MODERATE AND SMALL DEVIATIONS FOR THE RANGES OF ONE-DIMENSIONAL RANDOM WALKS

Xia Chen

We establish the moderate and small deviations for the ranges of the integer valued random walks. Our theorems apply to the limsup and the liminf laws of the iterated logarithm.

AMS 2000 Subject classifications: 60D05, 60F10, 60F15, 60G50.

Key words: range, intersection of ranges, random walks, moderate deviation, small deviation, law of the iterated logarithm.

Short title: Moderate and small deviations for the ranges

Research partially supported by NSF grant DMS-0405188. Address: Department of Mathemathics, University of Tennessee, Knoxville, TN 37996, USA e-mail: xchen@math.utk.edu URL: http://www.math.utk.edu/˜ xchen 1. Introduction.

Let {S(n)}_{n≥0} be an integer valued random walk. That is,

S(0) = 0 and S(n) = X(1) + ⋯ + X(n) n = 1, 2, ⋯

where {X; X(k); k ≥ 1} is an i.i.d. integer valued sequence. Throughout we assume that

𝔼 X = 0 and σ^{2} ≡ 𝔼 X^{2} < ∞.  (1.1)

We also assume that the smallest group that supports {S(n)}_{n≥0} is ℤ.

For any Δ ⊂ ℝ^{+}, we set

S(Δ) = {S(k); k ∈ Δ}.

Let {S_{1}(n)}_{n≥0}, ⋯, {S_{p}(n)}_{n≥0} be independent copies of {S(n)}_{n≥0}. In this paper we study the random sequences

#{S[0, n]} and #{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]} n = 1, 2, · · · .

This work appears as a part of recent effort made by the author and his coauthors ([1]-[3], [5]-[9]) in the study of exponential asymptotics arising from sample path intersections.

To simplify our notation we allow p = 1, in which case the range intersection S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n] is identified with the range S[0, n].

Throughout, {b_{n}} is a positive sequence satisfying

b_{n} ⟶ ∞ and b_{n} = o(n) (n → ∞).  (1.2)

Theorem 1. For any λ > 0 and integer p ≥ 1,

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} = −\frac{pλ^{2}}{2σ^{2}} .  (1.3)

Taking p = 1 in Theorem 1 we have

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S[0, n]\} ≥ λ\sqrt{nb_{n}}\} = −\frac{λ^{2}}{2σ^{2}} .  (1.4)

We point out the classic fact that the random walk S(n) satisfies the same moderate deviation under stronger moment condition. A striking fact about Theorem 1 is that the moderate deviation holds for the range and the intersection of the ranges under no moment assumption rather than (1.1). Theorem 2. For any λ > 0,

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} = −\frac{π^{2}σ^{2}}{2λ^{2}} .  (1.5)

The small deviation of this type does not hold for intersection of the ranges. Indeed, the recent paper by Kleke and Mörters (2004) on the lower tail of the intersections of the Brownian paths suggests that the probability

ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \}

does not have exponential decay as p > 1.

Theorem 1 and Theorem 2 apply, respectively, to the limsup and the liminf laws of the iterated logarithm given in the following theorem.

Theorem 3. For any integer p ≥ 1,

lim sup_{n→∞} \frac{1}{\sqrt{n log log n}}#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} = \sqrt{\frac{2}{p}}σ a.s.  (1.6)

In the case p = 1,

lim inf_{n→∞} \sqrt{\frac{log log n}{n}}#\{S[0, n]\} = \frac{π}{\sqrt{2}}σ a.s.  (1.7)

The study shows that the asymptotic behaviors of the range and of the intersection of the ranges have strong dimension dependence. It was proved in Dvoretzky and Erdös (1951) that the range #{S[0, n]} of a d-dimension centered, square integrable lattice-valued random walk {S(n)}_{n≥0} satisfies the strong law of large numbers as d ≥ 2, contrary to what is given in Theorem 3. The central limit theorem shows that #{S[0, n]} − 𝔼 #{S[0, n]} is asymptotically normal as d ≥ 3 (Jain and Pruitt (1971, 1974)), and is attracted by the renormalized self-intersection local time of a planar Brownian motion as d = 2 (Le Gall (1986a)). See Le Gall and Rosen (1991) for a central limit theorem when the random walk is in the domain of attraction of a stable law. The LIL for #{S[0, n]} − 𝔼 #{S[0, n]} can be found in Jain and Pruitt (1972) and Hamana (1998) as d ≥ 4; and in Bass and Kumagai (2002) as d = 2, 3.

