TRANSIENT NEAREST NEIGHBOR RANDOM WALK ON THE LINE

Endre Csáki^{1} Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. E-mail address: csaki@renyi.hu

Antónia Földes^{2} Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York 10314, U.S.A. E-mail address: foldes@mail.csi.cuny.edu

Pál Révész^{1} Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107 A-1040 Vienna, Austria. E-mail address: reveszp@renyi.hu

Abstract: We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over which the walk runs through (always from left to right) without ever returning.

AMS 2000 Subject Classification: Primary 60G50; Secondary 60F15, 60J55.

Keywords: transient random walk, local time, strong theorems. Running head: Transient NN random walk.

1Research supported by the Hungarian National Foundation for Scientific Research, Grant No. K 61052 and K 67961.

2Research supported by a PSC CUNY Grant, No. 68030-0037.

1. Introduction

Let X_{0} = 0, X_{1}, X_{2}, … be a Markov chain with

E_{i} := P(X_{n+1} = i + 1 | X_{n} = i) = 1 − P(X_{n+1} = i − 1 | X_{n} = i)  (1.1)
= \{ \begin{matrix}1 & if i = 0 \\ 1/2 + p_{i} & if i = 1, 2, …, \end{matrix}

where 0 ≤ p_{i} < 1/2, i = 1, 2, . . .. This sequence {X_{i}} describes the motion of a particle which starts at zero, moves over the nonnegative integers and going away from 0 with a larger probability than to the direction of 0. We will be interested in the case when {p_{i}, i = 1, 2...} goes to zero. That is to say 0 has a repelling power which becomes small if the particle is far away from 0. We intend to characterize the local time of this motion.

A slightly different but symmetric variation of the same motion can be defined as follows. Let X_{0}^{∗} = 0, X_{1}^{∗}, X_{2}^{∗}, … be a Markov chain with

E_{i}^{∗} := P(X_{n+1}^{∗} = i + 1 | X_{n}^{∗} = i) = 1 − P(X_{n+1}^{∗} = i − 1 | X_{n}^{∗} = i) =
= \{ \begin{matrix}1/2 & if i = 0, \\ 1/2 + p_{i} & if i = 1, 2, …, \\ 1/2 − p_{−i} & if i = −1, −2, … \end{matrix}

Our results can be rephrased with the obvious modification for this walk as well. However to be in line with the existing literature we will use the definition in (1.1).

The properties of this model, often called birth and death chain, connections with orthogonal polynomials in particular, has been treated extensively in the literature. See e.g. the classical paper by Karlin and McGregor [11], or more recent papers by Coolen-Schrijner and Van Doorn [3] and Dette [5].

As it will turn out in this paper, the properties of the walk and its local time is very sensitive even for small changes in {p_{i}}-s. There is a well-known result in the literature (cf. e.g. Chung [2]) characterizing those sequences {p_{i}} for which {X_{i}} is transient (resp. recurrent). Theorem A: ([2], page 74) Let X_{n} be a Markov chain with transition probabilities given in (1.1). Define

U_{i} := \frac{1 − E_{i}}{E_{i}} = \frac{1/2 − p_{i}}{1/2 + p_{i}}  i = 1, 2...  (1.2)

Then X_{n} is transient if and only if

∑_{k=1}^{∞} ∏_{i=1}^{k} U_{i} < ∞.

This criteria however does not reveal explicitly what are the transient/reccurent type of {p_{i}} sequences. Lamperti [13], [15] proved a more general theorem about recurrence and transience of real nonnegative processes (not necessarily Markov chains). Here we spell out his result in our setup only, which easily follows from Theorem A as well. Corollary: If for all i large enough,

p_{i} ≤ \frac{1}{4i} + O (\frac{1}{i^{1+δ}} )  δ > 0,  (1.3)

then {X_{i}} is recurrent. If instead, for some θ > 1

p_{i} ≥ \frac{θ}{4i}  (1.4)

for i large enough, then {X_{i}} is transient.

As we proceed to find the necessary tools for getting results about the local time, as a byproduct, we will get a much sharper version of this Corollary.

In this paper we concentrate only on the transient case.

