On the joint asymptotic behaviours of ranked heights of Brownian excursions

by

Endre CS^{}ÁKI^{1} and Yueyun HU^{2}

Dedicated to Professor Pál Révész on the occasion of his 65-th birthday

Summary. In this paper, we present some joint asymptotic behaviours of the heights of Brownian excursions.

Keywords. Ranked heights, Brownian excursions, joint behaviours.

1991 Mathematics Subject Classification. 60F15, 60G55.

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

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@ccr.jussieu.fr

1. Introduction

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

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

the ranked heights of all the excursions of the reflected Brownian motion |B| up to time t (including the meander height sup_{g_{t}≤u≤t} |B(u)|, where g_{t} is the last zero of B before t). This gives a natural way to order the countable many Brownian excursions. We refer to Csáki, Erdős and Révész [3] and Révész [17, Chap. XIII] for asymptotic studies on excursion lengths of random walk and Brownian motion, and to Pitman and Yor [12, 13] for the laws of excursion lengths. The ordered excursion heights were studied first by Pitman and Yor [14, 15], who also considered a general class of recurrent self-similar Markov processes.

Here, we are interested in the joint asymptotic behaviours of {M_{n}(t), n ≥ 1} as t → ∞.

Theorem 1.1. Let f(t) > 0 be a nondecreasing function. For fixed 0 ≤ r ≤ 1,

ℙ (M_{1}(t) > \sqrt{t} f(t) and M_{2}(t) > r \sqrt{t} f(t), i.o.) = \{ \begin{matrix}0 \\ 1 \end{matrix} ,

according as

∫ ^{∞} \frac{dt}{t}f(t) exp (−\frac{(1 + 2r)^{2}f^{2}(t)}{2}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,

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

Theorem 1.2. The set of vectors

(\frac{M_{1}(t)}{\sqrt{2t log log t}}, \frac{M_{2}(t)}{\sqrt{2t log log t}}) , t > 3,

is relatively compact in ℝ^{2} and its set of limit points is given by

{(x, y) ∈ R^{2} : 0 ≤ y ≤ x, x + 2y ≤ 1}.

It is an interesting open problem to give a Strassen’s [18] functional law for the vector of processes (2t log log t)^{−1/2} (M_{1}(ut), M_{2}(ut))_{0≤u≤1}.

The lower functions of M_{1}(t) and M_{n}(t) for n ≥ 2, are considerably different (cf. [4]), which leads us to consider the joint lower functions of (M_{1}(t), M_{2}(t)):

Theorem 1.3. Let f(t) > 0 be a nondecreasing function. For each 1 ≤ σ ≤ ∞,

ℙ (M_{1}(t) < σ \frac{\sqrt{t}}{f(t)} and M_{2}(t) < \frac{\sqrt{t}}{f(t)}, i.o.) = \{ \begin{matrix}0 \\ 1 \end{matrix} ,

according as

(1.2)  ∫ ^{∞} \frac{dt}{t}f^{2}(t) exp ( − \frac{π^{2}f^{2}(t)}{8}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  if σ = 1,
(1.3)  ∫ ^{∞} \frac{dt}{t}f^{4}(t) exp ( − \frac{π^{2}f^{2}(t)}{8}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  if 1 < σ < 2,
(1.4)  ∫ ^{∞} \frac{dt}{t}f^{6}(t) exp ( − \frac{π^{2}f^{2}(t)}{8}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  if σ = 2,
(1.5)  ∫ ^{∞} \frac{dt}{t}f^{2}(t) exp ( − \frac{π^{2}f^{2}(t)}{2σ^{2}}) \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  if 2 < σ < ∞,
(1.6)  ∫ ^{∞} \frac{dt}{tf(t)} \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} ,  if σ = ∞.

Notice that (1.2) is the well-known Chung’s test, whereas (1.6) is taken from [4]. See [2] for 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. In particular, the above result merely says that if M_{1}(t), M_{2}(t) are small, the critical ratio of M_{2}(t)/M_{1}(t) is 1/2. We shall study the ratio M_{n+1}(t)/M_{n}(t) for n ≥ 1, as t → ∞, and the result reads as follows:

Theorem 1.4. Let f > 0 be a nondecreasing function. For n ≥ 1, we have

(1.7)  ℙ (\frac{M_{n+1}(t)}{M_{n}(t)} < \frac{1}{f(t)}, i.o.) = \{ \begin{matrix}0 \\ 1 \end{matrix} ⟺ ∫ ^{∞} \frac{dt}{tf^{n}(t)} \{ \begin{matrix}< ∞ \\ = ∞ \end{matrix} .

