motion case.

Corollary 2.1. Recall ν =^{def} (2 − d)/2 ∈ (0, 1) and Remark 2.1. Let n ≥ 2. For λ > 0 and x_{1} ≥ ... ≥ x_{n−1} ≥ 1, we have

𝔼[e^{− \frac{λ^{2}}{2} H_{n}^{(d)}(1)} | (M_{i}^{(d)}(H_{n}^{(d)}(1)) = x_{i})_{1≤i≤n−1}] = ℓ_{ν}(λ) [\frac{ℓ_{−ν}(λ)}{ℓ_{ν}(λ)}]^{n} ∏_{i=1}^{n−1} ℓ_{ν}^{2}(λx_{i}), (2.11)
H_{n}^{(d)}(1) =^{law} ∑_{i=1}^{n} σ_{i}^{(d)} + ∑_{j=1}^{n−1} ξ_{j}^{(d)},  (2.12)

where all the variables {σ_{i}^{(d)}, 1 ≤ i ≤ n; ξ_{j}^{(d)}, 1 ≤ j ≤ n − 1} are independent, and

σ_{i}^{(d)} =^{law} T_{0→1}^{(−ν)} for 1 ≤ i ≤ n, ξ_{j}^{(d)} =^{law} T_{1→0}^{(−ν)} for 1 ≤ j ≤ n − 1.

Proof of Corollary 2.1. By applying the strong Markov property of R successively at the stopping times H_{1}^{(d)}(1), ..., H_{n−1}^{(d)}(1), (2.12) follows. We also point out that in view of (2.6)–(2.7), (2.12) is in agreement with a direct computation based on (2.11) and (2.3).

To show (2.11), using (2.5) and (2.4) to write the LHS of (2.11) as

ℓ_{ν}(λ) ∏_{i=1}^{n−1} ℓ_{ν}^{2}(λx_{i}) 𝔼[ ∏_{j≥n+1} ℓ_{ν}^{2}(λM_{j}^{(d)}(H_{n}^{(d)}(1)))]

= ℓ_{ν}(λ) ∏_{i=1}^{n−1} ℓ_{ν}^{2}(λx_{i}) [1 + 2ν ∫_{0}^{1} \frac{dx}{x^{1+2ν}} (1 − ℓ_{ν}^{2}(λx))]^{−n},

yielding (2.11) by using the fact that \frac{d}{dx}(\frac{x^{−2ν} ℓ_{ν}(x)}{ℓ_{−ν}(x)}) = −2νx^{−1−2ν}ℓ_{ν}^{2}(x), which can be obtained from the fact that (\frac{I_{−ν}}{I_{ν}})′(x) = −\frac{2 sin(νπ)}{π} \frac{1}{xI_{ν}^{2}(x)} (cf. Abramovitz and Stegun [1, pp. 375]). ⊔ ⊓

Remark 2.2. Taking d = 1 (i.e. ν = 1/2) in (2.12) (and in (2.6)–(2.7)), we recover Pitman and Yor [20]’s formula:

𝔼 exp ( − \frac{λ^{2}}{2} H_{n}^{(1)}(1)) = [ cosh λ]^{−n} e^{−λ(n−1)},  λ > 0.  (2.13)

Let us end this section with two preliminary results:

Lemma 2.1. Let Y_{1}, …, Y_{k} be independent random variables and assume that for i = 1, …, k, there exist some constants β_{i} > 0 and α_{i} ∈ ℝ such that

ℙ(Y_{i} < ϵ) ≍ ϵ^{α_{i}} exp ( − \frac{β_{i}}{ϵ} ),  ϵ → 0.  (2.14)

Then we have

ℙ(Y_{1} + … + Y_{k} < ϵ) ≍ ϵ^{α_{1}+...+α_{k}−(k−1)/2} exp ( − \frac{(\sqrt{β_{1}} + … \sqrt{β_{k}})^{2}}{ϵ} ),  ϵ → 0. (2.15)

Proof of Lemma 2.1. Follows from elementary computations by using integration by parts and Laplace method. ⊔ ⊓

Lemma 2.2. Recall 0 < ν = (2 − d)/2 < 1. We have (n being fixed)

ℙ(H_{n}^{(d)}(1) < ϵ) ≍ ϵ^{ν} exp ( − \frac{(2n − 1)^{2}}{2ϵ} ),  ϵ → 0, n ≥ 1,  (2.16)
ℙ(H_{n}^{(d)}(1) > x) ∼ \frac{n − 1}{2^{ν}Γ(1 + ν)} x^{−ν},  x → ∞, n ≥ 2.  (2.17)