As for the intersection of the independent ranges, it is well known (Dvoretzky and Erdös (1951), (1954)) that p independent paths of the centered and squared integrable ℤ^{d}-valued random walks {S_{1}(n)}_{n≥0}, ⋯, {S_{p}(n)}_{n≥0} intersect infinitely often if and only if p(d − 2) ≤ d. Under this condition, the laws of weak convergence were established in Le Gall (1986a, 1986b). The moderate deviations for #{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]}, or the related laws of the iterated logarithm, can be found in Marcus and Rosen (1997) for d = 4 and p = 2; in Rosen (1997) for d = p = 3; and in the author’s recent paper (Chen (2004)) for d = 2 and p ≥ 2 and for d = 3 and p = 2.

The case d = 1 is radically different from the multi-dimensional case where the range and the intersection of the independent ranges have different orders of upper tail asymptotics. While the asymptotic behaviors of the range are closely related to that of the intersection local time as d ≥ 2, the 1-dimensional case is largely determined by the relation

S[0, n] ⊂ [ min_{0≤k≤n} S(k), max_{0≤k≤n} S(k)]  (1.8)

where, as a convention in this paper, the right hand side represents a interval containing only integers. Indeed, the equality holds in the case of the simple random walks. In the general case, it is known (Le Gall (1986a)) that the possible “holes” within the interval on the right hand side of (1.8) are insignificant in the sense of weak convergence. The case of the moderate deviations appears to be more delicate. Indeed, the moderate deviation for the connected range max_{0≤k≤n} S(k) − min_{0≤k≤n} S(k) can fail without assuming an extra moment condition as strong as exponential integrability. This fact suggests that in general, the range is not exponentially close to the connected range.

2. Upper bound in Theorem 1.

In this section we prove that

lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} ≤ −\frac{pλ^{2}}{2σ^{2}} .  (2.1)

By Chebyshev inequality, we only need to show that

lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\}\} ≤ \frac{σ^{2}θ^{2}}{2p} .  (2.2)

To this end we first prove Lemma 2.1. For any θ > 0,

sup_{n} 𝔼 exp \{\frac{θ}{\sqrt{n}}#\{S[0, n]\}\} < ∞.  (2.3)

Proof. Recall ((4.8), Chen (2004)) that for any a, b > 0,

ℙ{#{S[0, n]} ≥ a + b} ≤ ℙ{#{S[0, n]} ≥ a}ℙ{#{S[0, n]} ≥ b}.

We therefore have that for any C > 0 and the integer m ≥ 1,

ℙ\{#\{S[0, n]\} ≥ Cm\sqrt{n}\} ≤ (ℙ\{#\{S[0, n]\} ≥ C\sqrt{n}\})^{m}.

By the fact that 𝔼 #\{S[0, n]\} = O(\sqrt{n}) one can take C > 0 large enough so that

sup_{n} ℙ\{#\{S[0, n]\} ≥ C\sqrt{n}\} ≤ e^{−2}.

Therefore, (2.3) holds for θ = C^{−1}. We now show that it holds for all θ > 0. Indeed, take δ > 0 such that δ < (Cθ)^{−2} and write k_{n} = [δn]. The desired conclusion follows from the following estimate:

𝔼 exp \{\frac{θ}{\sqrt{n}}#\{S[0, n]\}\} ≤ (𝔼 exp \{\frac{θ}{\sqrt{n}}#\{S[0, k_{n}]\}\})^{[δ^{−1}]+1}

≤ (𝔼 exp \{\frac{C^{−1}}{\sqrt{k_{n}}} #\{S[0, k_{n}]\}\})^{[δ^{−1}]+1}.

We first establish (2.2) in the case p = 1. Let t > 0 be arbitrary but fixed and write t_{n} = [tn/b_{n}] and γ_{n} = [n/t_{n}]. Notice that

#{S[0, n]} ≤ ∑_{i=1}^{γ_{n}+1} #{S[(i − 1)t_{n}, it_{n}]}

and that the sequence

#{S[(i − 1)t_{n}, it_{n}]} i = 1, ⋯, γ_{n} + 1

is an i.i.d. So for any θ > 0,

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S[0, n]\}\} ≤ (𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S[0, t_{n}]\}\})^{γ_{n}+1}.

According to Theorem 4.1 in Le Gall (1986a),

\sqrt{\frac{b_{n}}{n}} #\{S[0, t_{n}]\} ⟶^{d} σ\sqrt{t}( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s))  (2.4)

as n → ∞, where W (s) is an one dimensional Brownian motion. By Lemma 2.1 and the dominated convergence theorem,

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S[0, t_{n}]\}\} ⟶ 𝔼 exp \{σθ\sqrt{t}( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s))\} (n → ∞).

Hence,

lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S[0, n]\}\}

≤ \frac{1}{t} log 𝔼 exp \{σθ\sqrt{t}( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s))\}.

(2.5)

Let

H = {x ∈ C_{0}[0, 1]; x(s) is absolutely continuous on [0, 1] and ∫_{0}^{1} |ẋ(s)|^{2}ds < ∞}.