The rest of this paper is organized as follows: In Section 2, we compute some Laplace transforms needed to obtain the joint small and large deviations for the heights. The proofs of Theorems 1.1 and 1.2 are given in Section 3, whereas Section 4 is devoted to prove Theorem 1.3, and Theorem 1.4 is proved in Section 5.

We write f(x) ∼ g(x) (resp: f(x) ≍ g(x) for x ∈ I, with I being some interval of ℝ) as x → x_{0} if lim_{x→x_{0}} f(x)/g(x) = 1 (resp: 0 < C_{1} ≤ inf_{x∈I} f(x)/g(x) ≤ sup_{x∈I} f(x)/g(x) ≤ C_{2} < ∞). Unless stated otherwise, (C_{i}, 1 ≤ i ≤ 15) denote some (unimportant) positive constants.

Acknowledgements: We are very grateful to Professor Marc Yor for helpful discussions and for references. The cooperation between the authors was supported by the joint French-Hungarian Intergovernmental Grant “Balaton” (grants no. F25/97 and F-32/2000).

2. Preliminaries

Define

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

Recall the following characterization of the law of the sequence {M_{i}(H_{n}(1)), i ≥ 1; H_{n}(1)}:

Proposition 2.1 ([4]). The two sequences {M_{i}(H_{n}(1)), i ≤ n} and {M_{j}(H_{n}(1)), j ≥ n + 1} are independent. Furthermore, we have

(2.2) For n ≥ 2, \{0 ≤ \frac{1}{M_{1}(H_{n}(1))} ≤ ... ≤ \frac{1}{M_{n−1}(H_{n}(1))} ≤ 1\} has the same law as the rear-

ranged nondecreasing sequence of n − 1 i.i.d. uniform variables in [0, 1]; (2.3) The law of {M_{j}(H_{n}(1)), j ≥ n + 1} is characterized as follows: for every measurable

function f ≥ 0, we have

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

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

𝔼(e^{− \frac{λ^{2}}{2} H_{n}(1)} | \{M_{i}(H_{n}(1)) = x_{i} > 0, i ≠ n\}) = \frac{λ}{sinh(λ)} ∏_{i≠n} (\frac{λx_{i}}{sinh(λx_{i})})^{2} .

Lemma 2.2. Fix n ≥ 1 and 0 < r ≤ 1. We have for λ > 0 that

(2.5) 𝔼e^{− \frac{λ^{2}}{2} H_{n}(1)}1l_{(H_{n}(1)<H_{n+1}(r))} = e^{−λ(n−1)} (\frac{sinh(λr)}{sinh(λ) cosh(λr)})^{n} ,
(2.6) 𝔼e^{− \frac{λ^{2}}{2} H_{n+1}(r)}1l_{(H_{n}(1)>H_{n+1}(r))} = cosh^{−(n+1)}(λr) (e^{−λrn} − e^{−λn}[\frac{sinh(λr)}{sinh(λ)}]^{n}) .

Proof. By conditioning on {M_{i}(H_{n}(1), i ≠ n}, it follows from (2.4) that the LHS of (2.5) equals

= \frac{λ}{sinh(λ)} 𝔼 ∏_{i≠n} [\frac{λM_{i}(H_{n}(1))}{sinh(λM_{i}(H_{n}(1)))}]^{2}1l_{(M_{n+1}(H_{n}(1))<r)}

= \frac{λ}{sinh(λ)} 𝔼 ∏_{i≤n−1} [\frac{λM_{i}(H_{n}(1))}{sinh(λM_{i}(H_{n}(1)))}]^{2}

× 𝔼 ∏_{i≥n+1} [\frac{λM_{i}(H_{n}(1))}{sinh(λM_{i}(H_{n}(1)))}]^{2}1l_{(M_{n+1}(H_{n}(1))<r)}, (2.7)

by using the fact that {M_{i}(H_{n}(1), i < n} and {M_{i}(H_{n}(1), i > n} are independent. It follows from (2.2) that (2.8)