There are many results in the literature about the limiting behavior of {X_{n}}, depending on the sequence {p_{i}}. Lamperti [14] determined the limiting distribution of X_{n}. Theorem B: ([14]) If lim_{i→∞} ip_{i} = B/4 > 0, then

lim_{n→∞} P (\frac{X_{n}}{\sqrt{n}} < x) = \frac{1}{2^{B/2−1/2}Γ(B/2 + 1/2)} ∫_{0}^{x} u^{B}e^{−u^{2}/2} du.

In fact, Lamperti [14] (see also Rosenkrantz [17]) proved weak convergence of X_{n}/\sqrt{n} to a Bessel process as well. We intend to give further connections (strong invariance, etc.) between X_{n} and Bessel process in a subsequent paper.

The law of the iterated logarithm for X_{n} was given by Brézis et al. [1], Székely [19], Gallardo [8], Voit [20]. Their somewhat more general results specialized in our setup, reads as follows. Theorem C: ([1], [19], [8], [20])

If lim_{i→∞} ip_{i} = c > 0, then

lim sup_{n→∞} \frac{X_{n}}{\sqrt{2n log log n}} = 1 a.s.

Voit [21] has proved a law of large numbers for certain Markov chains, which we quote in our setup only.

Theorem D: ([21]) If lim_{i→∞} i^{α}p_{i} = c > 0 for some 0 < α < 1, then

lim_{n→∞} \frac{X_{n}}{n^{1/(1+α)}} = 2c(1 + α) a.s.

Our main concern in this paper is to study the local time of {X_{n}}, defined by

ξ(x, n) := #{k : 0 ≤ k ≤ n, X_{k} = x}, x = 0, 1, 2, …,  (1.5)

and

ξ(x, ∞) := lim_{n→∞} ξ(x, n).  (1.6)

2. Lemmas and Notations

For U_{i} defined in (1.2) we get by elementary calculation that Fact 1.

U_{i} = \frac{1 − E_{i}}{E_{i}} = \frac{1/2 − p_{i}}{1/2 + p_{i}} = 1 − 4p_{i} + O(p_{i}^{2})
= exp(−4p_{i} + O(p_{i}^{2})) (i = 0, ±1, ±2, …).  (2.1)

Introduce the notation

D(m, n) := \{ \begin{matrix}0 & if n = m, \\ 1 & if n = m + 1, \\ 1 + ∑_{j=1}^{n−m−1} ∏_{i=1}^{j} U_{m+i} = \\ 1 + ∑_{j=1}^{n−m−1} exp (−(1 + o_{m}(1))4 ∑_{i=m+1}^{m+j} p_{i}) & if n ≥ m + 2. \end{matrix}

Denote

lim_{n→∞} D(m, n) =: D(m, ∞).

Lemma 2.1. If p_{i} ↓ 0 when i → ∞, then for m large enough

D(m, ∞) ≥ \frac{C}{p_{m}} ,

where C is an absolute constant. Consequently,

lim_{m→∞} D(m, ∞) = +∞.

Proof: Let \frac{1}{p_{m}} ≤ j ≤ \frac{2}{p_{m}} , then from (2.1) for m big enough we have

∑_{i=m}^{m+j}(p_{i} + Cp_{i}^{2}) ≤ \frac{2}{p_{m}} (p_{m} + Cp_{m}^{2}) ≤ 2(1 + Cp_{m}) ≤ 2(1 + C).

Consequently

exp (− ∑_{i=m}^{m+j}(p_{i} + Cp_{i}^{2})) ≥ exp(−2(1 + C))

and

D(m, ∞) ≥ 1 + ∑_{j=0}^{∞} exp (− ∑_{i=m}^{m+j}(p_{i} + Cp_{i}^{2}))

≥ 1 + ∑_{j=[ \frac{1}{pm}]}^{[ \frac{2}{pm}]} exp(−2(1 + C)) ≥ 1 + \frac{1}{p_{m}} exp(−2(1 + C)).

□ For 0 ≤ a ≤ b ≤ c define

p(a, b, c) :=
= P(min{j : j > m, X_{j} = a} < min{j : j > m, X_{j} = c} | X_{m} = b),

i.e. p(a, b, c) is the probability that a particle starting from b hits a before c.

Lemma 2.2. For 0 ≤ a ≤ b ≤ c

p(a, b, c) = 1 − \frac{D(a, b)}{D(a, c)}.

