Favourite sites, favourite values and jumping sizes for random walk and Brownian motion

ENDRE CSÁKI

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

127, H-1364, Hungary. E-mail: csaki@math-inst.hu

PÁL RÉVÉSZ

Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107, A-1040 Vienna, Austria. E-mail: revesz@ci.tuwien.ac.at

ZHAN SHI

Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05,

France. E-mail: zhan@proba.jussieu.fr

We determine: (a) the joint almost sure asymptotic behaviour of the most visited site and the maximum local time of one-dimensional simple random walk or Brownian motion; (b) the maximal jumping size of the most visited site. These solve two open problems of Erdős and Révész.

Keywords: Brownian motion; favourite site; local time; random walk

1. Introduction

Consider a one-dimensional simple symmetric random walk {S_{n}}_{n⩾0}, starting from S_{0} = 0. Let

ξ(n, x) = #{0 ⩽ k ⩽ n : S_{k} = x}.  (1.1)

Thus, ξ(n, x) records the number of times the random walk visits the site x in the first n steps. Define

𝕌(n) = {x ∈ ℤ : ξ(n, x) = sup_{y∈ℤ} ξ(n, y)}.

In words, 𝕌(n) is the set of the most visited sites (or favourite sites) of the random walk at time n. Let

U(n) = max_{x∈𝕌(n)} x.  (1.2)

In Erdős and Révész (1984), U(n) was called the (largest) favourite site of {S_{k}}_{0⩽k⩽n}. The choice of U(n) is irrelevant in the sense that all the results for U(n) stated in this paper remain true if “max” is replaced for example by “min” in (1.2).

The study of asymptotic properties of 𝕌(n) and U(n) was initiated by Erdős and Révész (1984) and Bass and Griffin (1985), and followed by several groups of mathematicians. Nevertheless, little is known so far, due mainly to the fact that the process n ↦ U(n) has some very peculiar asymptotic behaviours. For example, a surprising theorem of Bass and Griffin (1985) confirms the transience of U(n), in the sense that lim_{n→∞} |U(n)| = ∞ almost surely. However, it is an open problem to determine the rate of escape of U(n). Another interesting problem is from Erdős and Révész (1984), who conjectured that almost surely for all sufficiently large n, 𝕌(n) contains at most two points.

We are interested in the limsup asymptotics of U(n). First, we recall the law of the iterated logarithm (LIL) for U(n) which was proved by Erdős and Révész (1984) and Bass and Griffin (1985), using different methods:

lim sup_{n→∞} \frac{U(n)}{ϕ(n)} = 1,  a.s.,  (1.3)

where

ϕ(n) = (2n log log n)^{1/2} = (2n log_{2} n)^{1/2}.  (1.4)

(However, the problem of characterizing the upper class functions of U(n) via an integral test in the sense of P. Lévy remains open. For a list of ten other open problems concerning favourite sites, see pp. 130–131 of Révész 1990).

We also recall the LIL for maximum local time: with

ξ^{∗}(n) = sup_{x∈ℤ} ξ(n, x),  (1.5)

we have

lim sup_{n→∞} \frac{ξ^{∗}(n)}{ϕ(n)} = 1,  a.s.  (1.6)

This was first proved by Kesten (1965), and it was extended to the form of an integral test by Csáki (1989).

In their proof of (1.3), Erdős and Révész (1984) noticed that, for any ε > 0, almost surely there exist infinitely many n such that simultaneously, U(n) ⩾ (1 − ε)ϕ(n) and ξ^{∗}(n) ⩾ c ϕ(n), for some positive constant c depending on ε. This led them to ask the following question: what can be said about the joint asymptotics of U(n) and ξ^{∗}(n)? When U(n) is very close to its maximal possible value, how large can ξ^{∗}(n) be? This question also appeared later in Erdős and Révész (1987), and it was also stated as Open problem 1 on p. 130 in Révész (1990).

If U(n) and ξ^{∗}(n) were asymptotically independent, then one would expect that the limit set of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))} should be {(x, y) : x^{2} + y^{2} ⩽ 1}. However, things do not quite go like this. Namely we have the following result in this regard.

Theorem 1.1. With probability one, the random sequence

\{ ( \frac{U(n)}{ϕ(n)}, \frac{ξ^{∗}(n)}{ϕ(n)} ) \}_{n⩾3}

is relatively compact, whose limit set is identical to the simplex

A = { (x, y) : y ⩾ 0, |x| + y ⩽ 1 }.

Remark. Our Theorem 1.1 shows that favourite value and favourite site are somewhat negatively dependent, i.e., if the most visited site is small then the time spent in it is large, and if it is large then the time spent in it is correspondingly smaller.

The second question we are interested in is also due to Erdős and Révész (1984, 1987) (Open problem #4 on page 130 of Révész 1990): what is the maximal jumping size of U(n)? The jumping size of U(n) is defined as U(n + 1) − U(n) if it does not vanish. Concerning its large values, they asked whether it is possible to determine the function f(n) such that lim sup_{n→∞}(U(n + 1) − U(n))/f(n) = 1 almost surely.

In view of (1.3), one might expect U(n + 1) − U(n) to be a constant multiple of ϕ(n) along a random subsequence. What might be somewhat surprising is that it turns out to be as large as (1 − ε)ϕ(n), for any ε > 0.

Theorem 1.2. We have,

lim sup_{n→∞} \frac{U(n + 1) − U(n)}{ϕ(n)} = 1, a.s.

In the study of favourite sites, it is more frequent to encounter questions than answers, and this may help explain why there are relatively few known results compared with the huge literature on random walk. We mention the work of Tóth and Werner (1997) who were interested in favourite edges instead of favourite sites, and of Csáki and Shi (1998) who studied how close a favourite site can be to the frontier of the range of random walk.

It will be clear from the proofs in Sections 4 and 5 that both Theorems 1.1 and 1.2 have their counterparts for Brownian motion, which we state as follows. Similarly to the case of random walk, we can define the set of favourite sites of Brownian motion as:

𝕍(t) = {x ∈ ℝ : L(t, x) = sup_{y∈ℝ} L(t, y)},

where L denotes the local time of a Brownian motion. It is known (Eisenbaum 1989 and 1997, Leuridan 1997) that, with probability one, 𝕍(t) contains at most two points for all t > 0. Without loss of generality, let us choose

V (t) = max_{x∈𝕍(t)} x.

According to Eisenbaum (1989), {V (t); t > 0} is a process with limits on the left almost surely.

Theorem 1.3. Let L^{∗}(t) = sup_{x∈ℝ} L(t, x). The limit set of {(V (t)/ϕ(t), L^{∗}(t)/ϕ(t))}_{t⩾3} is almost surely {(x, y) : y ⩾ 0, |x| + y ⩽ 1}.

Theorem 1.4. We have,

lim sup_{t→∞} \frac{V (t) − V (t−)}{ϕ(t)} = 1, a.s.

We say a few words about the organization of this paper. Looking at Theorem 1.1, that {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3} is almost surely relatively compact follows from (1.3) and (1.6), so we only have to show that its limit set equals A. This is done in two steps. Namely, we prove that, almost surely,

(a) any (x, y) ∈ A is a limit point of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3};
(b) all the limit points of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3} are contained in A.

Parts (a) and (b) are proved in Sections 4 and 5 respectively; furthermore, Section 4 contains the proof of Theorem 1.2. Before that, in Section 2 we present some probability estimates for multi-dimensional Brownian motion, which may be of independent interest. These estimates are applied to Brownian local times in Section 3, and enable us to obtain the key ingredient in the proofs of the theorems.

Notation. Throughout the paper, a(x) ∼ b(x) (x → x_{0}) denotes lim_{x→x_{0}} a(x)/b(x) = 1 for positive a(x) and b(x).

2. Probability estimates for Brownian motion

This section contains some probability estimates for d-dimensional Brownian motion. Although only the special case d = 2 is needed in the proof of the theorems, we present the estimates for any dimension d ⩾ 1.

In the rest of the section, {B(t); t ⩾ 0} denotes an ℝ^{d}-valued Brownian motion (d ⩾ 1) and

R(t) = ∥B(t)∥, t ⩾ 0,

is the Euclidean modulus of B. Note that R is a d-dimensional Bessel process. For x ⩾ 0, let ℙ^{x} denote the probability measure under which R starts from x. We assume that under ℙ, R(0) = 0. Thus ℙ^{0} = ℙ. We also write 𝔼^{x} for the expectation with respect to ℙ^{x}.

The square integral of R plays an important role in our approach. We write, for brevity,

X(t) = ∫_{0}^{t} R^{2}(s) ds, t ⩾ 0.

We start by recalling two useful results. The first is an intuitively clear comparison criterion for R. It can easily be proved by applying the diffusion comparison theorem stated in Revuz and Yor (1999, Theorem IX.3.7) to squared Bessel processes. The second is a special case of de Bruijn’s exponential-type Tauberian theorem (see Bingham et al. 1987, p. 254).