𝔼 ∏_{i≤n−1} [\frac{λM_{i}(H_{n}(1))}{sinh(λM_{i}(H_{n}(1)))}]^{2} = (∫_{0}^{1} du (\frac{λ/u}{sinh(λ/u)})^{2})^{n−1} = (\frac{λ}{sinh(λ)} e^{−λ})^{n−1} ,

whereas by means of (2.3), the second expectation term in (2.7) is equal to

𝔼 ∏_{i≥n+1} [\frac{λM_{i}(H_{n}(1))}{sinh(λM_{i}(H_{n}(1)))}1l_{(M_{i}(H_{n}(1))<r)}]^{2} = (1 + ∫_{0}^{1} \frac{dx}{x^{2}}[1 − (\frac{λx}{sinh(λx)}1l_{(x<r)})^{2} ])^{−n}
= (λ coth(λr))^{−n} ,

which in view of (2.7) and (2.8), implies (2.5). The proof of (2.6) is quite similar: by using Brownian scaling and applying Proposition 2.1 to H_{n+1}(1) instead of H_{n}(1), we obtain the desired result. ⊔ ⊓

Corollary 2.3. For 0 < r ≤ 1, we have

(2.9)  𝔼e^{− \frac{λ^{2}}{2} H_{1}(1)∧H_{2}(r)} = \frac{sinh(λ(1 − r))}{cosh^{2}(λr) sinh λ} + \frac{tanh(λr)}{sinh λ},
(2.10)  𝔼e^{− \frac{λ^{2}}{2} H_{1}(1)∨H_{2}(r)} = \frac{2 sinh(λ(1 − r))}{sinh(2λ) cosh(λr)} + \frac{sinh(λr) e^{−λ}}{sinh λ cosh^{2}(λr)},
(2.11)  ℙ(M_{n+1}(H_{n}(1)) < r) = r^{n},  n ≥ 1.

Proof. By letting r = 1 in (2.5), we recover Pitman and Yor’s formula [14]: for λ > 0,

(2.12)  𝔼 exp (−\frac{λ^{2}}{2} H_{n}(1)) = e^{−λ(n−1)} cosh^{−n}(λ).

By taking λ = 0 in (2.5), we get (2.11). Adding (2.5) and (2.6) gives (2.9), and (2.10) follows from (2.9) and (2.12). ⊔ ⊓

We end this section by two elementary results:

Lemma 2.4. Let X_{1}, X_{2} be two independent nonnegative random variables such that

(2.13)  ℙ(X_{i} > t) ≍ t^{γ_{i}} exp(−c_{i} t),  ∀t ≥ 1, i = 1, 2,

for some constants c_{1} ≥ c_{2} > 0 and γ_{1}, γ_{2} > −1. Then

(2.14)  ℙ(X_{1} + X_{2} > t) ≍ \{ \begin{matrix}t^{γ_{2}} exp(−c_{2} t), & if 0 < c_{2} < c_{1}, \\ t^{γ_{1}+γ_{2}+1} exp(−c_{2} t), & if c_{1} = c_{2}, \end{matrix} t ≥ 1.

Lemma 2.5. Suppose Y_{1} and Y_{2} are two independent positive random variables such that

(2.15) ℙ(Y_{i} < ϵ) ≍ ϵ^{α_{i}} exp (−\frac{β_{i}}{ϵ} ) , 0 < ϵ < 1,

for some constants β_{i} > 0 and α_{i} ∈ ℝ. Then

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

3. Joint upper functions

Lemma 3.1. Fix 0 ≤ r ≤ 1. Uniformly for λ ≥ 1,

(3.1)  ℙ (M_{1}(1) > λ and M_{2}(1) > r λ) ≍ \frac{1}{λ} exp (−\frac{(1 + 2r)^{2}λ^{2}}{2} ) .

Proof. Recall (2.1):

(3.2)  ℙ (M_{1}(1) > λ and M_{2}(1) > r λ) = ℙ (H_{1}(1) ∨ H_{2}(r) < x) ,

with x =^{def} λ^{−2} ≤ 1. For 0 < a < b, denote by T_{a→b}^{(3)} a positive r.v. such that

(3.3) 𝔼 exp (−\frac{λ^{2}}{2} T_{a→b}^{(3)}) = \frac{b sinh(λa)}{a sinh(λb)}, λ > 0.