By the large deviation principle for Brownian motions (Theorem 5.2.3 in Dembo and Zeitouni (1998)),

lim_{t→∞} \frac{1}{t} log 𝔼 exp \{σθ\sqrt{t}( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s))\}

= sup_{x∈H} \{σθ( max_{0≤s≤1} x(s) − min_{0≤s≤1} x(s)) − \frac{1}{2} ∫_{0}^{1} |ẋ(s)|^{2}ds\}.

(2.6)

Applying the mean value theorem and Cauchy’s inequality gives

max_{0≤s≤1} x(s) − min_{0≤s≤1} x(s) ≤ ( ∫_{0}^{1} |ẋ(s)|^{2}ds)^{1/2}.

Therefore, the right hand side of (2.6) is bounded by

sup_{λ>0} \{σθλ − \frac{1}{2}λ^{2}\} = \frac{1}{2}σ^{2}θ^{2}.

Letting t → ∞ in (2.5) we have

lim sup_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S[0, n]\}\} ≤ \frac{1}{2}σ^{2}θ^{2}.  (2.7)

We now prove (2.2) in the case p > 1. By the fact

#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} = ∑_{x∈ℤ} ∏_{j=1}^{p} 1_{\{x∈S_{j}[0,n]\}} ≤ ∑_{x∈ℤ} \frac{1}{p} ∑_{j=1}^{p} 1_{\{x∈S_{j}[0,n]\}} = \frac{1}{p} ∑_{j=1}^{p} #\{S_{j}[0, n]\}

and by independence,

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} #\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\}\} ≤ (𝔼 exp \{\frac{θ}{p}\sqrt{\frac{b_{n}}{n}} #\{S[0, n]\}\})^{p}.

By (2.7) (with θ being replaced by θp^{−1}) we have (2.2).

3. Lower bound in Theorem 1.

In this section we prove that

lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} ≥ −\frac{pλ^{2}}{2σ^{2}} .  (3.1)

Let M > 0 be a deterministic number and write S̄(0) = 0 and S̄(n) = X(1)1_{{|X(1)|≤M}} + ⋯ + X(n)1_{{|X(n)|≤M}} n = 1, 2, · · · .

We may define the independent copies {X_{1}(k)}_{k≥1}, ⋯, {X_{p}(k)}_{k≥1} of {X(k)}_{k≥1} and let

S̄_{j}(n) = X_{j}(1) + ⋯ + X_{j}(n) n = 1, 2, ⋯

S̄_{j}(0) = 0 and S̄_{j}(n) = X_{j}(1)1_{{|X_{j}(1)|≤M}} + ⋯ + X_{j}(n)1_{{|X_{j}(n)|≤M}} n = 1, 2, ⋯

for each j = 1, ⋯, p. Lemma 3.1. For any t > 0 and n ≥ 1,

\begin{matrix}ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ t\} \\ ≥ (1 + ℙ\{|X| > M\})^{−np}ℙ\{#\{S̄_{1}[0, n] ∩ ⋯ ∩ S̄_{p}[0, n]\} ≥ t\}. \end{matrix}

Proof. Write [1, n] = {1, ⋯ n}. For any A ⊂ [1, n], let {k_{1}, ⋯, k_{n−|A|}} = [1, n] \ A and for any 1 ≤ j ≤ p, write

S_{j}^{A}(0) = 0 and S_{j}^{A}(i) = X_{j}(k_{1}) + ⋯ X_{j}(k_{i}) i = 1, ⋯, n − |A|

where |A| = #(A). We have ℙ{#{S̄_{1}[0, n] ∩ ⋯ ∩ S̄_{p}[0, n]} ≥ t}

= ∑_{A_{1},···,A_{p}⊂[1,n]} ℙ{|X_{1}(l_{1})| > M, ⋯, |X_{p}(l_{p})| > M ∀(l_{1}, ⋯, l_{p}) ∈ A_{1} × ⋯ × A_{p};

|X_{1}(m_{1})| ≤ M, ⋯, |X_{p}(m_{p})| ≤ M ∀(m_{1}, ⋯, m_{p}) ∈ ([1, n] \ A_{1}) × ⋯ × ([1, n] \ A_{1});

#{S_{1}^{A_{1}}[0, n − |A_{1}|] ∩ ⋯ ∩ S_{p}^{A_{p}}[0, n − |A_{p}|]} ≥ t}

≤ ∑_{A_{1},···,A_{p}⊂[1,n]} ℙ{|X_{1}(l_{1})| > M, ⋯, |X_{p}(l_{p})| > M ∀(l_{1}, ⋯, l_{p}) ∈ A_{1} × ⋯ × A_{p};

#{S_{1}^{A_{1}}[0, n − |A_{1}|] ∩ ⋯ ∩ S_{p}^{A_{p}}[0, n − |A_{p}|]} ≥ t}

= ∑_{A_{1},···,A_{p}⊂[1,n]} (ℙ{|X| > M})^{|A_{1}|+···+|A_{p}|}ℙ{#{S_{1}[0, n − |A_{1}|] ∩ ⋯ ∩ S_{p}[0, n − |A_{p}|]} ≥ t}

≤ (1 + ℙ{|X| > M})^{np}ℙ{#{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]} ≥ t}.