Proof of Lemma 2.2. The proof is based on (2.12). First, from (2.7), the density of ξ_{1}^{(d)} =^{law} T_{1→0}^{(−ν)} can be obtained by inverting the Laplace transform; here we adopt an argument of Bessel time reversal (cf. [27]), which implies that ξ_{1} has the same law as the last exit time at 1 by a transient Bessel process BES_{0}^{ν} (of dimension 4 − d). Hence, it follows from Getoor [12] that

ℙ(ξ_{1}^{(d)} ∈ dt)/dt = \frac{1}{2^{ν}Γ(ν)} t^{−(1+ν)} exp ( − \frac{1}{2t}),  t > 0.  (2.18)

The tail of σ_{1}^{(d)} =^{law} T_{0→1}^{(−ν)} is given by Gruet and Shi [13], more generally, for all μ > −1, they have obtained that:

ℙ(T_{0→1}^{(μ)} < ϵ) ∼ \frac{2^{1−μ}}{Γ(1 + μ)} ϵ^{−μ} exp ( − \frac{1}{2ϵ}),  ϵ → 0,  μ > −1.  (2.19)

From (2.18), we have that

ℙ(ξ_{1}^{(d)} < ϵ) ∼ \frac{2^{1−ν}}{Γ(ν)} ϵ^{1−ν}e^{−1/(2ϵ)}, ϵ → 0,

which in view of (2.19) with μ = −ν, yields (2.16) by applying Lemma 2.1 to (2.12).

Now, applying (2.6) and (2.7) to (2.12) for the Laplace transform of H_{n}^{(d)}(1), we deduce that for n ≥ 2,

∫_{0}^{∞} dte^{−λt} ℙ(H_{n}^{(d)}(1) > t) = \frac{1}{λ} (1 − 𝔼e^{−λH_{n}^{(d)}(1)}) ∼ (n − 1) 2^{−ν} λ^{−(1+ν)}, λ → 0,

which, according to a Tauberian theorem (cf. Feller [11, pp. 423]) yields (2.17). ⊔ ⊓

3. PROOF OF THEOREM 1.1

From now on, let us go back to the reflected Brownian motion |B|. Recall (1.1) and define for n ≥ 1,

H_{n}(r) =^{def} inf{t > 0 : M_{n}(t) > r},  r ≥ 0.  (3.1)

Let us fix n ≥ 2 in this section. Taking ν = 1/2 in Lemma 2.2 and using self-similarity yields that for a fixed t > 0, we have as λ → ∞

ℙ(M_{n}(t) ≥ λ\sqrt{t}) = ℙ(H_{n}(1) < λ^{−2}) ≍ λ^{−1} exp ( − \frac{(2n − 1)^{2}λ^{2}}{2} ),  (3.2)
ℙ(M_{n}(t) ≤ \frac{\sqrt{t}}{λ}) = ℙ(H_{n}(1) > λ^{2}) ∼ (n − 1)\sqrt{\frac{2}{π}} λ^{−1}.  (3.3)

From (3.2), (3.3) and the monotonicity of M_{n}(t), it is a routine to prove the convergent parts of the tests in (1.4) and (1.5), see e.g. Erdős [10]. We omit here the details and only prove the divergent parts.

Proof of Theorem 1.1: the divergent part of (1.4). Let f be a nondecreasing function such that the integral in (1.4) diverges. Without any loss of generality, we can limit our attention to the “critical” case:

\frac{1}{2n − 1}\sqrt{log log t} ≤ f(t) ≤ \frac{2}{2n − 1}\sqrt{log log t},  t ≥ t_{0},  (3.4)

for some constant t_{0} > 0 (see e.g. Erdős [10] for a rigorous justification). Let i ≥ i_{0}, where here and in the sequel, i_{0} denotes some fixed but sufficiently large constant whose value may change from one line to the next. Define

r_{i} =^{def} exp(i/ log i),  t_{i} = r_{i}^{2}/f̂^{2}(r_{i}),  f̂(t) =^{def} f(t^{3}),  t ≥ t_{0}.