We simply write T_{b}^{(3)} for T_{0→b}^{(3)} (whose Laplace transform can be obtained from (3.3) by letting a → 0). In fact, T_{a→b}^{(3)} is distributed as the first hitting of b by a three-dimensional Bessel process starting from a (cf. [10] and [16]). It follows from (2.10) that

ℙ (H_{1}(1) ∨ H_{2}(r) < x) = (1 − r)ℙ (T_{1−r→2}^{(3)} + r^{2}T_{1}^{(1)} < x) +

+ r ℙ (T_{r→1}^{(3)} + r^{2} (T_{1}^{(1)} + T̂_{1}^{(1)}) + T_{1}(B) < x) , (3.4) where, T_{1}^{(1)} (resp: T_{1}(B)) is distributed as the first hitting time of 1 by |B| (resp: B), and T̂_{1}^{(1)} =^{law} T_{1}^{(1)}, and all the variables in (3.4) are mutually independent.

For 0 < a < b, the following formula (3.5) is taken from Borodin and Salminen [1, pp. 339]):

(3.5)  ℙ(T_{a→b}^{(3)} ∈ dt)/dt = \frac{b}{a} ∑_{k=−∞}^{∞} \frac{b − a + 2kb}{\sqrt{2πt^{3}}} exp (−\frac{(b − a + 2kb)^{2}}{2t} )

= \frac{4π b}{a} ∑_{k=1}^{∞} (−1)^{k−1} k sin (\frac{kaπ}{b} ) exp (−\frac{k^{2}π^{2} t}{2b^{2}}) , (3.6)

where (3.6) follows from Poisson’s summation formula (cf. Feller [8, pp.592]). Recall that

(3.7)  ℙ(T_{1}^{(1)} < x) ≍ ℙ(T_{1}(B) < x) ≍ x^{1/2} exp ( − \frac{1}{2x}),  0 < x < 1,

see also Gruet and Shi [9]. Applying Lemma 2.5 to (3.4) and using (3.7), (3.5) for the small deviation of T_{a→b}^{(3)}, (3.1) follows. ⊔ ⊓

Proof of Theorem 1.1. From (3.1), scaling and monotonicity, it is routine to prove the convergence part of Theorem 1.1 (cf. the pioneer paper of Erdős [7]), the details are omitted.

To show the divergent part, we only need to consider the “critical” case: there exists some t_{0} > 0 such that

(3.8)  \frac{1}{3}\sqrt{log log t} ≤ f(t) ≤ 3\sqrt{log log t},  t ≥ t_{0},

see e.g. [7] for a rigorous justification. We only consider the case 0 < r ≤ 1, since the case of r = 0 is described by the classical Erdős-Feller-Kolmogorov-Petrowsky test. Recall (2.1) and define

t_{i} =^{def} exp (θ \frac{i}{log i}),
λ_{i} =^{def} \sqrt{t_{i}}f(t_{i}),
A_{i} =^{def} {t_{i−1} < H_{1}(λ_{i}) ∨ H_{2}(rλ_{i}) < t_{i}},

where θ > 0 is a constant whose value will be determined later. It suffices to prove that

(3.9)  ℙ(A_{i}; i.o.) = 1.

Observe that from (3.1) and using scaling, we can find some sufficiently large θ > 0 such that

ℙ(A_{i}) = ℙ(H_{1}(λ_{i}) ∨ H_{2}(rλ_{i}) < t_{i}) − ℙ(H_{1}(λ_{i}) ∨ H_{2}(rλ_{i}) < t_{i−1})

≍ \frac{1}{f(t_{i})} exp (−\frac{(1 + 2r)^{2}f^{2}(t_{i})}{2}) , (3.10)

which in view of the divergence of the integral test yields that

(3.11)  ∑_{i} ℙ(A_{i}) = ∞.