By the assumption (1.1), we can find a positive deterministic sequence {α_{n}} such that

α_{n} ⟶ 0, \frac{n}{b_{n}} ℙ\{|X| > α_{n}\sqrt{\frac{n}{b_{n}}} \} ⟶ 0, \sqrt{\frac{n}{b_{n}}} 𝔼 |X|1_{\{|X|>α_{n}\sqrt{nb_{n}^{−1}}\}} ⟶ 0  (3.2)

as n → ∞. In the rest of this section, we take M = M_{n} = α_{n}\sqrt{nb_{n}^{−1}} and let the random walks S̄(k); S̄_{1}(k), ⋯, S̄_{p}(k) be defined as before. By Lemma 3.1 and (3.2),

\begin{matrix}ℙ\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} ≥ \\ ≥ \{1 + o(\frac{b_{n}}{n} )\}^{−np}ℙ\{#\{S̄_{1}[0, n] ∩ ⋯ ∩ S̄_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\}. \end{matrix}

To prove (3.1), therefore, we need only to establish that

lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S̄_{1}[0, n] ∩ ⋯ ∩ S̄_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} ≥ −\frac{pλ^{2}}{2σ^{2}} .

Write t_{n} = [n/b_{n}] and γ_{n} = [n/t_{n}]. By the fact that γ_{n}t_{n} ≤ n, it suffices to show

lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S̄_{1}[0, γ_{n}t_{n}] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}]\} ≥ λ\sqrt{nb_{n}}\} ≥ −\frac{pλ^{2}}{2σ^{2}} .  (3.3)

Notice that

[ max_{1≤j≤p} min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k), min_{1≤j≤p} max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k)] \ (S̄_{1}[0, γ_{n}t_{n}] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}])

⊂ ⋃_{j=1}^{p} ([ min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k), max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k)] \ S̄_{j}[0, n]).

Consequently,

( min_{1≤j≤p} max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) − max_{1≤j≤p} min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k)) − #{S̄_{1}[0, γ_{n}t_{n}] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}]}

≤ ∑_{j=1}^{p} ({ max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) − min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k)} − #{S̄_{j}[0, γ_{n}t_{n}]}).

Hence, ℙ\{#\{S̄_{1}[0, γ_{n}t_{n}|] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}]\} ≥ λ\sqrt{nb_{n}}\}

+ pℙ\{\{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\} ≥ p^{−1}ϵ\sqrt{nb_{n}}\}

≥ ℙ\{ min_{1≤j≤p} max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) − max_{1≤j≤p} min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) ≥ (λ + ϵ)\sqrt{nb_{n}}\}.

By the fact that

min_{1≤j≤p} max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) − max_{1≤j≤p} min_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) ≥ min_{1≤j≤p} max_{0≤k≤γ_{n}t_{n}} S̄_{j}(k) ≥ min_{1≤j≤p} S̄_{j}(γ_{n}t_{n})

we have ℙ\{#\{S̄_{1}[0, γ_{n}t_{n}|] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}]\} ≥ λ\sqrt{nb_{n}}\}

+ pℙ\{\{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\} ≥ p^{−1}ϵ\sqrt{nb_{n}}\}

≥ (ℙ\{S̄(γ_{n}t_{n}) ≥ (λ + ϵ)\sqrt{nb_{n}}\})^{p}.

Hence,

max \{ lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S̄_{1}[0, γ_{n}t_{n}] ∩ ⋯ ∩ S̄_{p}[0, γ_{n}t_{n}]\} ≥ λ\sqrt{nb_{n}}\},

lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{\{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\} ≥ p^{−1}ϵ\sqrt{nb_{n}}\}\}

≥ p lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{S̄(γ_{n}t_{n}) ≥ (λ + ϵ)\sqrt{nb_{n}}\}.

(3.4)

Notice that for any real number θ,

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (S̄(γ_{n}t_{n}) − 𝔼 S̄(γ_{n}t_{n}))\}

= (𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (X1_{\{|X|≤M_{n}\}} − 𝔼 X1_{\{|X|≤M_{n}\}})\})^{γ_{n}t_{n}}

= \{1 + \frac{b_{n}}{2n}σ^{2}θ + o(\frac{b_{n}}{n} )\}^{γ_{n}t_{n}} (n → ∞)

where the last step follows from (3.2) and Taylor’s expansion. Therefore,

lim_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (S̄(γ_{n}t_{n}) − 𝔼 S̄(γ_{n}t_{n}))\} = \frac{1}{2}σ^{2}θ^{2}.