Fact 2.1. Let 0 ⩽ x < y. The Bessel process {R(t); t ⩾ 0} is stochastically smaller under ℙ^{x} than under ℙ^{y}.

Fact 2.2. Let Λ be a random variable with ℙ(Λ ⩾ 0) = 1, and let a > 0 be a constant. The following conditions are equivalent:

lim_{r→∞} \frac{1}{\sqrt{r}} log 𝔼(e^{−rΛ}) = −a,  (i)
lim_{ε→0} ε log ℙ(Λ < ε) = −\frac{a^{2}}{4} .  (ii)

We now prove a few preliminary estimates.

Lemma 2.3. Fix θ ∈ (0, 1), t > 0, and r > r_{1} > 0. Then

lim_{ε→0} ε log ℙ(θε ⩽ X(t) ⩽ ε, r_{1} ⩽ R(t) ⩽ r) = −\frac{(r_{1}^{2} + td)^{2}}{8} .  (2.1)

Proof. The conditional Laplace transform of X(t) given (R(0), R(t)) is known (cf. Pitman and Yor 1982):

𝔼^{x}( e^{− \frac{λ^{2}}{2} X(t)} | R(t) = y )
= \frac{λt}{sinh(λt)} \frac{I_{(d−2)/2}(λxy/ sinh(λt))}{I_{(d−2)/2}(xy/t)} exp( −\frac{(x^{2} + y^{2})(λt coth(λt) − 1)}{2t} ), (2.2)

where I_{(d−2)/2}(·) denotes the modified Bessel function of index (d − 2)/2. For convenience, we recall here the asymptotics of the modified Bessel function, (cf. Abramowitz and Stegun 1965, pp. 375 and 377 respectively):

I_{p}(z) ∼ (z/2)^{p}/Γ(p + 1),  z → 0,  (2.3)
I_{p}(z) ∼ e^{z}/\sqrt{2πz},  z → ∞.  (2.4)

In view of (2.3), we can let x go to 0 in (2.2) to see that

𝔼( e^{− \frac{λ^{2}}{2} X(t)} | R(t) = y ) = ( \frac{λt}{sinh(λt)} )^{d/2} exp( −\frac{y^{2}(λt coth(λt) − 1)}{2t} ).  (2.5)

Thus,

𝔼( e^{− \frac{λ^{2}}{2} X(t)} | r_{1} ⩽ R(t) ⩽ r )

= ∫_{r_{1}}^{r} \frac{ℙ(R(t) ∈ dy)}{ℙ(r_{1} ⩽ R(t) ⩽ r)} ( \frac{λt}{sinh(λt)} )^{d/2} exp( −\frac{y^{2}(λt coth(λt) − 1)}{2t} ).

When λ → ∞, the expression on the right-hand side is exp(−(1 + o(1))\frac{r_{1}^{2}+td}{2}λ). Applying de Bruijn’s Tauberian theorem (Fact 2.2) gives that, as ε → 0,

ℙ( X(t) < ε | r_{1} ⩽ R(t) ⩽ r ) = exp( −(1 + o(1))\frac{(r_{1}^{2} + td)^{2}}{8ε} ).

Therefore,

ℙ( X(t) < ε, r_{1} ⩽ R(t) ⩽ r ) = exp( −(1 + o(1))\frac{(r_{1}^{2} + td)^{2}}{8ε} ).  (2.6)

Similarly, using (2.5) and Fact 2.2, we have that, for any y > 0, when ε → 0,

ℙ( X(t) < ε | R(t) = y ) = exp( −(1 + o(1))\frac{(y^{2} + td)^{2}}{8ε} ).  (2.7)

This estimate will be of use later.

The probability expression on the left-hand side of (2.1) can be written as

ℙ(X(t) < ε, r_{1} ⩽ R(t) ⩽ r) − ℙ(X(t) < θε, r_{1} ⩽ R(t) ⩽ r).

In view of (2.6), this yields Lemma 2.3. ⊔ ⊓

Lemma 2.4. For any x > 0,

ℙ^{x}( X(t) < ε ) = exp( −(1 + o(1))\frac{(x^{2} + td)^{2}}{8ε} ),  ε → 0.  (2.8)

Proof. Recall the probability transition density of R (see Revuz and Yor 1999, Chap. XI): for y > 0,

ℙ^{x}(R(t) ∈ dy) = \frac{y}{t} ( \frac{y}{x} )^{(d−2)/2} I_{(d−2)/2}( \frac{xy}{t} ) exp( −\frac{x^{2} + y^{2}}{2t} ) dy.  (2.9)

Writing

𝔼^{x}( e^{− \frac{λ^{2}}{2} X(t)} ) = ∫_{0}^{∞} ℙ^{x}(R(t) ∈ dy) 𝔼^{x}( e^{− \frac{λ^{2}}{2} X(t)} | R(t) = y ),

and using (2.2)–(2.4) and (2.9), we easily arrive at:

log 𝔼^{x}( e^{− \frac{λ^{2}}{2} X(t)} ) = −(1 + o(1))\frac{(x^{2} + td)λ}{2}, λ → ∞.

Lemma 2.4 now follows from an application of Fact 2.2. ⊔ ⊓ Lemma 2.5. Fix θ ∈ (0, 1), t > 0, ν > 1/2 and r > r_{1} > 0. Assuming

(r^{2} + td)^{2}θ < (r_{1}^{2} + td)^{2},  (2.10)

we have

lim_{ε→0} inf_{x∈[r_{1}, r]} ε log ℙ^{x}(θε ⩽ X(t) ⩽ ε, R(t) < ε^{ν}) = −\frac{(r^{2} + td)^{2}}{8} .

Proof. Recall that R(t) = ∥B(t)∥. Under ℙ^{x}, B can be realized as an ℝ^{d}-valued Brownian motion with ∥B(0)∥ = x. Define B_{0}(s) = B(s)− \frac{s}{t}B(t) (for s ∈ [0, t]), which is a Brownian bridge on [0, t], independent of B(t). Let R_{0}(s) = ∥B_{0}(s)∥. Note that {R_{0}(s); s ∈ [0, t]} is a so-called Bessel bridge starting from x (under ℙ^{x}), “conditioned to hit 0 at time t”, independent of R(t).

By the triangular inequality, |R(s) − R_{0}(s)| ⩽ R(t) for all s ∈ [0, t]. By Minkowski’s inequality, |\sqrt{X(t)} − (∫_{0}^{t} R_{0}^{2}(s) ds)^{1/2}| ⩽ \sqrt{t} R(t). Therefore, for any y > 0 and q > 0,

\begin{aligned}ℙ^{x}( X(t) ⩽ q | R(t) = y ) & ⩾ ℙ^{x}( ( ∫_{0}^{t} R_{0}^{2}(s) ds )^{1/2} ⩽ \sqrt{q} − \sqrt{t} y ) \\ & = ℙ(\sqrt{X(t)} ⩽ \sqrt{q} − \sqrt{t} y | R(t) = x), \end{aligned}

the last equality following from the time inversion: {R_{0}(t − s); s ∈ [0, t]} is again a Bessel bridge, starting from 0, conditioned to hit x at time t. Hence, by Fact 2.1, when ε is so small that \sqrt{t} ε^{ν} < \sqrt{ε},

inf_{x∈[r_{1}, r]} ℙ^{x}( X(t) ⩽ ε, R(t) < ε^{ν} )

= ℙ^{r}( X(t) ⩽ ε, R(t) < ε^{ν} )
⩾ ℙ^{r}(R(t) ⩽ ε^{ν}) ℙ(X(t) ⩽ (\sqrt{ε} − \sqrt{t} ε^{ν})^{2} | R(t) = r ).  (2.11)

By (2.7), when ε → 0, the second probability term on the right-hand side of (2.11) is exp(−(1 + o(1))(r^{2} + td)^{2}/8ε), whereas ℙ^{r}(R(t) < ε^{ν}) is of polynomial order of ε^{ν} (this is an immediate consequence of (2.9) and (2.3), or can be better understood using the fact that R^{2}(t) under ℙ^{r} has a non-centered chi-square law). Consequently,

lim inf_{ε→0} inf_{x∈[r_{1}, r]} ε log ℙ^{x}( X(t) ⩽ ε, R(t) < ε^{ν} ) ⩾ −\frac{(r^{2} + td)^{2}}{8} .  (2.12)

On the other hand, by Fact 2.1,

sup_{x∈[r_{1}, r]} ℙ^{x}(X(t) < θε, R(t) < ε^{ν}) ⩽ ℙ^{r_{1}}(X(t) < θε)
= exp( −(1 + o(1))\frac{(r_{1}^{2} + td)^{2}}{8θε} ),  (2.13)

the last identity following from (2.8). Combining (2.12) with (2.13), and in view of (2.10), we obtain the lower bound in the lemma.