Especially

p(0, 1, n) = 1 − \frac{1}{D(0, n)}, p(n, n + 1, ∞) = 1 − \frac{1}{D(n, ∞)}.  (2.2)

Proof: The proof of this lemma is fairly standard, we give it for completeness. Clearly we have

p(a, a, c) = 1,
p(a, c, c) = 0,
p(a, b, c) = E_{b}p(a, b + 1, c) + (1 − E_{b})p(a, b − 1, c).

Consequently

p(a, b + 1, c) = \frac{1}{E_{b}} p(a, b, c) − \frac{1 − E_{b}}{E_{b}} p(a, b − 1, c)

and

p(a, b + 1, c) − p(a, b, c) = \frac{1 − E_{b}}{E_{b}} (p(a, b, c) − p(a, b − 1, c)) =
= U_{b}(p(a, b, c) − p(a, b − 1, c)).

By iteration we get

p(a, b + 1, c) − p(a, b, c) =  (2.3)
= U_{b}U_{b−1}(p(a, b − 1, c) − p(a, b − 2, c))
= … = U_{b}U_{b−1} ⋯ U_{a+1}(p(a, a + 1, c) − p(a, a, c)) =
= U_{b}U_{b−1} ⋯ U_{a+1}(p(a, a + 1, c) − 1).

Starting with the trivial identity

p(a, a + 1, c) − p(a, a, c) = p(a, a + 1, c) − 1

and adding to it the above equations for b = a + 1, …, c − 1 we get

−1 = p(a, c, c) − p(a, a, c) = D(a, c)(p(a, a + 1, c) − 1),

i.e.

p(a, a + 1, c) = 1 − \frac{1}{D(a, c)}.  (2.4)

Hence (2.3) and (2.4) imply

p(a, b + 1, c) − p(a, b, c) = −\frac{1}{D(a, c)}U_{b}U_{b−1} ⋯ U_{a+1}.

Adding these equations we obtain

p(a, b + 1, c) − 1 = p(a, b + 1, c) − p(a, a, c) =
= −\frac{1}{D(a, c)}(1 + U_{a+1} + U_{a+1}U_{a+2} + ⋯ + U_{a+1}U_{a+2} ⋯ U_{b}) =
= −\frac{D(a, b + 1)}{D(a, c)}.

Hence we have the lemma. □ Introduce the following notations:

λ(0, i) = 1,
λ(1, i) = i,
λ(2, i) = λ(1, i) log i, …,
λ(k, i) = λ(k − 1, i) log_{k−1} i (k = 3, 4, …),

where

log_{0} i = i,
log_{1} i = log i, …,
log_{k} i = log log_{k−1} i,

and

Λ(0, i) = 0,
Λ(K, i) = ∑_{k=1}^{K} \frac{1}{λ(k, i)}, (K = 1, 2, …)
Λ(K, i, B) = Λ(K − 1, i) + \frac{B}{λ(K, i)} (B > 0).

Note that

Λ(1, i, B) = \frac{B}{i} ,
Λ(2, i, B) = \frac{1}{i} + \frac{B}{i log i},
Λ(3, i, B) = \frac{1}{i} + \frac{1}{i log i} + \frac{B}{i log i log log i}.

For some K = 1, 2, …, B > 0 fixed, define

i_{0} = min \{i : \frac{1}{4}Λ(K, i, B) < \frac{1}{2}\}

and let

p_{i} = \{ \begin{matrix}p_{i_{0}}, & if 1 ≤ i ≤ i_{0} \\ \frac{1}{4}Λ(K, i, B) & if i > i_{0}. \end{matrix}  (2.5)

Now we are interested in the case {p_{i}} above. In fact, in the future for convenience, when we say that

p_{i} = \frac{1}{4}Λ(K, i, B)

we actually mean that p_{i} is defined by (2.5).