By Gärtner-Ellis theorem (Theorem 2.3.6 in Dembo and Zeitouni (1998)),

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{S̄(γ_{n}t_{n}) − 𝔼 S̄(γ_{n}t_{n}) ≥ λ\sqrt{nb_{n}}\}

= − sup_{θ} \{λθ − \frac{1}{2}σ^{2}θ^{2}\} = −\frac{λ^{2}}{2σ^{2}} (λ > 0).

(3.5)

On the other hand, from (3.2) we have

\begin{matrix}|𝔼 S̄(γ_{n}t_{n})| = n|𝔼 X1_{\{|X|≤α_{n}\sqrt{nb_{n}^{−1}}\}}| \\ = n|𝔼 X1_{\{|X|>α_{n}\sqrt{nb_{n}^{−1}}\}}| ≤ n𝔼 |X|1_{\{|X|>α_{n}\sqrt{nb_{n}^{−1}}\}} = o(\sqrt{nb_{n}}) (n → ∞). \end{matrix}

Hence, (3.5) remains true as S̄(γ_{n}t_{n}) − 𝔼 S̄(γ_{n}t_{n}) is replaced by S̄(γ_{n}t_{n}). In particular,

lim_{n→∞} \frac{1}{b_{n}} log ℙ\{S̄(γ_{n}t_{n}) ≥ (λ + ϵ)\sqrt{nb_{n}}\} = −\frac{(λ + ϵ)^{2}}{2σ^{2}} .  (3.6)

In view of (3.4) and (3.6), we will have (3.3) if

lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{\{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\} ≥ ϵ\sqrt{nb_{n}}\} = −∞ (3.7)

holds for any ϵ > 0.

Indeed,

[ min_{0≤k≤γ_{n}t_{n}} S̄(k), max_{0≤k≤γ_{n}t_{n}} S̄(k)] \ S̄[0, γ_{n}t_{n}]

⊂ ⋃_{i=1}^{[b_{n}]} ([ min_{(i−1)γ_{n}≤k≤iγ_{n}} S̄(k), max_{(i−1)γ_{n}≤k≤iγ_{n}} S̄(k)] \ S̄[(i − 1)γ_{n}, iγ_{n}]).

Consequently,

{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)} − #{S̄[0, γ_{n}t_{n}]}

≤ ∑_{i=1}^{γ_{n}} ({ max_{(i−1)t_{n}≤k≤it_{n}} S̄(k) − min_{(i−1)t_{n}≤k≤it_{n}} S̄(k)} − #{S̄[(i − 1)t_{n}, it_{n}]})

= ∑_{i=1}^{γ_{n}} ξ_{i} (say).

Notice that ξ_{1}, ⋯ ξ_{γ_{n}} are i.i.d. non-negative random variables. For any θ > 0,

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} \{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\}\}

≤ (𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (\{ max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)\} − #\{S̄[0, t_{n}]\})\})^{γ_{n}}.

(3.8)

By the fact that

#{S[0, t_{n}]} ≤ max_{0≤k≤t_{n}} S(k) − min_{0≤k≤t_{n}} S(k),

by the invariant principle

\sqrt{\frac{b_{n}}{n}} 𝔼 ( max_{0≤k≤t_{n}} S(k) − min_{0≤k≤t_{n}} S(k)) ⟶ 𝔼 ( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s)) (n → ∞)

and by (4.g) in Le Gall (1986a) which leads to

\sqrt{\frac{b_{n}}{n}} 𝔼 #\{S[0, t_{n}]\} ⟶ 𝔼 ( max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s)) (n → ∞),

we must have

\sqrt{\frac{b_{n}}{n}} (\{ max_{0≤k≤t_{n}} S(k) − min_{0≤k≤t_{n}} S(k)\} − #\{S[0, t_{n}]\}) ⟶^{P} 0.

Notice that for any ϵ > 0,

ℙ\{\{ max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)\} − #\{S̄[0, t_{n}]\} ≥ ϵ\sqrt{\frac{n}{b_{n}}} \}

≤ ℙ\{\{ max_{0≤k≤t_{n}} S(k) − min_{0≤k≤t_{n}} S(k)\} − #\{S[0, t_{n}]\} ≥ ϵ\sqrt{\frac{n}{b_{n}}} \} + \frac{n}{b_{n}} ℙ\{|X| > α_{n}\sqrt{\frac{n}{b_{n}}} \}.

In view of (3.2) we obtain

\sqrt{\frac{b_{n}}{n}} (\{ max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)\} − #\{S̄[0, t_{n}]\}) ⟶^{P} 0.  (3.9)

By the facts that |X| ≤ C\sqrt{n/b_{n}} and that \sqrt{b_{n}/n}S̄(t_{n}) is stochastically bounded, we have (Theorem 2.5, de Acosta (1980)) that

sup_{n} 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} max_{0≤k≤t_{n}} |S̄(k)|\} < ∞

for all θ > 0. This, combined with (3.9), the fact that

0 ≤ { max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)} − #{S̄[0, t_{n}]}

≤ { max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)} ≤ 2 max_{0≤k≤t_{n}} |S̄(k)|

and the dominated convergence theorem, implies that

𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} (\{ max_{0≤k≤t_{n}} S̄(k) − min_{0≤k≤t_{n}} S̄(k)\} − #\{S̄[0, t_{n}]\})\} ⟶ 1 (n → ∞).