The upper bound is easy. Indeed, just as for (2.13), we have

\begin{aligned}inf_{x∈[r_{1}, r]} ℙ^{x}(θε ⩽ X(t) ⩽ ε, R(t) < ε^{ν}) & ⩽ ℙ^{r}(X(t) ⩽ ε) \\ & = exp( −(1 + o(1))\frac{(r^{2} + td)^{2}}{8ε} ), \end{aligned}

as desired. ⊔ ⊓

Lemma 2.6. For r > 0 and t > 0,

lim sup_{ε→0} sup_{x∈[0,r]} ε log ℙ^{x}( X(t) < ε, sup_{0⩽s⩽t} R(s) ⩾ r ) ⩽ −\frac{(r^{2} + td)^{2}}{8} .  (2.14)

Proof. Let

H(r) = inf{t > 0 : R(t) > r}.  (2.15)

For each x ∈ [0, r], the probability term on the left-hand side of (2.14) is

⩽ ℙ^{x}( H(r) ⩽ t, X(t) − X(H(r)) < ε ) ⩽ ℙ^{r}(X(t) < ε),

the second inequality following from the strong Markov property. Now (2.14) follows from Lemma 2.4. ⊔ ⊓

Lemma 2.7. Let 0 < x_{1} < x_{2} < r_{1}^{∗} < r_{2}^{∗} and r_{1} < r_{2} < r_{1}^{∗} be such that

r_{2}^{∗} − r_{1}^{∗} > r_{1}^{∗} − r_{1}.  (2.16)

Then

lim inf_{t→0} inf_{x∈[x_{1},x_{2}]} t log ℙ^{x}( sup_{0⩽s⩽t} R(s) ∈ [r_{1}^{∗}, r_{2}^{∗}], R(t) ∈ [r_{1}, r_{2}] )

⩾ −(r_{1}^{∗} − x_{1})^{2} − (r_{1}^{∗} − r_{1})^{2}.

Proof. We first recall the tail estimate of R, which follows from a well-known tail estimate for general Gaussian processes (see for example Marcus and Shepp 1972): for r > 0,

ℙ( sup_{0⩽s⩽t} R(s) > r ) = exp( −(1 + o(1))\frac{r^{2}}{2t} ),  t → 0.  (2.17)

Now look at the probability

ℙ^{r_{1}^{∗}}(R(t) ∈ [r_{1}, r_{2}]) = ∫_{r_{1}}^{r_{2}} ℙ^{r_{1}^{∗}}(R(t) ∈ dy).

From (2.9) and (2.4), it is easily seen that for bounded (from above and below) y and r, and t ∈ (0, 1],

\frac{ℙ^{r}(R(t) ∈ dy)}{dy} ⩾ \frac{c_{1}}{\sqrt{t}} exp( −\frac{(r − y)^{2}}{2t} ),  (2.18)

where c_{1} > 0 is a constant depending only on d and on the bounds for (y, r). Thus, for t ∈ (0, 1],

ℙ^{r_{1}^{∗}}(R(t) ∈ [r_{1}, r_{2}]) ⩾ \frac{c_{1} (r_{2} − r_{1})}{\sqrt{t}} exp( −\frac{(r_{1}^{∗} − r_{1})^{2}}{2t} ).  (2.19)

On the other hand, under ℙ^{r_{1}^{∗}}, R is the modulus of an ℝ^{d}-valued Brownian motion B with ∥B(0)∥ = r_{1}^{∗}. By the triangular inequality,

ℙ^{r_{1}^{∗}}( sup_{0⩽s⩽t} R(s) > r_{2}^{∗} ) ⩽ ℙ( sup_{0⩽s⩽t} R(s) > r_{2}^{∗} − r_{1}^{∗} ),

which, according to (2.17), is exp(−(1+o(1))(r_{2}^{∗}−r_{1}^{∗})^{2}/2t), when t goes to 0. This, together with (2.19) and (2.16), implies that

ℙ^{r_{1}^{∗}}( sup_{0⩽s⩽t} R(s) ⩽ r_{2}^{∗}, R(t) ∈ [r_{1}, r_{2}] ) ⩾ exp( −(1 + o(1))\frac{(r_{1}^{∗} − r_{1})^{2}}{2t} ),  (2.20)

when t goes to 0.

Recall H(r_{1}^{∗}) from (2.15). For x ∈ [x_{1}, x_{2}], we have, by the strong Markov property,

ℙ^{x}( sup_{0⩽s⩽t} R(s) ∈ [r_{1}^{∗}, r_{2}^{∗}], R(t) ∈ [r_{1}, r_{2}] )

= ℙ^{x}( H(r_{1}^{∗}) ⩽ t, sup_{0⩽s⩽t} R(s) ⩽ r_{2}^{∗}, R(t) ∈ [r_{1}, r_{2}] )

= ∫_{0}^{t} ℙ^{x}(H(r_{1}^{∗}) ∈ du) ℙ^{r_{1}^{∗}}( sup_{0⩽s⩽t−u} R(s) ⩽ r_{2}^{∗}, R(t − u) ∈ [r_{1}, r_{2}] )

⩾ ∫_{0}^{t/2} ℙ^{x}(H(r_{1}^{∗}) ∈ du) ℙ^{r_{1}^{∗}}( sup_{0⩽s⩽t−u} R(s) ⩽ r_{2}^{∗}, R(t − u) ∈ [r_{1}, r_{2}] )

⩾ ∫_{0}^{t/2} ℙ^{x}(H(r_{1}^{∗}) ∈ du) exp( −(1 + o(1))\frac{(r_{1}^{∗} − r_{1})^{2}}{t} ),

the last equality following from (2.20), where o(1) is uniform in x ∈ [x_{1}, x_{2}]. Lemma 2.7 will be proved if we can verify

inf_{x∈[x_{1},x_{2}]} ℙ^{x}(H(r_{1}^{∗}) ⩽ t/2) ⩾ exp( −(1 + o(1))\frac{(r_{1}^{∗} − x_{1})^{2}}{t} ),  t → 0.  (2.21)

To this end, observe that for x ∈ [x_{1}, x_{2}],
ℙ^{x}(H(r_{1}^{∗}) ⩽ t/2) ⩾ ℙ^{x}(R(t/2) > r_{1}^{∗}) = ∫_{r_{1}^{∗}}^{∞} ℙ^{x}(R(t/2) ∈ dy).

By (2.18), there exists a constant c_{1} depending only on (r_{1}^{∗}, r_{2}^{∗}, x_{1}, x_{2}, d), such that for all x ∈ [x_{1}, x_{2}],

ℙ^{x}(H(r_{1}^{∗}) ⩽ t/2) ⩾ ∫_{r_{1}^{∗}}^{r_{2}^{∗}} ℙ^{x}(R(t/2) ∈ dy)

⩾ \frac{2c_{1}}{t} ∫_{r_{1}^{∗}}^{r_{2}^{∗}} exp( −\frac{(y − x)^{2}}{t} ) dy

⩾ \frac{2c_{1}}{t} ∫_{r_{1}^{∗}}^{r_{2}^{∗}} exp( −\frac{(y − x_{1})^{2}}{t} ) dy

= exp( −(1 + o(1))\frac{(r_{1}^{∗} − x_{1})^{2}}{t} ),

which yields (2.21), thus completes the proof of Lemma 2.7. ⊔ ⊓

Lemma 2.8. Let a ∈ (0, 1), b_{2} > b_{1} > δ > 0, ϱ > 1, ν > 1/2 and 0 < c < (ϱ − 1)/b_{2}. Assume

\sqrt{b_{2}} − \sqrt{b_{1}} > \sqrt{b_{1}} − \sqrt{b_{1} − δ}.

Let

E_{1}(ε) = { ε ⩽ X(1 + cε) ⩽ ϱε, R(1 + cε) < ε^{ν}, b_{1} ⩽ sup_{s∈[a,a+cε]} R^{2}(s) ⩽ b_{2},

sup_{s∈[a,a+cε]} R^{2}(s) ⩾ δ + sup_{s∈[0,1+cε]\[a,a+cε]} R^{2}(s) }.

Then

lim inf_{ε→0} ε log ℙ(E_{1}(ε)) ⩾ −\frac{(2(b_{1} − δ) + d)^{2}}{8} − \frac{5(\sqrt{b_{1}} − \sqrt{b_{1} − δ})^{2}}{c} .