Notice that H_{n}(r_{i}) < t_{i} implies that M_{n}(t_{i}) ≥ r_{i} = \sqrt{t_{i}}f̂(r_{i}) = \sqrt{t_{i}}f(r_{i}^{3}) > \sqrt{t_{i}}f(t_{i}) by (3.4). Therefore, if we would have proven that

ℙ(H_{n}(r_{i}) < t_{i}, i.o.) > 0,  (3.5)

we would obtain that ℙ(M_{n}(t) > \sqrt{t}f(t), i.o.) > 0, which, from Kolmogorov’s 0–1 law, equals in fact 1 and ends the proof.

It remains to show (3.5). The idea consists of working with “favorite” events which are given as follows:

E_{i} =^{def} {H_{n}(r_{i}) < t_{i}} ∩ {M_{1}(H_{n}(r_{i})) < r_{i+1}},  i ≥ i_{0}.  (3.6)

It suffices to show ℙ(E_{i}, i.o.) > 0. First, let us estimate ℙ(E_{i}). Taking d = 1 in (2.11) gives the following first equality (x_{1} ≥ ... ≥ x_{n−1} > 1)

𝔼(e^{− \frac{λ^{2}}{2} H_{n}(1)}|\{M_{i}(H_{n}(1)) = x_{i} > 1, 1 ≤ i ≤ n − 1\})

= \frac{λ}{sinh λ} (\frac{tanh λ}{λ})^{n} ∏_{i=1}^{n−1} (\frac{λx_{i}}{sinh(λx_{i})})^{2}

= 𝔼 exp ( − \frac{λ^{2}}{2} (η_{0} + ζ^{(n)} + ∑_{i=1}^{n−1} x_{i}(η_{i} + η̃_{i}))),  (3.7)

where all the variables {η_{0}, ζ^{(n)}, η_{i}, η̃_{i}, 1 ≤ i ≤ n−1} in (3.7) are positive and independent, with η_{0} =^{law} η_{i} =^{law} η̃_{i} and

𝔼 exp ( − \frac{λ^{2}}{2} η_{0}) = \frac{λ}{sinh λ},  𝔼 exp ( − \frac{λ^{2}}{2} ζ^{(n)}) = (\frac{tanh λ}{λ})^{n},  λ > 0.

Here we used the well-known fact that both λ/(sinh λ) and (tanh λ)/λ are Laplace transforms, at λ^{2}/2, of some positive random variables. From (3.7), we see that conditioning on {M_{i}(H_{n}(1)) = x_{i} > 1, 1 ≤ i ≤ n − 1}, H_{n}(1) is stochastically smaller than η_{0} + ζ^{(n)} + x_{1} ∑_{i=1}^{n−1}(η_{i} + η̃_{i}). It turns out that for x > 1

ℙ(H_{n}(r_{i}) < t_{i} | M_{1}(H_{n}(r_{i})) = xr_{i}) = ℙ(H_{n}(1) < \frac{1}{f̂^{2}(r_{i})} | M_{1}(H_{n}(1)) = x)

≥ ℙ(η_{0} + ζ^{(n)} + x ∑_{i=1}^{n−1}(η_{i} + η̃_{i}) < \frac{1}{f̂^{2}(r_{i})})

≥ ℙ(η_{0} + ζ^{(n)} + ∑_{i=1}^{n−1}(η_{i} + η̃_{i}) < \frac{1}{xf̂^{2}(r_{i})}),

= ℙ(Σ^{(n)} + ∑_{i=1}^{n−1} η_{i} < \frac{1}{xf̂^{2}(r_{i})}),  (3.8)

where Σ^{(n)} =^{def} η_{0} + ζ^{(n)} + ∑_{i=1}^{n−1} η̃_{i} is independent of (η_{i}, 1 ≤ i ≤ n − 1). We shall estimate the small deviation in (3.8). Observe that 𝔼e^{− \frac{λ^{2}}{2} Σ^{(n)}} = cosh^{−n}(λ), it turns out that Σ^{(n)} =^{law} ∑_{i=1}^{n} σ_{i}, with σ_{i} being i.i.d. and σ_{i} =^{law} T_{0→1}^{(−1/2)} (recalling the notations in Remark 2.1), whereas η_{i} =^{law} T_{0→1}^{(1/2)} has the same law of the first hitting time at 1 by a threedimensional Bessel process starting from 0. By using (2.19) with μ = −1/2 for the small deviation of σ_{i} and with μ = 1/2 for that of η_{i}, we have