By (3.8), therefore,

lim_{n→∞} \frac{1}{b_{n}} log 𝔼 exp \{θ\sqrt{\frac{b_{n}}{n}} \{ max_{0≤k≤γ_{n}t_{n}} S̄(k) − min_{0≤k≤γ_{n}t_{n}} S̄(k)\} − #\{S̄[0, γ_{n}t_{n}]\}\} = 0

for every θ > 0.

Finally, (3.7) follows from a standard application of Chebyshev’s inequality.

4. Proof of Theorem 2.

By the fact that

#{S[0, n]} ≤ max_{0≤k≤n} S(k) − min_{0≤k≤n} S(k) ≤ 2 max_{0≤k≤n} |S(k)|

we have

ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ≥ ℙ\{ max_{0≤k≤n} |S(k)| ≤ \frac{λ}{2}\sqrt{\frac{n}{b_{n}}} \}.

According to Theorem 4.5 in de Acosta (1983),

lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{ max_{0≤k≤n} |S(k)| ≤ \frac{λ}{2}\sqrt{\frac{n}{b_{n}}} \} ≥ −\frac{σ^{2}π^{2}}{2λ^{2}} .

We thus have

lim inf_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ≥ −\frac{σ^{2}π^{2}}{2λ^{2}} .  (4.1)

On the other hand, let t > 0 be arbitrary but fixed and write t_{n} = [tn/b_{n}] and γ_{n} = [n/t_{n}]. We have

\begin{matrix}ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ≤ ℙ\{#\{S[0, γ_{n}t_{n}]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} \\ ≤ ℙ\{ max_{1≤i≤γ_{n}} #\{S[(i − 1)t_{n}, it_{n}]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} = (ℙ\{#\{S[0, t_{n}]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \})^{γ_{n}}. \end{matrix}

From (2.4) we have

ℙ\{#\{S[0, t_{n}]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ⟶ ℙ\{ max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s) ≤ \frac{λ}{\sqrt{t}}\}.

Consequently,

lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ≤ \frac{1}{t} log ℙ\{ max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s) ≤ \frac{λ}{\sqrt{t}}\}.  (4.2)

The exact distribution of the Brownian range was first found in Feller (1951), which gives that

ℙ\{ max_{0≤s≤1} W (s) − min_{0≤s≤1} W (s) ≤ \frac{λ}{\sqrt{t}}\} ∼ \frac{8σ^{2}t}{λ^{2}} exp \{ − \frac{π^{2}σ^{2}}{2λ^{2}} t\} (t → ∞).

Letting t ⟶ ∞ on the right hand side of (4.2) gives

lim sup_{n→∞} \frac{1}{b_{n}} log ℙ\{#\{S[0, n]\} ≤ λ\sqrt{\frac{n}{b_{n}}} \} ≤ −\frac{π^{2}σ^{2}}{2λ^{2}} .  (4.3)

Finally, Theorem 2 follows from (4.1) and (4.3).

5. Proof of Theorem 3.

Let θ > 1 be fixed and write n_{k} = [θ^{k}] (k = 1, 2, ⋯). Taking b_{n} = log log n in Theorem 2 gives that

∑_{k} ℙ\{#\{S[0, n_{k}]\} ≤ λ\sqrt{\frac{n_{k}}{log log n_{k}}} \} < ∞

for any λ < πσ/\sqrt{2}. By Borel-Cantelli lemma,

lim inf_{k→∞} \sqrt{\frac{log log n_{k}}{n_{k}}} #\{S[0, n_{k}]\} ≥ \frac{π}{\sqrt{2}}σ a.s.

For any n_{k} ≤ n ≤ n_{k+1},

\sqrt{\frac{log log n}{n}}#\{S[0, n]\} ≥ \sqrt{\frac{log log n_{k+1}}{n_{k+1}}} #\{S[0, n_{k}]\} ≥ (1 + o(1))θ^{−1/2}\sqrt{\frac{log log n_{k}}{n_{k}}} #\{S[0, n_{k}]\}.

So we have

lim inf_{n→∞} \sqrt{\frac{log log n}{n}}#\{S[0, n]\} ≥ θ^{−1/2} \frac{π}{\sqrt{2}}σ a.s.