Proof. Fix r ∈ (0, \sqrt{b_{1} − δ}) such that \sqrt{b_{2}} − \sqrt{b_{1}} > \sqrt{b_{1}} − r. Define

β = \frac{r^{2} + (1 − a)d}{2r^{2} + d}.  (2.22)

Since β ∈ (0, 1) and c < (ϱ − 1)/b_{2}, it is possible to choose 0 < β_{∗} < β < β^{∗} < 1 such that

1 − β_{∗} + b_{2} c + β^{∗} ⩽ ϱ.  (2.23)

Let r^{∗} ∈ (r, \sqrt{b_{1} − δ}). We also choose r_{∗} ∈ (0, r) so that

\sqrt{b_{1}} − r_{∗} ⩽ 2(\sqrt{b_{1}} − r),  (2.24)
\sqrt{b_{1}} − r_{∗} < \sqrt{b_{2}} − \sqrt{b_{1}} ,  (2.25)
(r^{2} + (1 − a)d)^{2} \frac{β}{β^{∗}} < (r_{∗}^{2} + (1 − a)d)^{2}.  (2.26)

Consider the measurable events

E_{2}(ε) = \{ (1 − β)ε ⩽ X(a) ⩽ (1 − β_{∗})ε, sup_{0⩽s⩽a} R(s) < \sqrt{b_{1} − δ} , r ⩽ R(a) ⩽ r^{∗} \},
E_{3}(ε) = \{ \sqrt{b_{1}} ⩽ sup_{a⩽s⩽a+cε} R(s) ⩽ \sqrt{b_{2}} , r_{∗} ⩽ R(a + cε) ⩽ r \},
E_{4}(ε) = { βε ⩽ X(1 + cε) − X(a + cε) ⩽ β^{∗}ε,

sup_{a+cε⩽s⩽1+cε} R(s) ⩽ \sqrt{b_{1} − δ} , R(1 + cε) < ε^{ν} \}.

Observe that on E_{3}(ε), we have X(a + cε) − X(a) ∈ [0, b_{2} c ε]. In view of (2.23), on E_{2}(ε) ∩ E_{3}(ε) ∩ E_{4}(ε), we have X(1 + cε) ∈ [ε, ϱε]. Thus

E_{1}(ε) ⊃ E_{2}(ε) ∩ E_{3}(ε) ∩ E_{4}(ε),

By the Markov property,

ℙ(E_{1}(ε)) ⩾ ℙ(E_{2}(ε) ∩ E_{3}(ε) ∩ E_{4}(ε))
⩾ ℙ(E_{2}(ε)) inf_{x∈[r,r^{∗}]} p_{1}(x, ε) inf_{y∈[r_{∗},r]} p_{2}(y, ε),  (2.27)

where, for x ∈ [r, r^{∗}] and y ∈ [r_{∗}, r],

\begin{aligned}p_{1}(x, ε) & = ℙ^{x}( \sqrt{b_{1}} ⩽ sup_{0⩽s⩽cε} R(s) ⩽ \sqrt{b_{2}} , r_{∗} ⩽ R(cε) ⩽ r ), \\ p_{2}(y, ε) & = ℙ^{y}( βε ⩽ X(1 − a) ⩽ β^{∗}ε, sup_{0⩽s⩽1−a} R(s) ⩽ \sqrt{b_{1} − δ} , R(1 − a) < ε^{ν} ). \end{aligned}

We now estimate ℙ(E_{2}(ε)), p_{1}(x, ε) and p_{2}(y, ε) respectively. First, since

\begin{aligned}ℙ(E_{2}(ε)) & ⩾ ℙ((1 − β)ε ⩽ X(a) ⩽ (1 − β_{∗})ε, r ⩽ R(a) ⩽ r^{∗}) \\ & − ℙ( X(a) ⩽ (1 − β_{∗})ε, sup_{0⩽s⩽a} R(s) ⩾ \sqrt{b_{1} − δ} ), \end{aligned}

it follows from Lemmas 2.3 and 2.6 that (noting that b_{1} − δ > r^{2})

ℙ(E_{2}(ε)) ⩾ exp( −(1 + o(1))\frac{(r^{2} + ad)^{2}}{8(1 − β_{∗})ε} ),  ε → 0.  (2.28)

It is easy to treat p_{1}(x, ε). Indeed, in view of (2.25), we can apply Lemma 2.7 to see that, when ε → 0,

inf_{x∈[r,r^{∗}]} p_{1}(x, ε) ⩾ exp( −(1 + o(1))\frac{(\sqrt{b_{1}} − r)^{2} + (\sqrt{b_{1}} − r_{∗})^{2}}{cε} ).  (2.29)

Finally, we observe that, for all y ∈ [r_{∗}, r],

p_{2}(y, ε) ⩾ inf_{y∈[r_{∗},r]} ℙ^{y}(βε ⩽ X(1 − a) ⩽ β^{∗}ε, R(1 − a) < ε^{ν})

− sup_{y∈[0,\sqrt{b_{1}−δ}]} ℙ^{y}(X(1 − a) ⩽ β^{∗}ε, sup_{0⩽s⩽1−a} R(s) > \sqrt{b_{1} − δ} ).

Recall (2.26). Applying Lemmas 2.5 and 2.6 yields (recalling b_{1} − δ > r^{2}) that, when ε → 0,

inf_{y∈[r_{∗},r]} p_{2}(y, ε) ⩾ exp( −(1 + o(1))\frac{(r^{2} + (1 − a)d)^{2}}{8β^{∗}ε} ).  (2.30)

Assembling (2.27)–(2.30) gives

lim inf_{ε→0} ε log ℙ(E_{1}(ε)) ⩾ −\frac{(r^{2} + ad)^{2}}{8(1 − β_{∗})} − \frac{(\sqrt{b_{1}} − r)^{2} + (\sqrt{b_{1}} − r_{∗})^{2}}{c} − \frac{(r^{2} + (1 − a)d)^{2}}{8β^{∗}}

⩾ −\frac{(r^{2} + ad)^{2}}{8(1 − β)} − \frac{5(\sqrt{b_{1}} − r)^{2}}{c} − \frac{(r^{2} + (1 − a)d)^{2}}{8β} ,

where in the last inequality, we have used β^{∗} > β > β_{∗} and the relation (2.24). In view of (2.22), we obtain:

lim inf_{ε→0} ε log ℙ(E_{1}(ε)) ⩾ −\frac{(2r^{2} + d)^{2}}{8} − \frac{5(\sqrt{b_{1}} − r)^{2}}{c} .

Since r can be as close to \sqrt{b_{1} − δ} as possible, this completes the proof of Lemma 2.8. ⊔ ⊓

Here is the main estimate which will be used in the proof of Theorem 1.1.

Lemma 2.9. Let a ∈ (0, 1), b_{2} > b_{1} > 0, ϱ > 1, ν > 1/2 and 0 < c < (ϱ − 1)/b_{2}. Let δ ∈ (0, b_{1}) be such that

\sqrt{b_{2}} − \sqrt{b_{1}} > \sqrt{b_{1}} − \sqrt{b_{1} − δ}.

Then

lim inf_{ε→0} ε log ℙ(E_{5}(ε)) ⩾ −\frac{(2(b_{1} − δ) + d)^{2}}{8} − \frac{5(\sqrt{b_{1}} − \sqrt{b_{1} − δ})^{2}}{c} ,  (2.31)

where

E_{5}(ε) = { ε ⩽ X(1) ⩽ ϱε, R(1) < ε^{ν}, b_{1} ⩽ sup_{s∈[0,1]} R^{2}(s) ⩽ b_{2},

sup_{s∈[a,a+cε]} R^{2}(s) ⩾ δ + sup_{s∈[0,1]\[a,a+cε]} R^{2}(s) }.

Proof. Let E_{1}(ε) be the measurable event defined in Lemma 2.8. Note that on E_{1}(ε), we have sup_{s∈[a,a+cε]} R^{2}(s) = sup_{s∈[0,1+cε]} R^{2}(s). By the scaling property,

ℙ(E_{1}(ε)) = ℙ( \frac{ε}{(1 + cε)^{2}} ⩽ X(1) ⩽ \frac{ϱε}{(1 + cε)^{2}} , R(1) < \frac{ε^{ν}}{(1 + cε)^{1/2}} ,

\frac{b_{1}}{1 + cε} ⩽ sup_{0⩽s⩽1} R^{2}(s) ⩽ \frac{b_{2}}{1 + cε},

sup_{s∈I(ε)} R^{2}(s) ⩾ \frac{δ}{1 + cε} + sup_{s∈[0,1]\I(ε)} R^{2}(s) ),

where I(ε) = [a/(1 + cε), (a + cε)/(1 + cε)]. Applying Lemma 2.8 gives (2.31). ⊔ ⊓

The proof of Theorem 1.2 requires a few more probability estimates. Lemma 2.10. For r > 0, ν > 1/2 and t > 0,

lim inf_{ε→0} inf_{x∈[0,r]} ε log ℙ^{x}( sup_{0⩽s⩽t} R(s) > r, X(t) < ε, R(t) < ε^{ν} )

⩾ −\frac{(3r^{2} + td)^{2}}{8} .

Proof. Let

p_{3}(x, ε) = ℙ^{x}( sup_{0⩽s⩽t} R(s) > r, X(t) < ε, R(t) < ε^{ν} ).