ℙ(σ_{i} < ϵ) ≍ ϵ^{1/2} exp ( − \frac{1}{2ϵ}) and ℙ(η_{i} < ϵ) ≍ ϵ^{−1/2} exp ( − \frac{1}{2ϵ}), ϵ → 0,

which in view of Lemma 2.1 yields that

ℙ(Σ^{(n)} + ∑_{i=1}^{n−1} η_{i} < ϵ) ≍ ϵ^{−n+3/2} exp ( − \frac{(2n − 1)^{2}}{2ϵ} ),  ϵ → 0.  (3.9)

Notice that f̂(r_{i}) ≍ \sqrt{log i}, and \frac{r_{i+1}}{r_{i}} − 1 ∼ \frac{1}{log i} ≥ C_{3}f̂^{−2}(r_{i}) for some universal constant C_{3} > 0. The law of M_{1}(H_{n}(1)) follows from (2.3) with ν = 1/2:

ℙ(M_{1}(H_{n}(r_{i})) < r_{i}x) = (1 − \frac{1}{x})^{n−1},  x ≥ 1.  (3.10)

Combining (3.8)–(3.10) with 1 < x < 1 + C_{3}f̂^{−2}(r_{i}), we obtain that

ℙ(E_{i}) ≥ ℙ(σ^{(n)} + ∑_{i=1}^{n−1} η_{i} < (1 + C_{3}f̂^{−2}(r_{i}))^{−1} f̂^{−2}(r_{i})) ℙ(M_{1}(H_{n}(r_{i})) < r_{i}(1 + C_{3}f̂^{−2}(r_{i})))

≥ C_{4}(f̂(r_{i}))^{2n−3} exp ( − \frac{(2n − 1)^{2}f̂^{2}(r_{i})}{2}) (f̂^{−2}(r_{i}))^{n−1}

= C_{4}f̂^{−1}(r_{i}) exp ( − \frac{(2n − 1)^{2}f̂^{2}(r_{i})}{2}),

for some constant C_{4} > 0. In light of (3.2), we have shown

ℙ(E_{i}) ≍ ℙ(H_{n}(r_{i}) < t_{i}) ≍ f̂^{−1}(r_{i}) exp ( − \frac{(2n − 1)^{2}f̂^{2}(r_{i})}{2}).  (3.11)

From the assumption of divergence of the integral in (1.4), it is elementary to see that

∑_{i} ℙ( E_{i} ) = ∞.  (3.12)

Let us estimate the second moment ℙ(E_{i} ∩ E_{j}). Define

D_{i} =^{def} inf{t > H_{n}(r_{i}) : B(t) = 0},

the first return time to 0 of B (i.e. of |B|) after the stopping time H_{n}(r_{i}). Let B̂(t) =^{def} B(t+ D_{i}), t ≥ 0; By the strong Markov property at the stopping time D_{i}, B̂ is a standard Brownian motion starting from 0, and independent of F_{D_{i}}, where (F_{t}, t ≥ 0) denotes the natural σ-fields generated by B. Define the processes M̂_{n}, Ĥ_{n} related to B̂ exactly in the same way as M_{n}, H_{n} do to B. Consider j > i > i_{0} and ω ∈ E_{i} ∩ E_{j}, if additionally M_{1}(D_{i})(ω) < r_{j}, all excursions having heights larger than r_{j} live after the time D_{i}(ω), which implies that Ĥ_{n}(r_{j})(ω) < t_{j}. Therefore using (3.11) gives that

ℙ(E_{i} ∩ E_{j} ∩ {M_{1}(D_{i}) < r_{j}}) ≤ ℙ(E_{i} ∩ {Ĥ_{n}(r_{j}) < t_{j}})
= ℙ(E_{i})ℙ(Ĥ_{n}(r_{j}) < t_{j})
≤ C_{5}ℙ(E_{i})ℙ(E_{j}).  (3.13)

Observe that E_{i} ∩ E_{j} ∩ {M_{1}(D_{i}) ≥ r_{j}} ⊂ E_{i} ∩ {r_{j+1} > M_{1}(D_{i}) ≥ r_{j}} ∩ {Ĥ_{n−1}(r_{j}) < t_{j}}. Furthermore M_{1}(D_{i}) is independent of E_{i}, since M_{1}(D_{i}) in fact is the maximum of |B(H_{n}(r_{i})+·)| considered up to its first hitting time at 0, hence ℙ(M_{1}(D_{i}) ∈ [r_{j}, r_{j+1})) = r_{i}/r_{j} − r_{i}/r_{j+1}. It follows that