Letting θ → 1^{+} gives that

lim inf_{n→∞} \sqrt{\frac{log log n}{n}}#\{S[0, n]\} ≥ \frac{π}{\sqrt{2}}σ a.s.  (5.1)

Similarly, by Theorem 1 and Borel-Cantelli lemma we can prove that

lim sup_{n→∞} \frac{1}{\sqrt{n log log n}}#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≤ \sqrt{\frac{2}{p}}σ a.s.  (5.2)

On the other hand, write m_{k} = k^{k} (k = 1, 2, ⋯). By Theorem 2 we have that for any λ > πσ/\sqrt{2}

∑_{k} ℙ\{#\{S[m_{k}, m_{k+1}]\} ≤ λ\sqrt{\frac{m_{k+1}}{log log m_{k+1}}} \}

= ∑_{k} ℙ\{#\{S[0, m_{k+1} − m_{k}]\} ≤ λ\sqrt{\frac{m_{k+1}}{log log m_{k+1}}} \} = ∞.

Notice that {#{S[m_{k}, m_{k+1}]}}_{k} is an independent sequence. By Borel-Cantelli lemma,

lim inf_{k→∞} \sqrt{\frac{log log m_{k+1}}{m_{k+1}}} #\{S[m_{k}, m_{k+1}]\} ≤ \frac{π}{\sqrt{2}}σ a.s.

Consider the inequality,

#{S[0, m_{k+1}]} ≤ #{S[m_{k}, m_{k+1}]} + #{S[0, m_{k}]}.

Since

\sqrt{m_{k} log log m_{k}} = o(\sqrt{\frac{m_{k+1}}{log log m_{k+1}}} ) (k → ∞)

by (5.2) we have

lim sup_{k→∞} \sqrt{\frac{log log m_{k+1}}{m_{k+1}}} #\{S[0, m_{k}]\} = 0 a.s.

Hence,

lim inf_{k→∞} \sqrt{\frac{log log m_{k+1}}{m_{k+1}}} #\{S[0, m_{k+1}]\} ≤ \frac{π}{\sqrt{2}}σ a.s.

Consequently,

lim inf_{n→∞} \sqrt{\frac{log log n}{n}}#\{S[0, n]\} ≤ \frac{π}{\sqrt{2}}σ a.s.  (5.3)

Hence, (1.7) follows from (5.1) and (5.3).

We now come to the proof of the lower bound of (1.6). That is, we prove

lim sup_{n→∞} \frac{1}{\sqrt{n log log n}}#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ \sqrt{\frac{2}{p}}σ a.s.  (5.4)

As p = 1, the proof of Theorem can be carried out in a way similar to the one for (5.3). The case p > 1 is more delicate for lack of subadditivity.

For given x̄ = (x_{1}, ⋯, x_{p}) ∈ (ℤ)^{p}, we introduce the notation ℙx̄ for the probability induced by the random walks S_{1}(n), ⋯, S_{p}(n) in the case when S_{1}(n), ⋯, S_{p}(n) start at x_{1}, ⋯, x_{p}, respectively. The notation 𝔼 x̄ denote the expectation correspondent to ℙx̄. To be consistent with the notation we used before, we have ℙ^{(0,···,0)} = ℙ and 𝔼^{(0,···,0)} = 𝔼. Write

||x̄|| = max_{1≤j≤p} |x_{j}| x̄ = (x_{1}, ⋯, x_{p}) ∈ (ℝ)^{p}.

By the argument used in the proof of (5.6), Chen (2004), we will have (5.4) if

lim inf_{n→∞} \frac{1}{b_{n}} log inf_{||x̄||≤\sqrt{n/b_{n}}} ℙx̄\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} ≥ −\frac{pλ^{2}}{2σ^{2}} .  (5.5)

for any λ > 0.

Take t_{n} = [n/b_{n}] and T_{x} = inf\{n ≥ 0; S(n) = x\}. Then for any x̄ ∈ ℤ^{p} with ||x̄|| ≤ \sqrt{n/b_{n}},

ℙx̄\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\}

= ℙ\{#\{ ⋂_{j=1}^{p} (x_{j} + S_{j}[0, n])\} ≥ λ\sqrt{nb_{n}}\}

≥ ∑_{k_{1},···,k_{p}=0}^{t_{n}} ℙ\{T_{−x_{1}} = k_{1}, ⋯, T_{−x_{p}} = k_{p}; #\{ ⋂_{j=1}^{p} (x_{j} + S_{j}[k_{j}, n])\} ≥ λ\sqrt{nb_{n}}\}

= ∑_{k_{1},···,k_{p}=0}^{t_{n}} ( ∏_{j=1}^{p} ℙ\{T_{−x_{j}} = k_{j}\})ℙ\{#\{S_{1}[0, n − k_{1}] ∩ ⋯ ∩ S_{p}[0, n − k_{p}]\} ≥ λ\sqrt{nb_{n}}\}

≥ ( inf_{|x|≤\sqrt{n/b_{n}}} ℙ\{T_{x} ≤ t_{n}\})^{p}ℙ\{#\{S_{1}[0, n − t_{n}] ∩ ⋯ ∩ S_{p}[0, n − t_{n}]\} ≥ λ\sqrt{nb_{n}}\}.