Lemma 2.3. Let p_{i} = \frac{1}{4}Λ(K, i, B). Then

D(0, ∞) \{ \begin{matrix}= ∞ & if B ≤ 1, \\ < ∞ & if B > 1, \end{matrix}  (2.6)

p(0, 1, ∞) \{ \begin{matrix}= 1 & if B ≤ 1, \\ < 1 & if B > 1. \end{matrix}  (2.7)

For n ≥ m + 2, B ≠ 1 and m big enough

D(m, n) = (1 + o_{m}(1))λ(K − 1, m)(log_{K−1} m)^{B} ×  (2.8)
× \frac{1}{B − 1} (\frac{1}{(log_{K−1} m)^{B−1}} − \frac{1}{(log_{K−1} n)^{B−1}} ) .

If B > 1,

D(m, ∞) = (1 + o_{m}(1))\frac{λ(K, m)}{B − 1},  (2.9)

p(m, m + 1, ∞) = 1 − (1 + o_{m}(1))\frac{(B − 1)}{λ(K, m)}.  (2.10)

Proof: To prove (2.8), observe that from (2.1) we have for n ≥ m + 2

D(m, n) = 1 + ∑_{j=1}^{n−m−1} ∏_{i=1}^{j} U_{m+i}  (2.11)

= 1 + ∑_{j=1}^{n−m−1} exp (− ∑_{i=m+1}^{m+j} (Λ(K, i, B))) exp (O(1) ∑_{i=m+1}^{m+j} Λ^{2}(K, i, B))

= 1 + (1 + o_{m}(1)) ∑_{j=1}^{n−m−1} exp (− ∑_{i=m+1}^{m+j} Λ(K, i, B))
=: 1 + (1 + o_{m}(1))A(m, n, K).

Now we give a lower bound for A(m, n, K).

A(m, n, K) ≥ ∑_{j=1}^{n−m−1} exp (− ∫_{m}^{m+j} Λ(K, x, B) dx)  (2.12)

= ∑_{j=1}^{n−m−1} \frac{λ(K − 1, m)(log_{K−1} m)^{B}}{λ(K − 1, m + j)(log_{K−1}(m + j))^{B}}

= λ(K − 1, m)(log_{K−1} m)^{B} ∑_{ℓ=m+1}^{n−1} \frac{1}{λ(K − 1, ℓ)(log_{K−1} ℓ)^{B}}
≥ λ(K − 1, m)(log_{K−1} m)^{B} ∫_{m+1}^{n} \frac{1}{λ(K − 1, x)(log_{K−1} x)^{B}} dx
= λ(K − 1, m)(log_{K−1} m)^{B} (\frac{(log_{K−1} m)^{1−B} − (log_{K−1} n)^{1−B}}{B − 1} ) .

It is easy to see that the proof of the upper bound goes the same way, resulting the same expression as in (2.12) with m replaced by m + 1 which combined with (2.11) proves (2.8). The proof of (2.6) is similar, and the rest of the lemma follows from these two. □

Consequence: If for any K = 1, 2...

p_{i} = \frac{Λ(K, i, B)}{4},

then the Markov chain is recurrent if B ≤ 1 and transient if B > 1.

Now we would like to consider the case when p_{i} is essentially \frac{B}{4i^{α}}, which should be understood in the same way as it was defined in (2.5). Namely, let

i_{0} = min \{i : \frac{B}{4i^{α}} < \frac{1}{2}\}

and let

p_{i} = \{ \begin{matrix}p_{i_{0}}, & if 1 ≤ i ≤ i_{0} \\ \frac{B}{4i^{α}} & if i > i_{0}. \end{matrix}  (2.13)

Lemma 2.4. In case p_{i} = \frac{B}{4i^{α}} (0 < α < 1) we have

D(m, ∞) = (1 + o_{m}(1))\frac{m^{α}}{B} ,  (2.14)
1 − p(m, m + 1, ∞) = (1 + o_{m}(1))\frac{B}{m^{α}} .  (2.15)

Proof: Consider the case 0 < α < 1/2 first. By (2.1)

∏_{i=1}^{j} U_{m+i} ≤ exp (−B ∑_{ν=m+1}^{m+j} ν^{−α} + ∑_{ν=m+1}^{m+j} Cν^{−2α}) =

≤ (1 + o_{m}(1)) exp (\frac{−B}{1 − α} [(m + j)^{1−α} − (m)^{1−α}] + \frac{C}{1 − 2α} [(m + j)^{1−2α} − (m)^{1−2α}]) .