ℙ(E_{i} ∩ E_{j} ∩ {M_{1}(D_{i}) ≥ r_{j}}) ≤ ℙ(E_{i} ∩ {r_{j} ≤ M_{1}(D_{i}) < r_{j+1}} ∩ {Ĥ_{n−1}(r_{j}) < t_{j}})

= ℙ(E_{i}) ℙ(r_{j} ≤ M_{1}(D_{i}) < r_{j+1}) ℙ(Ĥ_{n−1}(r_{j}) < t_{j})

≤ C_{6}ℙ(E_{i}) \frac{r_{i}}{r_{j}} f̂(r_{j}) exp ( − \frac{(2n − 3)^{2}f̂^{2}(r_{j})}{2})

≤ C_{7}ℙ(E_{i}) exp ( − \frac{j − i}{log j}) j^{−C_{8}},  (3.14)

where we have used the facts that r_{i}/r_{j} ≤ exp ( − (j − i)/ log j) and f̂^{2}(r_{j}) ≍ log j, j ≥ i_{0}. From (3.11) and (3.13)–(3.14), some elementary calculations show that

lim inf_{n→∞} \frac{∑_{i_{0}≤i,j≤n} ℙ(E_{i} ∩ E_{j})}{( ∑_{i_{0}≤i≤n} ℙ(E_{i}))^{2}} ≤ C_{5},

which according to Kochen and Stone’s version [17] of the Borel-Cantelli lemma, together with (3.12) yields that

ℙ(E_{i}, i.o.) ≥ 1/C_{5} > 0,

resulting (3.5) and ending the proof of the divergent part of (1.4). ⊔ ⊓

Proof of Theorem 1.1: the divergent part of (1.5). Since H_{n}(r) ≥ H_{2}(r), it suffices to prove this part for n = 2. Assume that the integral in (1.5) diverges and we work again with the “critical” case when

\sqrt{log t} ≤ f(t) ≤ log^{2} t,  t ≥ t_{0}.  (3.15)

Let

f̃(t) =^{def} f(t^{3}),  r_{i} =^{def} 2^{i},  t_{i} =^{def} r_{i}^{2}f̃^{2}(r_{i}).

Our aim is to prove that

ℙ(H_{2}(r_{i}) > t_{i}, i.o.) > 0,  (3.16)

which would imply that for these i.o. i such that H_{2}(r_{i}) > t_{i}, we have M_{2}(t_{i}) < r_{i} = \sqrt{t_{i}}/f̃(r_{i}) ≤ \sqrt{t_{i}}/f(t_{i}), so that ℙ(M_{2}(t) < \sqrt{t} f(t), i.o.) > 0, and the desired result follows from Kolmogorov’s 0–1 law.

Now, let d_{H_{2}(r)} =^{def} inf{t > H_{2}(r) : B(t) = 0} be the first return time to 0 after H_{2}(r). Define m(r) =^{def} sup{|B(s)| : H_{2}(r) ≤ s ≤ d_{H_{2}(r)}} the maximal height of the excursion straddling H_{2}(r), m(r) is independent of H_{2}(r_{i}) by strong Markov property at H_{2}(r_{i}). Consider the event

F_{i} =^{def} {t_{i+1} > H_{2}(r_{i}) > t_{i}} ∩ {m(r_{i}) < r_{i+1}} ∩ {d_{H_{2}(r_{i})} < t_{i+2}}.  (3.17)

It follows from (3.3) and the self-similarity that

ℙ(F_{i}) = ℙ(2f̃^{2}(r_{i}) > H_{2}(1) > f̃^{2}(r_{i})) ℙ({m(1) < 2} ∩ {d_{H_{2}(1)} < 4f̃^{2}(r_{i})})
≍ f̃^{−1}(r_{i}),  (3.18)

which, in light of the fact that ∫^{∞} dt/(tf̃(t)) = ∞, implies

∑_{i} ℙ(F_{i}) = ∞.  (3.19)