Recall the hitting time H(·) from (2.15). For any x ∈ [0, r] and any u ∈ [0, t],

\begin{aligned}p_{3}(x, ε) & ⩾ ℙ^{x}(H(r) ⩽ u, X(t) < ε, R(t) < ε^{ν}) \\ & ⩾ ℙ^{x}( H(r) ⩽ u, ∫_{H(r)}^{t} R^{2}(v) dv < ε − r^{2}u, R(t) < ε^{ν} ). \end{aligned}

By the strong Markov property, this leads to:

p_{3}(x, ε) ⩾ ∫_{0}^{u} ℙ^{x}(H(r) ∈ ds) ℙ^{r}(X(t − s) < ε − r^{2}u, R(t − s) < ε^{ν})

= ∫_{0}^{u} ℙ^{x}(H(r) ∈ ds) ℙ^{r/\sqrt{t−s}}( X(1) < \frac{ε − r^{2}u}{(t − s)^{2}} , R(1) < \frac{ε^{ν}}{\sqrt{t − s}} )

⩾ ℙ^{x}(H(r) ⩽ u) inf_{s∈[0,u]} ℙ^{r/\sqrt{t−s}}( X(1) < \frac{ε − r^{2}u}{(t − s)^{2}} , R(1) < \frac{ε^{ν}}{\sqrt{t − s}} )

⩾ ℙ^{x}(H(r) ⩽ u) inf_{y∈[r/\sqrt{t}, r/\sqrt{t−u}]} ℙ^{y}( X(1) < \frac{ε − r^{2}u}{t^{2}}, R(1) < \frac{ε^{ν}}{\sqrt{t}} ). (2.32)

This holds for all u ∈ [0, t]. We now choose u = χ ε, for some constant χ ∈ (0, r^{−2}) whose value will be determined later. In view of Fact 2.1, we have

inf_{x∈[0,r]} ℙ^{x}(H(r) ⩽ u) = ℙ(H(r) ⩽ u)

= ℙ( sup_{0⩽v⩽u} R(v) > r )

= exp( −(1 + o(1))\frac{r^{2}}{2χε} ),  ε → 0,  (2.33)

the last equality following from (2.17). On the other hand, Lemma 2.5 readily yields that, for our choice of u,

inf_{y∈[r/\sqrt{t}, r/\sqrt{t−u}]} ℙ^{y}( X(1) < \frac{ε − r^{2}u}{t^{2}}, R(1) < \frac{ε^{ν}}{\sqrt{t}} )

= exp( −(1 + o(1))\frac{(r^{2} + td)^{2}}{8(1 − r^{2}χ)ε} ).  (2.34)

Assembling (2.32)–(2.34) gives

lim inf_{ε→0} inf_{x∈[0,r]} ε log p_{3}(x, ε) ⩾ −\frac{r^{2}}{2χ} − \frac{(r^{2} + td)^{2}}{8(1 − r^{2}χ)},

for any χ ∈ (0, r^{−2}). Picking up χ = 2/(3r^{2} + td) completes the proof of Lemma 2.10. ⊔ ⊓

Now we are ready to establish the main probability estimate which will be of use in the proof of Theorem 1.2.

Lemma 2.11. Fix a ∈ (0, 1), r > 0, δ ∈ (0, r^{2}/3) and ν > 1/2. Let

E_{6}(ε) = { sup_{0⩽v⩽1} R(v) < r, X(1) < ε, R(1) < ε^{ν},

sup_{s∈[a,1]} R^{2}(s) > sup_{s∈[0,a]} R^{2}(s) + δ }.

Then

lim inf_{ε→0} ε log ℙ(E_{6}(ε)) ⩾ −\frac{(3δ + d)^{2}}{8} .

Proof. Let b ∈ (0, r^{2}/3 − δ) and c ∈ (0, 1). Note that

\begin{aligned}ℙ(E_{6}(ε)) & ⩾ ℙ( sup_{s∈[0,a]} R^{2}(s) < b, X(a) < cε, \\ & b + δ < sup_{s∈[a,1]} R^{2}(s) < r^{2}, X(1) − X(a) < (1 − c)ε, R(1) < ε^{ν} ). \end{aligned}

By the Markov property, this leads to:

ℙ(E_{6}(ε)) ⩾ p_{4}(ε) inf_{x∈[0,\sqrt{b}]} p_{5}(x, ε),  (2.35)

where

\begin{aligned}p_{4}(ε) & = ℙ( sup_{s∈[0,a]} R^{2}(s) < b, X(a) < cε ), \\ p_{5}(x, ε) & = ℙ^{x}( b + δ < sup_{s∈[0,1−a]} R^{2}(s) < r^{2}, X(1 − a) < (1 − c)ε, R(1 − a) < ε^{ν} ). \end{aligned}

By Lemmas 2.3 and 2.6, for any b_{1} ∈ (0, b),

p_{4}(ε) ⩾ ℙ( b_{1} ⩽ sup_{s∈[0,a]} R^{2}(s) ⩽ b, X(a) < cε ) ⩾ exp( −(1 + o(1))\frac{(b_{1} + ad)^{2}}{8cε} ),

as ε goes to 0. Therefore

p_{4}(ε) ⩾ exp( −(1 + o(1))\frac{(b + ad)^{2}}{8cε} ), ε → 0. (2.36)

On the other hand, observe that

inf_{x∈[0,\sqrt{b}]} p_{5}(x, ε) ⩾ inf_{x∈[0,\sqrt{b+δ}]} ℙ^{x}( sup_{s∈[0,1−a]} R^{2}(s) > b + δ,
X(1 − a) < (1 − c)ε, R(1 − a) < ε^{ν} )

− sup_{x∈(0,r]} ℙ^{x}( sup_{s∈[0,1−a]} R^{2}(s) ⩾ r^{2}, X(1 − a) < (1 − c)ε ).

Applying respectively Lemmas 2.10 and 2.6 to the two probability terms on the right hand side yields

inf_{x∈[0,\sqrt{b}]} p_{5}(x, ε) ⩾ exp( −(1 + o(1))\frac{(3(b + δ) + (1 − a)d)^{2}}{8(1 − c)ε} )

− exp( −(1 + o(1))\frac{(r^{2} + (1 − a)d)^{2}}{8(1 − c)ε} ).

Since 3(b + δ) < r^{2}, this leads to:

inf_{x∈[0,\sqrt{b}]} p_{5}(x, ε) ⩾ exp( −(1 + o(1))\frac{(3(b + δ) + (1 − a)d)^{2}}{8(1 − c)ε} ).  (2.37)

Combining (2.35)–(2.37) yields that

lim inf_{ε→0} ε log ℙ(E_{6}(ε)) ⩾ −\frac{(b + ad)^{2}}{8c} − \frac{(3(b + δ) + (1 − a)d)^{2}}{8(1 − c)} ,

for any c ∈ (0, 1) and b ∈ (0, r^{2}/3 − δ). Take c = (b + ad)/(4b + 3δ + d) and then send b to 0^{+} to complete the proof of Lemma 2.11. ⊔ ⊓

3. Brownian local time

Let {W (t); t ⩾ 0} be a one-dimensional Brownian motion starting from 0. Let {L(t, x); t ⩾ 0, x ∈ ℝ} denote its jointly continuous local time process, in the sense that for any Borel function f ⩾ 0 and t ⩾ 0,

∫_{0}^{t} f(W (s)) ds = ∫_{−∞}^{∞} f(x)L(t, x) dx.  (3.1)

For brevity, we write

L^{∗}(t) = sup_{x∈ℝ} L(t, x), t ⩾ 0.

For each r > 0, define

T_{r} = inf{t > 0 : W (t) > r},  (3.2)

the hitting time associated with W. Here is the key probability estimate in the proof of Theorem 1.1.

Lemma 3.1. Let a ∈ (0, 1), b_{2} > b_{1} > 0, ϱ > 1 and 0 < c < (ϱ − 1)/b_{2}. Let δ ∈ (0, b_{1}) be such that

\sqrt{b_{2}} − \sqrt{b_{1}} > \sqrt{b_{1}} − \sqrt{b_{1} − δ}.

We have,

lim inf_{ε→0} ε log ℙ(F_{1}(ε)) ⩾ −\frac{(b_{1} − δ + 1)^{2}}{2} − \frac{5(\sqrt{b_{1}} − \sqrt{b_{1} − δ})^{2}}{c} ,

where

\begin{aligned}F_{1}(ε) & = \{ ε ⩽ T_{1} ⩽ ϱε, b_{1} ⩽ L^{∗}(T_{1}) ⩽ b_{2}, \\ & sup_{x∈[a−cε,a]} L(T_{1}, x) > δ + sup_{x∉[a−cε,a]} L(T_{1}, x) \}. \end{aligned}

Proof. The Ray–Knight theorem (Ray 1963, Knight 1963) says that, {L(T_{1}, 1−x); x ⩾ 0} is a continuous inhomogeneous Markov process. More precisely, when x ∈ [0, 1], it is a two-dimensional squared Bessel process, starting from 0, and when x ⩾ 1, it becomes a squared Bessel process of “dimension 0”.

Let {R(t); t ⩾ 0} be a two-dimensional Bessel process with R(0) = 0, and {Z(t); t ⩾ 0} a Bessel process of dimension 0, with Z(0) = 1. We assume that R and Z are independent. As before, we write

X(t) = ∫_{0}^{t} R^{2}(s) ds, t ⩾ 0.