Consequently, D(m, n)

≤ (1 + o_{m}(1)) exp (\frac{Bm^{1−α}}{1 − α} − \frac{Cm^{1−2α}}{1 − 2α} ) ∑_{k=m+1}^{n} exp((−\frac{Bk^{1−α}}{1 − α} + \frac{Ck^{1−2α}}{1 − 2α} )
≤ (1 + o_{m}(1)) exp (\frac{Bm^{1−α}}{1 − α} − \frac{Cm^{1−2α}}{1 − 2α} ) ∫_{m+1}^{n} exp (−\frac{Bx^{1−α}}{1 − α} (1 − C \frac{1 − α}{1 − 2α}x^{−α})) dx
≤ (1 + o_{m}(1)) exp (\frac{Bm^{1−α}}{1 − α} − \frac{Cm^{1−2α}}{1 − 2α} ) ∫_{m+1}^{n} exp (−\frac{Bx^{1−α}}{1 − α} h_{m}) dx,

where

h_{m} = 1 − C \frac{1 − α}{1 − 2α}(m + 1)^{−α}.

In the calculation above C is a positive constant the value of which is not important. In the future we will use C, C^{∗} or C_{1}, C_{2} … for which this remark applies, and their values might change from line to line. Using substitution and the asymptotic representation of the incomplete Gamma function (see e.g. Gradsteyn and Ryzhik [9] page 942, formula (8.357))

Γ(β, x) = ∫_{x}^{∞} t^{β−1}e^{−t} dt = x^{β−1}e^{−x} (1 + \frac{O(1)}{x}) as x → ∞,

we conclude that as m → ∞

D(m, ∞) ≤ (1 + O(\frac{1}{m^{1−α}})) \frac{m^{α}}{Bh_{m}} exp (\frac{Bm^{1−α}}{1 − α} − \frac{Cm^{1−2α}}{1 − 2α} ) exp (−\frac{h_{m}Bm^{1−α}}{1 − α} )
= (1 + O(\frac{1}{m^{1−α}})) \frac{m^{α}}{B} .

A similar calculation (which we omit) gives the same lower bound. The case of α = 1/2 goes along the same lines with obvious modifications. On the other hand, the case 1/2 < α < 1 can be worked out similarly, but it is obvious with less precise calculations as well. □

Lemma 2.5. In case p_{i} = \frac{B}{4(log i)^{α}} with α > 0, there exist 0 < K_{1} < K_{2} such that

K_{1}(log m)^{α} ≤ D(m, ∞) ≤ K_{2}(log m)^{α},  (2.16)
1 − p(m, m + 1, ∞) = \frac{O(1)}{(log m)^{α}} .  (2.17)

Proof: First we give the upper bound. For m ≥ m_{0}

∑_{i=m}^{m+j} (\frac{B}{(log i)^{α}} − \frac{C}{(log i)^{2α}} ) = ∑_{i=m}^{m+j} \frac{B}{(log i)^{α}} (1 − \frac{C^{∗}}{(log i)^{α}} )

≥ ∑_{i=m}^{m+j} \frac{B(1 − ε)}{(log i)^{α}} =: A(m, j, ε).

Then for

ℓ(log m)^{α} ≤ j < (ℓ + 1)(log m)^{α} (ℓ = 0, 1, 2, …)

we have

A(m, j, ε) ≥ \frac{B(1 − ε)ℓ(log m)^{α}}{(log[m + (ℓ + 1)(log m)^{α}])^{α}} =: H(m, ℓ, α).

It is easy to see now, that if (ℓ + 1)(log m)^{α} ≤ m then for an appropriate C_{1}

H(m, ℓ, α) ≥ \frac{B(1 − ε)ℓ(log m)^{α}}{(log(2m))^{α}} ≥ C_{1}ℓ.

On the other hand, if (ℓ + 1)(log m)^{α} ≥ m, then for an appropriate C_{2}

H(m, ℓ, α) ≥ \frac{B(1 − ε)ℓ(log m)^{α}}{(log(2(ℓ + 1)(log m)^{α})^{α}} ≥ C_{2}ℓ^{1/(2α)}.

Then with N = N(α) := [\frac{m}{(log m)^{α}}] − 1.