The second moment ℙ(F_{i} ∩F_{j}) is easy to estimate. Let j ≥ i+3, and define B̃(t) =^{def} B(t+ d_{H_{2}(r_{j})}), t ≥ 0. B is a Brownian motion independent of F_{i}. Define H̃_{1}(r), H̃_{2}(r) related to B̃ in the same way as H_{1}(r), H_{2}(r) do to B. Consider a path ω ∈ F_{i} ∩ F_{j}, we have H_{2}(r_{j})(ω) − d_{H_{2}(r_{i})}(ω) > t_{j} − t_{i+2} ≥ t_{j}/2. There are only two possibilities. First consider the case (M_{1}(H_{2}(r_{i}))(ω) < r_{j}), meaning that the two highest excursions before H_{2}(r_{j})(ω) are realized after the stopping time d_{H_{2}(r_{i})}(ω), hence we have H_{2}(r_{j})(ω) = H̃_{2}(r_{j})(ω) + d_{H_{2}(r_{i})}(ω). Assembling all this and using (3.3) and (3.18) yield

ℙ(F_{i} ∩ F_{j} ∩ {M_{1}(H_{2}(r_{i})) < r_{j}}) ≤ ℙ(F_{i} ∩ {H̃_{2}(r_{j}) > t_{j}/2}) ≤ C_{9}ℙ(F_{i})ℙ(F_{j}). (3.20)

It remains to consider the case (M_{1}(H_{2}(r_{i}))(ω) ≥ r_{j}), which means that there exists exactly one excursion living in the time interval [d_{H_{2}(r_{i})}, H_{2}(r_{j})](ω), whose height is larger than r_{j}; hence H_{2}(r_{j})(ω) = H̃_{1}(r_{j})(ω) + d_{H_{2}(r_{i})}(ω). It turns out that

ℙ(F_{i} ∩ F_{j} ∩ {M_{1}(H_{2}(r_{i})) > r_{j}}) ≤ ℙ(F_{i} ∩ {H̃_{1}(r_{j}) > t_{j}/2})

= ℙ(F_{i}) ℙ(H̃_{1}(1) > \frac{f̃^{2}(r_{j})}{2})

≤ ℙ(F_{i}) \frac{2}{f̃^{2}(r_{j})},  (3.21)

where the last inequality is due to Chebychev’s inequality and to the fact that 𝔼H̃_{1}(1) = 1, since H̃_{1}(1) is the first hitting time at 1 by |B| (in fact H̃_{1}(1) has a tail of exponential decay (cf. [4] for all Bessel hitting times), but here the rough estimate is sufficient for us). From (3.18), (3.20) and (3.21), it is elementary to show that

lim inf_{n→∞} \frac{∑_{i_{0}≤i,j≤n} ℙ(F_{i} ∩ F_{j})}{( ∑_{i_{0}≤i≤n} ℙ(F_{i}))^{2}} ≤ C_{9},

which in view of (3.19) and a version of the Borel-Cantelli lemma (cf. [17]), yield that ℙ(F_{i}, i.o.) ≥ 1/C_{9} > 0; hence we have proven (3.16) and finished the whole proof. ⊔ ⊓

4. LOW EXCURSIONS

In this Section, we shall study the asymptotic behavior of the height M_{[n(t)]}(t) with n(t) ↑ ∞ being a nondecreasing function. Our first preliminary result concerns the tail asymptotics:

Lemma 4.1. Fix r > 0 and assume that lim_{t→∞} n(t)/(log log t) = c with c ∈ [0, ∞]. We have

log ℙ(n(t)M_{[n(t)]}(1) > r\sqrt{log log t}) ∼ −r^{2}μ (\frac{c}{r^{2}} ) log log t,  t → ∞,  (4.1)

where μ(0) = 2, μ(∞) = 1/2 and for 0 < x < ∞ μ(x) is defined in Theorem 1.2.

Proof of Lemma 4.1. Define T (r) =^{def} inf{t > 0 : B(t) = r} for r ∈ ℝ. Recall (3.1). It follows from (2.12) that

H_{n}(1) =^{law} Σ^{(n)} + T (n − 1),  (4.2)

where Σ^{(n)} is independent of T (n − 1), and 𝔼 exp ( − \frac{λ^{2}}{2} Σ^{(n)}) = cosh^{−n}(λ). It follows from the well-known Gaussian tail that

ℙ(T (r) < ϵr^{2}) = ℙ( sup_{0≤s≤1} B(s) > \frac{1}{\sqrt{ϵ}}) ≍ \sqrt{ϵ} exp (−\frac{1}{2ϵ}) ,  ϵ → 0.  (4.3)

Let now x = x(t) = n(t)/(r^{2} log log t) → c/r^{2} ∈ [0, ∞], as t → ∞. Then it is easy to see that (4.1) is equivalent to

log ℙ (H_{n}(1) < x n) ∼ −n\frac{μ(x)}{x},  n → ∞,  (4.4)

with three possible cases in (4.4), namely: x ∈ (0, ∞) being fixed; x → 0; and x → ∞, as n → ∞.