To apply the Borel-Cantelli lemma, we shall estimate the second moment ℙ(A_{i} ∩ A_{j}) for j > i ≥ i_{0}, where i_{0} denotes some large constant. We first estimate

(3.12)  ℙ(M_{1}(t_{i}) > rλ_{j}) = ℙ(M_{1}(1) > rλ_{j}/\sqrt{t_{i}}) ≤ \frac{C_{3}\sqrt{t_{i}}}{rf(t_{j})\sqrt{t_{j}}} exp ( − \frac{f^{2}(t_{j})t_{j}}{2t_{i}} ),

where the inequality follows from (3.1) by taking r = 0 there. Define

β(t) =^{def} |B(H_{1}(λ_{i}) ∨ H_{2}(rλ_{i}) + t)|, t ≥ 0.

Using strong Markov property, the process β is a reflected Brownian motion, starting from |B(H_{1}(λ_{i}) ∨ H_{2}(rλ_{i}))| = λ_{i} or rλ_{i}, and independent of F_{H_{1}(λ_{i})∨H_{2}(rλ_{i})} (where, here and in the sequel, (F_{t}, t ≥ 0) denotes the natural filtration generated by B).

Observe that on the event A_{i} ∩ A_{j} ∩ {M_{1}(t_{i}) ≤ rλ_{j}}, there are at least two excursions of β before time t_{j} −t_{i−1}, whose heights are larger than rλ_{j} and λ_{j}. Write a =^{def} |B(H_{1}(λ_{i})∨ H_{2}(rλ_{i}))| = λ_{i} or rλ_{i}, we introduce a Brownian motion B̂ by β(t) = |a + B̂(t)| for t ≥ 0, B̂ is independent of F_{H_{1}(λ_{i})∨H_{2}(rλ_{i})}. Therefore,

A_{i} ∩A_{j} ∩{M_{1}(t_{i}) ≤ rλ_{j}} ⊂ A_{i} ∩{M̂_{1}(t_{j} −t_{i−1}) ≥ λ_{j} −a}∩{M̂_{2}(t_{j} −t_{i−1}) ≥ (rλ_{j} −a)^{+}},

with M̂_{1}(·), M̂_{2}(·) related to B̂ the same way as M_{1}(·), M_{2}(·) are related to B. It follows that

ℙ(A_{i} ∩ A_{j} ∩ {M_{1}(t_{i}) ≤ rλ_{j}})

≤ ℙ(A_{i})ℙ (\{M̂_{1}(1) ≥ \frac{λ_{j} − λ_{i}}{\sqrt{t_{j} − t_{i−1}}} \} ∩ \{M̂_{2}(1) ≥ \frac{(rλ_{j} − λ_{i})^{+}}{\sqrt{t_{j} − t_{i−1}}} \}) (3.13)

≤ ℙ(A_{i})\frac{C_{4}}{(λ_{j} − λ_{i}/r)^{+} + 1} exp (−\frac{(1 + 2r)^{2}((λ_{j} − λ_{i}/r)^{+})^{2}}{2(t_{j} − t_{i−1})} ) , (3.14)

by applying (3.1) to λ = (λ_{j} − λ_{i}/r)^{+}. Using (3.1) with r = 0, it follows from (3.13) that

ℙ(A_{i} ∩ A_{j} ∩ \{M_{1}(t_{i}) ≤ rλ_{j}\}) ≤ ℙ(A_{i}) ℙ (M_{1}(1) ≥ \frac{λ_{j} − λ_{i}}{\sqrt{t_{j} − t_{i−1}}} )

≤ C_{5}ℙ(A_{i}) \frac{\sqrt{t_{j} − t_{i−1}}}{λ_{j} − λ_{i}} exp (−\frac{(λ_{j} − λ_{i−1})^{2}}{2(t_{j} − t_{i−1})} ) . (3.15)

Using (3.12), (3.14) and (3.15), it is elementary to obtain that

(3.16)  ℙ(A_{i} ∩ A_{j}) ≤ \{ \begin{matrix}C_{6}ℙ(A_{i}) ℙ(A_{j}), & if  j − i ≥ log^{2} i, \\ C_{7}ℙ(A_{i})j^{−C_{8}}, & if  log i ≤ j − i < log^{2} i, \\ C_{9}ℙ(A_{i}) e^{−C_{10}(j−i)}, & if  2 ≤ j − i < log i. \end{matrix}