Observe that T_{1} = ∫_{0}^{∞} L(T_{1}, 1 − x) dx. By the Ray–Knight theorem and the scaling property for the Bessel process (writing x ∨ y for max(x, y)),

ℙ(F_{1}(ε)) = ℙ( ε ⩽ X(1) + R^{4}(1) ∫_{0}^{∞} Z^{2}(s) ds ⩽ ϱε,

b_{1} ⩽ sup_{0⩽s⩽1} R^{2}(s) ∨ R^{2}(1) sup_{s⩾0} Z^{2}(s) ⩽ b_{2},

sup_{s∈J(ε)} R^{2}(s) > δ + sup_{s∈[0,1]\J(ε)} R^{2}(s) ∨ R^{2}(1) sup_{s⩾0} Z^{2}(s) ),

where J(ε) = [1 − a, 1 − a + cε]. Let ϱ_{1} ∈ (1, ϱ) be such that c < (ϱ_{1} − 1)/b_{2}. For all sufficiently small ε > 0,

ℙ(F_{1}(ε)) ⩾ ℙ( ∫_{0}^{∞} Z^{2}(s) ds ⩽ 1, sup_{s⩾0} Z^{2}(s) ⩽ 2 ) ×

ℙ( R(1) < ε^{ν}, ε ⩽ X(1) ⩽ ϱ_{1}ε, b_{1} ⩽ sup_{0⩽s⩽1} R^{2}(s) ⩽ b_{2},

sup_{s∈J(ε)} R^{2}(s) > δ + sup_{s∈[0,1]\J(ε)} R^{2}(s) ).

The first probability term on the right-hand side is a positive constant. Lemma 3.1 now follows from Lemma 2.9. ⊔ ⊓

In the proof of Theorem 1.2, we need two other probability estimates for Brownian local time. Their proofs are in the same spirit as the proof of Lemma 3.1, based on the Ray–Knight theorem and using Lemmas 2.11 and 2.10 respectively in lieu of Lemma 2.9. The details are omitted.

Lemma 3.2. Let u ∈ (0, 1), b > 0 and δ ∈ (0, b/3). Then

lim inf_{ε→0} ε log ℙ( L^{∗}(T_{1}) < b, T_{1} < ε, sup_{x∈[0,u]} L(T_{1}, x) > sup_{x>u} L(T_{1}, x) + δ )

⩾ −\frac{(3δ + 2)^{2}}{8} .

Lemma 3.3. For b > 0,

lim inf_{ε→0} ε log ℙ( L^{∗}(T_{1}) > b, T_{1} < ε, inf_{t∈[0,T_{1}]} W (t) > −1 ) ⩾ −\frac{(3b + 2)^{2}}{8} .

4. Theorem 1.2 and Part (a) of Theorem 1.1

In this section, we prove Theorem 1.2 and Part (a) of Theorem 1.1. Let us first recall the following strong approximation theorem which holds simultaneously for local time and random walk.

Fact 4.1 (Révész 1990, pp. 105–107). After possible redefinitions of variables and processes, there exists a coupling for the simple random walk {S_{k}}_{k⩾0} and the Wiener process {W (t); t ⩾ 0}, such that for all ε > 0, as n goes to infinity,

sup_{x∈ℤ} max_{0⩽k⩽n} |ξ(k, x) − L(k, x)| = o(n^{1/4+ε}),  a.s.,  (4.1)
max_{0⩽k⩽n} |S_{k} − W (k)| = O(log n),  a.s.,  (4.2)

where ξ(n, x) and L(n, x) are defined in (1.1) and (3.1) respectively.

In the proof of Theorem 1.2 and of Part (a) of Theorem 1.1, we shall be working in the probability space such that the coupling for {S_{n}}_{n⩾0} and W in Fact 4.1 holds.

Proof of Theorem 1.2. Fix κ ∈ (0, 1), and define r_{n} = r_{n}(κ) = exp(n^{1+κ}). Let T be the first hitting time process for W, see (3.2).

Let u ∈ (0, 1/100) be such that

\frac{(u^{2} + 2(1 − u))^{2}}{4} + 49u < \frac{1 + 2κ}{1 + κ}.  (4.3)

Define

W^{(n)}(t) = W (t + T_{r_{n}}) − r_{n}, t ∈ [0, T_{r_{n+1}} − T_{r_{n}}].

By the strong Markov property, these are independent Brownian pieces. We can define T_{r}^{(n)} = inf{t > 0 : W^{(n)}(t) > r}, 0 < r ⩽ r_{n+1} − r_{n}, the first hitting time process for W^{(n)}, and also {L^{(n)}(t, x); 0 ⩽ t ⩽ T_{r_{n+1}−r_{n}}^{(n)}, x ∈ ℝ}, the local time of W^{(n)}.

Consider the following measurable events:

A_{1,n} = \{ sup_{x∈ℝ} L^{(n)}(T_{(1−u)r_{n+1}}^{(n)}, x) < ur_{n+1}, T_{(1−u)r_{n+1}}^{(n)} < \frac{(1 + 2κ)r_{n+1}^{2}}{2 log_{2} r_{n+1}} ,

sup_{x∈[0,ur_{n+1}]} L^{(n)}(T_{(1−u)r_{n+1}}^{(n)}, x) > sup_{x>ur_{n+1}} L^{(n)}(T_{(1−u)r_{n+1}}^{(n)}, x) + \frac{u^{2}r_{n+1}}{3} \},

A_{2,n} = { sup_{x∈ℝ}(L^{(n)}(T_{r_{n+1}−r_{n}}^{(n)}, x) − L^{(n)}(T_{(1−u)r_{n+1}}^{(n)}, x)) ⩾ 2ur_{n+1},

T_{r_{n+1}−r_{n}}^{(n)} − T_{(1−u)r_{n+1}}^{(n)} < \frac{(1 + 2κ)ur_{n+1}^{2}}{2 log_{2} r_{n+1}} ,

inf_{T_{(1−u)r_{n+1}}^{(n)} ⩽t⩽T_{r_{n+1}−rn}^{(n)}} W^{(n)}(t) > (1 − 2u)r_{n+1} }.

By the strong Markov property, {A_{1,n}, A_{2,n}}_{n⩾3} are mutually independent events. In particular,

ℙ(A_{1,n} ∩ A_{2,n}) = ℙ(A_{1,n})ℙ(A_{2,n}).  (4.4)

Since {W^{(n)}(t), t ∈ [0, T_{r_{n+1}} − T_{r_{n}}]} is again a Brownian motion,

ℙ(A_{1,n}) = ℙ( sup_{x∈ℝ} L(T_{(1−u)r_{n+1}}, x) < ur_{n+1}, T_{(1−u)r_{n+1}} < \frac{(1 + 2κ)r_{n+1}^{2}}{2 log_{2} r_{n+1}} ,

sup_{x∈[0,ur_{n+1}]} L(T_{(1−u)r_{n+1}}, x) > sup_{x>ur_{n+1}} L(T_{(1−u)r_{n+1}}, x) + \frac{u^{2}r_{n+1}}{3} )

= ℙ( sup_{x∈ℝ} L(T_{1}, x) < \frac{u}{1 − u}, T_{1} < \frac{1 + 2κ}{2(1 − u)^{2} log_{2} r_{n+1}} ,

sup_{x∈[0,u/(1−u)]} L(T_{1}, x) > sup_{x>u/(1−u)} L(T_{1}, x) + \frac{u^{2}}{3(1 − u)} ),

the second equality following from the scaling property. Applying Lemma 3.2 gives that, when n goes to infinity,

ℙ(A_{1,n}) ⩾ exp( −(1 + o(1))\frac{(u^{2} + 2(1 − u))^{2}}{4(1 + 2κ)} log_{2} r_{n+1} ).  (4.5)

Similarly,

ℙ(A_{2,n}) = ℙ( sup_{x∈ℝ} L(T_{ur_{n+1}−r_{n}}, x) ⩾ 2ur_{n+1}, T_{ur_{n+1}−r_{n}} < \frac{(1 + 2κ)ur_{n+1}^{2}}{2 log_{2} r_{n+1}} ,

inf_{0⩽t⩽T_{ur_{n+1}−rn}} W (t) > −ur_{n+1} )

= ℙ( sup_{x∈ℝ} L(T_{1}, x) ⩾ \frac{2ur_{n+1}}{ur_{n+1} − r_{n}} , T_{1} < \frac{(1 + 2κ)ur_{n+1}^{2}}{2(ur_{n+1} − r_{n})^{2} log_{2} r_{n+1}} ,

inf_{0⩽t⩽T_{1}} W (t) > −\frac{ur_{n+1}}{ur_{n+1} − r_{n}} )

⩾ ℙ( sup_{x∈ℝ} L(T_{1}, x) ⩾ 4, T_{1} < \frac{1 + 2κ}{2u log_{2} r_{n+1}} , inf_{0⩽t⩽T_{1}} W (t) > −1 ),

the last inequality following from the fact that ur_{n+1} − r_{n} ∈ [ur_{n+1}/2, ur_{n+1}] for all sufficiently large n. In light of Lemma 3.3, this implies

ℙ(A_{2,n}) ⩾ exp( −(1 + o(1))\frac{49u}{1 + 2κ} log_{2} r_{n+1} ).  (4.6)

In view of (4.3)–(4.6), we have ∑_{n} ℙ(A_{1,n} ∩ A_{2,n}) = ∞. By the Borel–Cantelli lemma, ℙ(A_{1,n} ∩ A_{2,n}, i.o.) = 1.