For fixed 0 < x < ∞, since from (4.2), H_{n}(1) is stochastically greater than Σ^{(n−1)} + T (n − 1), and smaller than Σ^{(n)} + T (n), (4.4) follows from Cramér’s theorem (cf. Dembo and Zeitouni [9, Theorem 2.2.3]). In the case when x → 0 or x → ∞ as n → ∞, first estimate ℙ(H_{n}(1) < x n) with x > 0 depending on n. The Laplace transform of H_{n}(1) is given by (2.13). Using Chebychev’s inequality

ℙ(H_{n}(1) < xn) ≤ e^{\frac{λ^{2}}{2} xn} 𝔼e^{− \frac{λ^{2}}{2} H_{n}(1)} = e^{\frac{λ^{2}}{2} xn} ( cosh λ)^{−n} e^{−λ(n−1)},  (4.5)

for all λ > 0. By putting λ = \frac{2n−1}{xn} in (4.5) we get

ℙ (H_{n}(1) < xn) ≤ 2^{n} exp (−\frac{(2n − 1)^{2}}{2xn} ) ,  (4.6)

giving an upper bound in (4.4) when x → 0.

On the other hand, by taking λ = (n − 1)/(xn) in (4.5) we get

ℙ (H_{n}(1) < xn) ≤ exp (−\frac{(n − 1)^{2}}{2xn} )  (4.7)

showing an upper bound in the case x → ∞.

To obtain a lower bound in the case when x → 0 notice that Σ^{(n)} is stochastically dominated by T (n), which implies in view of (4.2) and (4.3) that

ℙ(H_{n}(1) < xn) ≥ ℙ(T̂(n) + T (n − 1) < xn)
= ℙ(T (2n − 1) < xn)
≥ (C_{10} ∧ \sqrt{\frac{x}{n}}) e^{− \frac{(2n−1)^{2}}{2xn}} ,  (4.8)

where C_{10} > 0 is some universal constant, and T̂(r) is an independent copy of T (r). (4.6) and (4.8) together imply (4.4) in the case x → 0.

Since Σ^{(n)} can be written as the sum of n i.i.d. variables, of common law of Σ^{(1)}, with expectation 1, using the law of large numbers gives that for x ≫ 1,

ℙ(H_{n}(1) < xn) ≥ ℙ(Σ^{(n)} < 2n) ℙ(T (n − 1) < (x − 2)n)

≥ C_{11} ℙ(T (n − 1) < (x − 2)n)

≥ C_{12}\sqrt{\frac{x}{n}} exp ( − \frac{n}{2(x − 2)}),

which, together with (4.7), implies (4.4) in the case x → ∞. ⊔ ⊓

Proof of Theorem 1.2: First we show that x → μ(x) is decreasing for 0 < x < ∞. It is elementary to see that

μ′(x) = −λ_{0}^{2}x + log cosh λ_{0} + λ_{0} = −λ_{0} tanh λ_{0} + log cosh λ_{0} < 0,  (4.9)

for log cosh λ < λ tanh λ for all λ > 0, hence μ is decreasing and r_{c} is well defined in Theorem 1.2.

Now we prove the upper bound in (1.8) for fixed 0 < c < ∞. Fix a small ϵ > 0. Define t_{i} = (1 + ϵ)^{i} for i ≥ 1, then log log t_{i} ∼ log i. From (4.1) and using the monotonicity of μ, we have for all large i,

ℙ(n(t_{i}) M_{[n(t_{i})]}(t_{i+1}) > (ϵ + 1)r_{c} \sqrt{t_{i} log log t_{i}})

= ℙ(n(t_{i})M_{[n(t_{i})]}(1) > r_{c}\sqrt{ϵ + 1} \sqrt{log log t_{i}})

≤ exp (−(1 + \frac{ϵ}{2}) r_{c}^{2} μ(\frac{c}{(1 + ϵ)r_{c}^{2}} ) log log t_{i})

≤ exp (−(1 + \frac{ϵ}{2}) log log t_{i})