It follows that

lim inf_{n→∞} \frac{∑_{i_{0}≤i,j≤n} ℙ(A_{i} ∩ A_{j})}{(∑_{i_{0}≤i≤n} ℙ(A_{i}))^{2}} ≤ C_{6},

which in view of (3.11), according to Kochen and Stone’s version of the Borel-Cantelli lemma (cf. [11]) implies ℙ(A_{i}; i.o.) ≥ 1/C_{6} > 0. This probability in fact equals 1 from Kolmogorov’s 0-1 law. ⊔ ⊓

We present a lemma before proving Theorem 1.2:

Lemma 3.2. For each 0 < δ < 1/9, there exists a constant C_{11} = C_{11}(δ) > 0 such that for all x > 0,

(3.17)  ℙ(M_{1}(1) + 2M_{2}(1) > x) ≤ C_{11} exp ( − \frac{(1 − δ)^{3} x^{2}}{2} ).

Proof. The probability term in (3.17) is bounded by

(3.18)  ℙ(M_{1}(1) > (1 − δ)x) + ℙ(M_{2}(1) > \frac{1 − δ}{2} x) + ∑_{1≤k≤1/δ} I_{k},

with I_{k} =^{def} ℙ(kδx < M_{1}(1) ≤ (k + 1)δx; M_{2}(1) > \frac{1−(k+1)δ}{2} x). Using Lemma 3.1, the first

two probability terms in (3.18) are bounded by C_{12} exp ( − \frac{(1−δ)^{3} x^{2}}{2} ), and

I_{k} ≤ ℙ (M_{1}(1) > kδx; M_{2}(1) > \frac{1 − (k + 1)δ}{2} x) ≤ C_{13} exp ( − \frac{(1 − δ)^{3} x^{2}}{2} ),

which in view of (3.18), yields the desired estimate. ⊔ ⊓

Proof of Theorem 1.2. Denoting by K the limit set stated in Theorem 1.2, we first show that with probability one, for every point (a, b) ∈ K, there exists some sequence (\frac{M_{1}(t_{n})}{\sqrt{2t_{n} log log t_{n}}} , \frac{M_{2}(t_{n})}{\sqrt{2t_{n} log log t_{n}}} )_{n≥1} which converges to (a, b) as t_{n} → ∞. This is in fact a direct consequence of Strassen’s functional LIL for Brownian motion: Consider a continuous function f : [0, 1] → ℝ defined by

(3.19)  f(t) =^{def} \{ \begin{matrix}−t, & if 0 ≤ t ≤ b, \\ −b\frac{1+a}{1−b} + \frac{a+b}{1−b}t, & if b ≤ t ≤ 1. \end{matrix}

Therefore ∫_{0}^{1} f^{′2}(t)dt = b + (a + b)^{2}/(1 − b) ≤ 1 since (a, b) ∈ K. According to Strassen’s LIL, there exists a sequence (t_{k} = t_{k}(ω))_{k≥1} such that

sup_{0≤s≤1} |\frac{B(st_{k})}{\sqrt{2t_{k} log log t_{k}}} − f(s)| → 0, k → ∞,

yielding that

\frac{M_{1}(t_{k})}{\sqrt{2t_{k} log log t_{k}}} → a,  \frac{M_{2}(t_{k})}{\sqrt{2t_{k} log log t_{k}}} → b,  k → ∞,

as desired. To end this proof, we fix a small ϵ > 0 and write d(z, K) =^{def} inf_{z′∈K} |z − z′| for z ∈ ℝ^{2}. It suffices to show that with probability one,

(3.20)  d ((\frac{M_{1}(t)}{\sqrt{2t log log t}}, \frac{M_{2}(t)}{\sqrt{2t log log t}}) , K) → 0,  t → ∞.

Define r_{k} =^{def} exp (k^{1−ϵ/6}) for k ≥ 1. Using Lemma 3.2,

ℙ (d ((\frac{M_{1}(r_{k})}{\sqrt{2r_{k} log log r_{k}}} , \frac{M_{2}(r_{k})}{\sqrt{2r_{k} log log r_{k}}} ) , K) > ϵ)

≤ ℙ(M_{1}(r_{k}) + 2M_{2}(r_{k}) > \frac{ϵ}{2}\sqrt{2r_{k} log log r_{k}})

≤ C_{14}(ϵ) exp ( − (1 + ϵ/3) log log r_{k})