By Lemma 5.1 given below, there is a constant δ > 0 such that

\begin{matrix}inf_{||x̄||≤\sqrt{n/b_{n}}} ℙx̄\{#\{S_{1}[0, n] ∩ ⋯ ∩ S_{p}[0, n]\} ≥ λ\sqrt{nb_{n}}\} \\ ≥ δℙ\{#\{S_{1}[0, n − t_{n}] ∩ ⋯ ∩ S_{p}[0, n − t_{n}]\} ≥ λ\sqrt{nb_{n}}\} \end{matrix}

for all n ≥ 1. Hence, (5.5) follows from Theorem 1 with n being replaced by n − t_{n}.

We end this section with the following lemma.

Lemma 5.1. For any x ∈ ℤ, write

T_{x} = inf{n ≥ 0; S(n) = x}.

Then

lim inf_{n→∞} inf_{|x|≤C\sqrt{n}} ℙ\{T_{x} ≤ n\} > 0  (5.6)

for any C > 0 and

lim_{ϵ→0^{+}} lim inf_{n→∞} inf_{|x|≤ϵ\sqrt{n}} ℙ\{T_{x} ≤ n\} = 1.  (5.7)

Proof. Recall that the random walk {S(n)} is said to be aperiodic if the greatest common factor of the set

{n ≥ 1; ℙ{S(n) = 0} > 0}

is 1. We first prove Lemma 5.1 under the assumption of aperiodicity.

By Markov property,

ℙ{S(k) = x} = ∑_{j=0}^{k} ℙ{T_{x} = j, S(k) = x} = ∑_{j=0}^{k} ℙ{T_{x} = j}ℙ{S(k − j) = 0}.

Summing up on the both sides,

\begin{matrix}∑_{k=0}^{n} ℙ\{S(k) = x\} = ∑_{k=0}^{n} ∑_{j=0}^{k} ℙ\{T_{x} = j\}ℙ\{S(k − j) = 0\} \\ = ∑_{j=0}^{n} ℙ\{T_{x} = j\} ∑_{k=j}^{n} ℙ\{S(k − j) = 0\} ≤ ℙ\{T_{x} ≤ n\} ∑_{k=0}^{n} ℙ\{S(k) = 0\}. \end{matrix}

By the Remark in p. 661 of Le Gall and Rosen (1991) the aperiodicity of the random walk implies

sup_{x∈ℤ} |\sqrt{n}ℙ\{S(n) = x\} − \frac{1}{\sqrt{2π}σ} exp \{ − \frac{x^{2}}{2nσ^{2}} \}| ⟶ 0 (n → ∞).

which gives (5.6) and (5.7).

We now prove (5.6) and (5.7) without assuming aperiodicity. Let 0 < η < 1 be fixed and let {δ_{n}}_{n≥1} be i.i.d. Bernoulli random variables with the common law:

ℙ{δ_{1} = 0} = 1 − ℙ{δ_{1} = 1} = η.

We assume independence between {S(n)} and {δ_{n}}.

Define the renewal sequence {τ_{k}}_{k≥0} by τ_{0} = 0 and τ_{k+1} = inf{n > τ_{k}; δ_{n} = 1}.

Then {τ_{k} − τ_{k−1}}_{k≥1} is an i.i.d. sequence with common distribution

ℙ{τ_{1} = n} = (1 − η)η^{n−1} n = 1, 2, · · · .

Consider the random walk S̃(n) = S(τ_{n}). By the fact that

ℙ{S(τ_{1}) = 0} = (1 − η) ∑_{k=1}^{∞} η^{k−1}ℙ{S(k) = 0} > 0,

{S̃(n)} is aperiodic. Applying what we have proved to {S̃(n)},

lim inf_{n→∞} inf_{|x|≤C\sqrt{n}} ℙ\{T̃_{x} ≤ n\} > 0 and lim_{ϵ→0^{+}} lim inf_{n→∞} inf_{|x|≤ϵ\sqrt{n}} ℙ\{T̃_{x} ≤ n\} = 1  (5.8)

where

T̃_{x} = inf{n ≥ 0; S(τ_{n}) = x}.

Notice that

τT̃_{x} ≥ T_{x} x ∈ ℤ.

Take 0 < θ < 1 such that θ(1 − η)^{−1} < 1. Then for any x ∈ ℤ,

ℙ{T_{x} > n} ≤ ℙ{τT̃_{x} > n} ≤ ℙ{max_{k≤θn} τ_{k} > n} + ℙ{T̃_{x} > θn}.

By the classic law of large numbers, the first term on the right tends to zero as n → ∞. Therefore, (5.6) and (5.7) follows from (5.8) with n being replaced by [θn].

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