≤ i^{−1− \frac{ϵ}{3}} ,

being summable, which according to the convergent part of Borel-Cantelli lemma, implies that almost surely for all large i, we have M_{[n(t_{i})]}(t_{i+1}) ≤ (ϵ + 1)r_{c} \sqrt{t_{i} log log t_{i}} /n(t_{i}); hence for all large t, t ∈ [t_{i}, t_{i+1}), and by monotonicity M_{[n(t)]}(t) ≤ M_{[n(t_{i})]}(t_{i+1}) ≤ (ϵ + 1)r_{c} \sqrt{t_{i} log log t_{i}} /n(t_{i}) ≤ r_{c}(ϵ + 1) \sqrt{t log log t} /n(t), i.e.

lim sup_{t→∞} \frac{n(t)}{\sqrt{t log log t}} M_{[n(t)]}(t) ≤ (1 + ϵ)r_{c}, a.s.,

yielding the upper bound of (1.8) by letting ϵ → 0 along a countable sequence.

To obtain the lower bound of (1.8), fix a small δ > 0 and let t_{i} =^{def} i^{i}. Consider the event

G_{i} =^{def} \{n(t_{i})M_{[n(t_{i})]}(t_{i}) > (1 − δ)r_{c} \sqrt{t_{i} log log t_{i}}\}.  (4.10)

Write (F_{t}, t ≥ 0) for the natural σ-fields generated by B. Obviously, G_{i} is F_{t_{i}} measurable. If we would have proven that

∑_{i} ℙ(G_{i} | F_{t_{i−1}}) = ∞,  a.s.,  (4.11)

then, according to Lévy’s version of Borel-Cantelli’s lemma (cf. [25]), we would obtain that ℙ(G_{i}, i.o.) = 1, which shows lim sup_{t→∞} \frac{n(t)}{\sqrt{t log log t}} M_{[n(t)]}(t) ≥ (1 − δ)r_{c}, a.s., ending the proof of (1.8) by letting δ → 0.

To arrive at (4.11), define d_{t_{i−1}} = inf{t > t_{i−1} : B(t) = 0} the first return time to 0 of B after time t_{i−1}. Then B̂(t) =^{def} B(t + d_{t_{i−1}}), t ≥ 0, is a Brownian motion independent of F_{d_{t_{i−1}}} . Define M̂_{n}(r) in terms of B̂ in the same way as M_{n}(r) was defined in terms of B. Observe that conditioning on |B(t_{i−1})| = r > 0 and F_{t_{i−1}}, d_{t_{i−1}} − t_{i−1} =^{law} r^{2}|N |^{−2} with N a centered reduced Gaussian variable. Notice that \frac{t_{i−1}}{t_{i}} ≤ \frac{1}{i}. It follows that for all large i, ℙ(δt_{i}|N |^{−2} < δt_{i} − t_{i−1}) ≥ C_{13}, and we have

ℙ(G_{i} | F_{t_{i−1}}) ≥ ℙ(\{d_{t_{i−1}} < δt_{i}\} ∩ \{M̂_{[n(t_{i})]}(t_{i} − d_{t_{i−1}}) > \frac{(1 − δ)r_{c}}{n(t_{i})} \sqrt{t_{i} log log t_{i}}\} | F_{t_{i−1}} )

≥ C_{13} 1l_{(|B(t_{i−1})|≤\sqrt{δt_{i}})} ℙ(M̂_{[n(t_{i})]}((1 − δ)t_{i}) > \frac{(1 − δ)r_{c}}{n(t_{i})} \sqrt{t_{i} log log t_{i}})

≥ C_{13} 1l_{(|B(t_{i−1})|≤\sqrt{δt_{i}})} exp ( − (1 − \frac{δ}{2}) r_{c}^{2}μ(\frac{c}{(1 − δ)r_{c}^{2}} ) log log t_{i})

≥ C_{13} 1l_{(|B(t_{i−1})|≤\sqrt{δt_{i}})} i^{−1+δ/3},  (4.12)

by virtue of (4.1) and monotonicity of μ. Using the classical LIL for |B(t)| shows that almost surely for all but finite i, |B(t_{i−1})| ≤ \sqrt{3t_{i−1} log log t_{i−1}} ≤ \sqrt{δt_{i}}, which, combining with (4.12) implies (4.11), proving (1.8) in the case 0 < c < ∞.

The proof of (1.8) in other cases (c = 0 and c = ∞) is similar and therefore omitted. This completes the proof of Theorem 1.2. ⊔ ⊓

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