≤ C_{15}k^{−(1+ϵ)/4},

which is summable in k. The Borel-Cantelli lemma implies that with probability one,

for all large k, we have d ((\frac{M_{1}(r_{k})}{\sqrt{2r_{k} log log r_{k}}} , \frac{M_{2}(r_{k})}{\sqrt{2r_{k} log log r_{k}}} ) , K) ≤ ϵ. For r_{k} ≤ t < r_{k+1},

(1 − ϵ)\frac{M_{i}(r_{k})}{\sqrt{2r_{k} log log r_{k}}} ≤ \frac{M_{i}(t)}{\sqrt{2t log log t}} ≤ (1 + ϵ)\frac{M_{i}(r_{k+1})}{\sqrt{2r_{k+1} log log r_{k+1}}} with i = 1 or 2, and (3.20) follows. ⊔ ⊓

4. Joint lower functions

Lemma 4.1. Fix σ ∈ [1, ∞]. Uniformly for 0 < ϵ < 1,

(4.1)  ℙ(M_{1}(1) < σϵ and M_{2}(1) < ϵ) ≍ \{ \begin{matrix}exp ( − \frac{π^{2}}{8ϵ^{2}} ), & if σ = 1, \\ ϵ^{−2} exp ( − \frac{π^{2}}{2ϵ^{2}} ), & if 1 < σ < 2, \\ ϵ^{−4} exp ( − \frac{π^{2}}{2ϵ^{2}} ), & if σ = 2, \\ exp ( − \frac{π^{2}}{2ϵ^{2}σ^{2}} ), & if 2 < σ < ∞, \\ ϵ, & if σ = ∞. \end{matrix}

Proof. The case σ = ∞ was obtained in [4], whereas the case σ = 1 is the classical Chung’s small deviation (cf. [5]). We only consider 1 < σ < ∞. Recall (2.1), we deduce from the self-similarity that

(4.2)  ℙ(M_{1}(1) < σϵ and M_{2}(1) < ϵ) = ℙ(H_{1}(1) ∧ H_{2}(r) > t),

with t =^{def} ϵ^{−2} σ^{−2} and r = 1/σ < 1. Recall (3.3). It follows from (2.9) that for all t > 0, (4.3) ℙ(H_{1}(1) ∧ H_{2}(r) > t) = (1 − r)ℙ(T_{1−r→1}^{(3)} + r^{2}(T_{1}^{(1)} + T̂_{1}^{(1)}) > t) + rℙ(r^{2}Λ + T_{0→1}^{(3)} > t),

where all the r.v. in RHS of (3.6) are independent, and T̂_{1}^{(1)} =^{law} T_{1}^{(1)} is distributed as the first hitting time of 1 by |B|; the Laplace transform at λ^{2}/2 of Λ is \frac{tanh λ}{λ}. The density function of Λ can be obtained by inverting the Laplace transform (cf. [6, pp. 257]), which implies that

(4.4)  ℙ(Λ > x) ≍ exp ( − \frac{π^{2}}{8} x) ≍ ℙ(T_{1}^{(1)} > x),  x > 0.

where the last asymptotic equivalence is well-known. Applying Lemma 2.4 to (4.3) with (4.4), and (3.6) for the tail probability of T_{1−r→1}^{(3)}, the desired estimates in (4.1) follow. ⊔ ⊓

Proof of Theorem 1.3. Applying Lemma 4.1, the proof is similar to that of Theorem 1.1, we omit the details. ⊔ ⊓

5. Ratio.

Lemma 5.1. Fix n ≥ 1. Uniformly for 0 < ϵ < 1,

(5.1)  ℙ( inf_{H_{n}(1)≤t≤H_{n}(2)} \frac{M_{n+1}(t)}{M_{n}(t)} < ϵ) ≍ ϵ^{n}.

Proof. By monotonicity,

\frac{M_{n+1}(H_{n}(1))}{2} ≤ inf_{H_{n}(1)≤t≤H_{n}(2)} \frac{M_{n+1}(t)}{M_{n}(t)} ≤ M_{n+1}(H_{n}(2)) =^{law} 2M_{n+1}(H_{n}(1)),

yielding (5.1) in view of (2.11). ⊔ ⊓

Proof of Theorem 1.4. Observe that with probability one, for some large r_{0} = r_{0}(ω),

(5.2)  r ≤ H_{n}(r) ≤ r^{3},  r ≥ r_{0},

which can be easily obtained from the upper and lower functions for M_{n}(·) (cf. [4]). We omit the details of the proof of the convergent part of Theorem 1.4 which can be easily obtained by using (5.1) and (5.2).