Now let us see what happens on A_{1,n} ∩ A_{2,n}. First, we have, on A_{1,n} ∩ A_{2,n},

T_{r_{n+1}−r_{n}}^{(n)} < \frac{(1 + 2κ)(1 + u)r_{n+1}^{2}}{2 log_{2} r_{n+1}} .  (4.7)

By definition, for r ∈ (0, r_{n+1} − r_{n}] and x ∈ ℝ,

T_{r}^{(n)} = T_{r+r_{n}} − T_{r_{n}},  (4.8)
L^{(n)}(T_{r}^{(n)}, x) = L(T_{r+r_{n}}, x + r_{n}) − L(T_{r_{n}}, x + r_{n}).  (4.9)

Writing

\begin{aligned}s_{n} & = (1 − u)r_{n+1} + r_{n} ∈ (r_{n}, r_{n+1}), \\ p_{n} & = ur_{n+1} + r_{n} ∈ (r_{n}, s_{n}), \end{aligned}

we obtain, on A_{1,n} ∩ A_{2,n},

sup_{y∈ℝ}(L(T_{s_{n}}, y) − L(T_{r_{n}}, y)) < ur_{n+1},  (4.10)
sup_{y∈[r_{n},p_{n}]}(L(T_{s_{n}}, y) − L(T_{r_{n}}, y)) > sup_{y>p_{n}}(L(T_{s_{n}}, y) − L(T_{r_{n}}, y)) + \frac{u^{2}r_{n+1}}{3} , (4.11)
sup_{y∈ℝ}(L(T_{r_{n+1}}, y) − L(T_{s_{n}}, y)) ⩾ 2ur_{n+1},  (4.12)
inf_{T_{sn} ⩽t⩽T_{r_{n+1}}} W (t) > (1 − 2u)r_{n+1} + r_{n}.  (4.13)

On the other hand, by (4.7) and (4.8), we have, on A_{1,n} ∩ A_{2,n},

T_{r_{n+1}} − T_{r_{n}} < \frac{(1 + 2κ)(1 + u)r_{n+1}^{2}}{2 log_{2} r_{n+1}} .  (4.14)

Since ℙ(A_{1,n} ∩ A_{2,n}, i.o.) = 1, we know that there are almost surely infinitely many n satisfying (4.10)–(4.14). We shall now only be considering these infinitely many n.

Since r ↦ T_{r} is a subordinator of index (1/2), we have (Fristedt 1973): for any ε ∈ (0, 1/100), almost surely for large r,

T_{r} < r^{2+ε}.  (4.15)

Let ξ be the local time of the random walk {S_{k}}_{k⩾0} satisfying (4.1) and (4.2). In view of (4.1), almost surely for large n, (4.10) implies (recalling ξ^{∗}(n) from (1.5))

ξ^{∗}(⌊T_{s_{n}}⌋) ⩽ ur_{n+1} + T_{s_{n}}^{1/4+ε} + ξ^{∗}(⌊T_{r_{n}}⌋) + T_{r_{n}}^{1/4+ε},

which, in light of (4.15) and (1.6), yields

ξ^{∗}(⌊T_{s_{n}}⌋) ⩽ (1 + ε)ur_{n+1},

whereas (4.12) implies

ξ^{∗}(⌊T_{r_{n+1}}⌋) ⩾ (2 − ε)ur_{n+1}.

By the definition of favourite sites, these two estimates imply that, U(⌊T_{r_{n+1}}⌋), the favourite site at time ⌊T_{r_{n+1}}⌋, must be visited by the random walk during ( ⌊T_{s_{n}}⌋, ⌊T_{r_{n+1}}⌋], i.e.,

U(⌊T_{r_{n+1}}⌋) = S_{j},  for some j = j_{n}(ω) ∈ ( ⌊T_{s_{n}}⌋, ⌊T_{r_{n+1}}⌋].  (4.16)

On the other hand, (4.13) and (4.2) together confirm that

S_{i} > (1 − 2u)r_{n+1},  for all i ∈ [ ⌊T_{s_{n}}⌋, ⌊T_{r_{n+1}}⌋].  (4.17)

Now consider (4.11). Again, by means of (4.1), (4.15) and (1.6), it follows from (4.11) that

max_{y∈[r_{n},p_{n}]} ξ(⌊T_{s_{n}}⌋, y) > max_{y>p_{n}} ξ(⌊T_{s_{n}}⌋, y) + (\frac{1}{3} − ε)u^{2}r_{n+1},

which, by definition, implies U(⌊T_{s_{n}}⌋) ⩽ p_{n}. Since p_{n} < (1 − 2u)r_{n+1}, comparing this with (4.16) and (4.17), we conclude that, there exists a random integer m = m_{n}(ω) ∈ [ ⌊T_{s_{n}}⌋, ⌊T_{r_{n+1}}⌋] such that U(m) = U(⌊T_{s_{n}}⌋) and that U(m + 1) > (1 − 2u)r_{n+1}. For this m,

U(m + 1) − U(m) > (1 − 2u)r_{n+1} − p_{n} ⩾ (1 − 4u)r_{n+1},  (4.18)

and, of course, m ∈ [T_{r_{n}} − 1, T_{r_{n+1}}].

To complete the proof of Theorem 1.2, it suffices to note that by (4.14) and (4.15), T_{r_{n+1}} < (1 + 3κ)(1 + u)r_{n+1}^{2}/2 log_{2} r_{n+1}, which implies

r_{n+1} > \sqrt{\frac{2T_{r_{n+1}} log_{2} T_{r_{n+1}}}{(1 + 4κ)(1 + u)}} ⩾ \sqrt{\frac{2m log_{2} m}{(1 + 4κ)(1 + u)}}.

Going back to (4.18), we see that

U(m + 1) − U(m) > (1 − 4u)\sqrt{\frac{2m log_{2} m}{(1 + 4κ)(1 + u)}}.

We have therefore proved that

lim sup_{n→∞} \frac{U(n + 1) − U(n)}{\sqrt{2n log_{2} n}} ⩾ \frac{1 − 4u}{\sqrt{(1 + 4κ)(1 + u)}}, a.s.

Since both u and κ can be as close to 0 as possible, this yields the lower bound in Theorem 1.2.

The upper bound is trivial. Indeed, U(n + 1) − U(n) ⩽ max_{0⩽k⩽n} S_{k} − min_{0⩽k⩽n} S_{k}, and the upper bound immediately follows from the usual LIL for the range of random walk (see p. 44 of Révész 1990). ⊔ ⊓

Proof of Theorem 1.1: Part (a). Again, let us fix κ ∈ (0, 1), and define r_{n} = exp(n^{1+κ}). Recall T from (3.2). Let a ∈ (0, 1) and b > 0.

Let ϱ > 1, b^{∗} > b and 0 < c < (ϱ − 1)/b^{∗}. We can choose δ ∈ (0, b) such that \sqrt{b^{∗}} − \sqrt{b} > \sqrt{b} − \sqrt{b − δ} and that

\frac{(b − δ + 1)^{2}}{2} + \frac{5(\sqrt{b} − \sqrt{b − δ})^{2}}{c} < (1 + κ)\frac{(b + 1)^{2}}{2} .  (4.19)

Fix γ > 0 and let

I_{n} = [ ar_{n} − \frac{cγr_{n}}{log_{2} r_{n}} , ar_{n} ],
A_{n} = \{ \frac{γr_{n}^{2}}{log_{2} r_{n}} ⩽ T_{r_{n}} ⩽ \frac{ϱγr_{n}^{2}}{log_{2} r_{n}} , br_{n} ⩽ L^{∗}(T_{r_{n}}) ⩽ b^{∗}r_{n},

sup_{x∈I_{n}} L(T_{r_{n}}, x) > δr_{n} + sup_{x∉I_{n}} L(T_{r_{n}}, x) }.

By scaling, Lemma 3.1 and (4.19), for all sufficiently large n,

ℙ(A_{n}) ⩾ exp( −(1 + κ)\frac{(b + 1)^{2} log_{2} r_{n}}{2γ} ).

Therefore, as long as γ ⩾ (1+κ)^{2}(b+1)^{2}/2, we have ∑_{n} ℙ(A_{n}) = ∞. Rigorously speaking, we cannot directly apply the Borel–Cantelli lemma here, since the events (A_{n}) are not independent. However, as in the proof of Theorem 1.2, we can consider the independent Brownian pieces W^{(n)} and construct events which are based on W^{(n)}, and use the same argument to get rid of the dependence difficulty. We choose to omit tedious discussions which are much alike to those in the proof of Theorem 1.2, and claim that ℙ(A_{n}, i.o.) = 1. Therefore, almost surely there exist infinitely many n such that simultaneously,

\frac{γr_{n}^{2}}{log_{2} r_{n}} ⩽ T_{r_{n}} ⩽ \frac{ϱγr_{n}^{2}}{log_{2} r_{n}} , br_{n} ⩽ L^{∗}(T_{r_{n}}) ⩽ b^{∗}r_{n},

sup_{x∈I_{n}} L(T_{r_{n}}, x) > δr_{n} + sup_{x∉I_{n}} L(T_{r_{n}}, x).