To prove the divergent part, we assume without loss of generality (cf. [7]) that there exists some t_{0} > 0 such that

(5.3)  (log t)^{1/n} ≤ f(t) ≤ (log t)^{2/n} ,  t ≥ t_{0}.

Define r_{i} = 2^{i} and f̂(t) =^{def} f(t^{3}). Observe that in view of (5.2), it suffices to show that with probability one,

(5.4)  E_{i} =^{def} \{M_{n+1}(H_{n}(r_{i})) < \frac{r_{i}}{f̂(r_{i})}\}, occurs infinitely often.

Using (2.11), ℙ(E_{i}) = (f̂(r_{i}))^{−n}, and therefore

(5.5)  ∑_{i} ℙ(E_{i}) = ∞,

since ∫^{∞} dt/(tf̂^{n}(t)) = ∞. For j > i ≥ i_{0}, we estimate ℙ(E_{i} ∩ E_{j}) to apply Borel-Cantelli lemma. Denote by d_{H_{n}(r_{i})} =^{def} inf{t > H_{n}(r_{i}) : B(t) = 0}, the first return time. Define

B̃(t) =^{def} B(d_{H_{n}(r_{i})} + t), t ≥ 0,

which is a Brownian motion independent of F_{d_{Hn(r_{i})}} (recalling that (F_{t}, t ≥ 0) is the natural filtration of B). Similarly, define (M̃_{n}(·), H̃_{n}(·), n ≥ 1) related to B̃. For j > i, the event {M_{1}(d_{H_{n}(r_{i})}) < r_{j}} implies H_{n}(r_{j}) = H̃_{n}(r_{j}) + d_{H_{n}(r_{i})}, and

ℙ(E_{i} ∩ E_{j} ∩ \{M_{1}(d_{H_{n}(r_{i})}) < r_{j}\}) ≤ ℙ (E_{i} ∩ \{M̃_{n+1}(H̃_{n}(r_{j})) < \frac{r_{j}}{f̂(r_{j})}\})

= ℙ(E_{i})ℙ(E_{j}), (5.6)

by means of the independence of B̃ and E_{i}. It remains to consider the event {M_{1}(d_{H_{n}(r_{i})}) ≥ r_{j}}. First, we deduce from (2.2) and scaling that

(5.7)  ℙ(M_{1}(H_{n}(r)) > xr) = ℙ( inf_{1≤k≤n−1} U_{k} < \frac{1}{x}) ≤ \frac{n − 1}{x},  x > 1, r > 0,

where (U_{k}, 1 ≤ k ≤ n−1) denote n−1 independent uniform variables. Since M_{1}(d_{H_{n}(r_{i})}) = M_{1}(H_{n}(r_{i})) ∨ sup_{H_{n}(r_{i})≤t≤d_{Hn(r_{i})}} |B(t)|, is independent of M_{n+1}(H_{n}(1)), and

ℙ(E_{i} ∩ {M_{1}(d_{H_{n}(r_{i})}) ≥ r_{j}})

= ℙ(E_{i})ℙ(M_{1}(d_{H_{n}(r_{i})}) ≥ r_{j})

≤ ℙ(E_{i})(ℙ(M_{1}(H_{n}(r_{i}) > r_{j}) + ℙ( sup_{H_{n}(r_{i})≤t≤d_{Hn(r_{i})}} |B(t)| > r_{j}))

≤ ℙ(E_{i})(\frac{(n − 1)r_{i}}{r_{j}} + \frac{r_{i}}{r_{j}} )

= 2^{−(j−i)} ℙ(E_{i}). (5.8)

It follows from (5.6) and (5.8) that

lim inf_{n→∞} \frac{∑_{i_{0}≤i,j≤n} ℙ(E_{i} ∩ E_{j})}{(∑_{i_{0}≤i≤n} ℙ(E_{i}))^{2}} ≤ 1,

which in view of (5.5) and Kochen and Stone’s version of Borel-Cantelli lemma, implies (5.4), as desired. ⊔ ⊓

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