Applying Fact 4.1 implies that, infinitely often,

\frac{γr_{n}^{2}}{log_{2} r_{n}} ⩽ T_{r_{n}} ⩽ \frac{ϱγr_{n}^{2}}{log_{2} r_{n}} , (1 − κ)br_{n} ⩽ ξ^{∗}(⌊T_{r_{n}}⌋) ⩽ (1 + κ)b^{∗}r_{n}, U(⌊T_{r_{n}}⌋) ∈ I_{n}.

When γr_{n}^{2}/ log_{2} r_{n} ⩽ T_{r_{n}} ⩽ ϱγr_{n}^{2}/ log_{2} r_{n}, we have (recalling ϕ from (1.4))

\begin{matrix}br_{n} & ⩾ \frac{(1 + o(1))b}{\sqrt{2ϱγ}}ϕ(T_{r_{n}}), \\ b^{∗}r_{n} & ⩽ \frac{(1 + o(1))b^{∗}}{\sqrt{2γ}} ϕ(T_{r_{n}}), \\ I_{n} & ⊂ [ \frac{(1 + o(1))a}{\sqrt{2ϱγ}}ϕ(T_{r_{n}}), \frac{(1 + o(1))a}{\sqrt{2γ}}ϕ(T_{r_{n}}) ]. \end{matrix}

Thus, for any κ ∈ (0, 1), b^{∗} > b > 0, ϱ > 1 and γ ⩾ (1 + κ)^{2}(b + 1)^{2}/2, with probability one, the random sequence {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3} visits

[ \frac{(1 − κ)a}{\sqrt{2ϱγ}}, \frac{(1 + κ)a}{\sqrt{2γ}} ] × [ \frac{(1 − κ)^{2}b}{\sqrt{2ϱγ}}, \frac{(1 + κ)^{2}b^{∗}}{\sqrt{2γ}} ]

infinitely often. Take γ = (1 + κ)^{2}(b + 1)^{2}/2. Since (κ, b^{∗}, ϱ) can be as close to (0, b, 1) as possible, we conclude that (a/(b + 1), b/(b + 1)) is almost surely a limit point of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3}. For any x > 0 and y > 0 with x + y < 1, we can choose a ∈ (0, 1) and b > 0 such that (x, y) = (a/(b + 1), b/(b + 1)). Thus (x, y) is a limit point of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3}. Of course, the positivity of x is irrelevant, since when x < 0, we can consider {−S_{n}}_{n⩾0} instead of {S_{n}}_{n⩾0}.

As a consequence, any (x, y) ∈ A is almost surely a limit point of {(U(n)/ϕ(n), ξ^{∗}(n)/ϕ(n))}_{n⩾3}. We can interchange the order of “any (x, y) ∈ A” and “almost surely” by considering the points in A whose coordinates are rationals. ⊔ ⊓

5. Proof of Theorem 1.1: Part (b)

Part (b) of Theorem 1.1 is a consequence of the following:

Theorem 5.1. For any a ∈ (0, 1),

lim sup_{n→∞} sup_{x⩾aϕ(n)} \frac{ξ(n, x)}{ϕ(n)} = 1 − a, a.s.

More precisely, Part (b) of Theorem 1.1 follows from the upper bound in Theorem 5.1.

In view of (4.1), it is equivalent to state Theorem 5.1 as follows.

Theorem 5.2. Let {L(t, x); t ⩾ 0, x ∈ ℝ} be a Brownian local time process, and let a ∈ (0, 1). Then,

lim sup_{t→∞} sup_{x⩾aϕ(t)} \frac{L(t, x)}{ϕ(t)} = 1 − a, a.s.

The main ingredient in the proof of Theorem 5.2 is the following tail estimate:

Lemma 5.3. For a > 0 and b > 0,

lim_{t→∞} \frac{1}{log_{2} t} log ℙ( sup_{x⩾aϕ(t)} L(t, x) > bϕ(t) ) = −(a + b)^{2}.

The proof of Theorem 5.2 relies on Lemma 5.3 and the Borel–Cantelli lemma, along the same lines as in Section 4. No new ingredient is needed. So we feel free to omit the details. The rest of the section is devoted to the proof of Lemma 5.3.

Proof of Lemma 5.3. Let T be the first hitting time process of the Brownian motion whose local time is L, see (3.2). By the strong Markov property, for any u > 0 and v > 0,

ℙ( sup_{x⩾u} L(t, x) > v ) = ∫_{0}^{t} ℙ(T_{u} ∈ ds) ℙ( sup_{x⩾0} L(t − s, x) > v )

= ∫_{0}^{t} \frac{u ds}{\sqrt{2πs^{3}}} exp( −\frac{u^{2}}{2s} ) ℙ( sup_{x⩾0} L(1, x) > \frac{v}{\sqrt{t − s}} ).

It is known (see Kesten 1965, Csáki 1989) that

ℙ( sup_{x⩾0} L(1, x) > λ ) = exp( −(1 + o(1)) \frac{λ^{2}}{2} ), λ → ∞.

Therefore,

\begin{matrix}ℙ( sup_{x⩾aϕ(t)} L(t, x) > bϕ(t) ) \\ = ∫_{0}^{t} \frac{aϕ(t) ds}{\sqrt{2πs^{3}}} exp( −\frac{(aϕ(t))^{2}}{2s} − (1 + o(1))\frac{(bϕ(t))^{2}}{2(t − s)} ). \end{matrix}

This completes the proof of Lemma 5.3 by means of an elementary argument in the spirit of Laplace’s method (cf., e.g., Widder 1946). ⊔ ⊓

Acknowledgements

The authors are indebted to the referees and an associate editor for helpful remarks. The first and second author’s research was supported in part by the Hungarian National Foundation for Scientific Research, Grant No. T 019346 and T 029621. Cooperation between the authors was supported by the joint French–Hungarian Intergovernmental Grant “Balaton” (grant no. F25/97).

References

Abramowitz, M. and Stegun, I.A. (1965) Handbook of Mathematical Functions. New York: Dover.

Bass, R.F. and Griffin, P.S. (1985) The most visited site of Brownian motion and simple random walk. Z. Wahrsch. Verw. Gebiete, 70, 417–436.

Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge: Cambridge University Press.

Csáki, E. (1989) An integral test for the supremum of Wiener local time. Probab. Th. Rel. Fields, 83, 207–217.

Csáki, E. and Shi, Z. (1998) Large favourite sites of simple random walk and the Wiener process. Electronic J. Probab., 3, paper no. 14, pp. 1–31.

Eisenbaum, N. (1989) Temps locaux, excursions et lieu le plus visité par un mouvement brownien linéaire. Thèse de Doctorat de l’Université Paris VII.

Eisenbaum, N. (1997) On the most visited sites by a symmetric stable process. Probab. Th. Rel. Fields, 107, 527–535.

Erdős, P. and Révész, P. (1984) On the favourite points of a random walk. Mathematical Structures – Computational Mathematics – Mathematical Modelling, 2, pp. 152–157. Sofia.

Erdős, P. and Révész, P. (1987) Problems and results on random walks. In: Mathematical Statistics and Probability (P. Bauer et al., eds.), Proceedings of the 6th Pannonian Symposium, Vol. B, pp. 59–65. Dordrecht: Reidel.

Fristedt, B. (1973) Sample functions of stochastic processes with stationary independent increments. Adv. Probab., 3, 241–396. New York: Dekker.

Kesten, H. (1965) An iterated logarithm law for the local time. Duke Math. J., 32, 447–456.

Knight, F.B. (1963) Random walks and the sojourn density process of Brownian motion. Trans. Amer. Math. Soc., 109, 56–86.

Leuridan, C. (1997) Le point d’un fermé le plus visité par le mouvement brownien. Ann. Probab., 25, 953–996.

Marcus, M.B. and Shepp, L.A. (1972) Sample behavior of Gaussian processes. In: Proc. Sixth Berkeley Symp. Math. Statist. Probab., 2, pp. 423–441. Berkeley: University of California Press.

Pitman, J.W. and Yor, M. (1982) A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete, 59, 425–457.

Ray, D. (1963) Sojourn times of a diffusion process. Illinois J. Math., 7, 615–630.

Révész, P. (1990) Random Walk in Random and Non–Random Environments. Singapore: World Scientific.

Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd edn. Berlin: Springer.

Tóth, B. and Werner, W. (1997) Tied favourite edges for simple random walk. Combin. Probab. Comput., 6, 359–369.

Widder, D.V. (1941) The Laplace Transform. Princeton, N.J.: Princeton University Press.