D(m, ∞) ≤ ∑_{ℓ=0}^{N} e^{−C_{1}ℓ} (log m)^{α} + ∑_{ℓ=N}^{∞} e^{−C_{2}ℓ^{\frac{1}{2α}}} (log m)^{α} = O(1)(log m)^{α}.

The lower bound follows from Lemma 2.1. □

3 Local time

We intend to study the limit properties of the local time ξ(R, ∞) in case of transient random walks. To this end we also define the number of upcrossings by

ξ(R, n, ↑) := #{k : 0 ≤ k ≤ n, X_{k} = R, X_{k+1} = R + 1}.  (3.1)
ξ(R, ∞, ↑) := lim_{n→∞} ξ(R, n, ↑).  (3.2)

It is a well known fact that the local time of a transient Markov chain follows geometric distribution. It was first formulated in the seminal paper of Erdős and Taylor ([7]) though they just stated it for random walk in dimension d ≥ 3. So we could only calculate the parameter of this geometric distribution. However we still provide a proof for completeness.

Lemma 3.1. For R = 0, 1, 2, …

P(ξ(R, ∞) = L) = \frac{1 + 2p_{R}}{2D(R, ∞)} (1 − \frac{1 + 2p_{R}}{2D(R, ∞)})^{L−1} , L = 1, 2, …  (3.3)

Moreover, the sequence

ξ(R, ∞, ↑), R = 0, 1, 2, …

is a Markov chain and

P(ξ(R, ∞, ↑) = L) = \frac{1}{D(R, ∞)} (1 − \frac{1}{D(R, ∞)})^{L−1} , L = 1, 2, …  (3.4)

Remark: Defining p_{0} = 1/2 which is in agreement with our definition E_{0} = 1 we see that the local time and the upcrossing of the site 0 have the same distribution. Proof: Clearly we have for L = 1, 2, …

P(ξ(R, ∞) = L) = (\frac{1}{2} + p_{R}) (1 − p(R, R + 1, ∞))×

× ∑_{j=0}^{L−1} \binom{L − 1}{j} (\frac{1}{2} − p_{R})^{j} ((\frac{1}{2} + p_{R}) p(R, R + 1, ∞))^{L−j−1} =

= (\frac{1}{2} + p_{R}) (1 − p(R, R + 1, ∞)) ×

× (1 − (\frac{1}{2} + p_{R}) (1 − p(R, R + 1, ∞)))^{L−1} ,

implying (3.3) by (2.2).

The other statements of the Lemma are obvious. □

Theorem 3.1. If p_{R} → 0, as R → ∞, then

lim_{R→∞} P (\frac{ξ(R, ∞)}{2D(R, ∞)} > x) = lim_{R→∞} P (\frac{ξ(R, ∞, ↑)}{D(R, ∞)} > x) = e^{−x},

that is to say, \frac{ξ(R, ∞)}{2D(R, ∞)} and \frac{ξ(R, ∞, ↑)}{D(R, ∞)} have exponential limiting distributions.

The proof is a trivial consequence of Lemma 3.1.

Theorem 3.2. Assume that p_{R} → 0 as R → ∞. Then with probability 1 we have

ξ(R, ∞) ≤ 2(1 + ε)D(R, ∞) log R  (3.5)

for any ε > 0 if R is large enough.

Moreover,

ξ(R, ∞) ≥ MD(R, ∞) i.o. a.s.  (3.6)

for any M > 0.

In case p_{R} = \frac{Λ(K, R, B)}{4} with B > 1, instead of (3.5) and (3.6) we have the much sharper Theorem 3.3. For p_{R} = \frac{Λ(K, R, B)}{4}, B > 1, we have

lim sup_{R→∞} \frac{ξ(R, ∞)}{2D(R, ∞) log log R} ≤ 1.  (3.7)

and

lim sup_{R→∞} \frac{ξ(R, ∞)}{2D(R, ∞) log_{K+1} R} ≥ 1.  (3.8)

Especially in case p_{R} = \frac{Λ(1, R, B)}{4} = \frac{B}{4R}, B > 1, being D(R, ∞) = \frac{R}{B − 1}, we have

lim sup_{R→∞} \frac{(B − 1)ξ(R, ∞)}{2R log log R} = lim sup_{R→∞} \frac{(B − 1)ξ(R, ∞, ↑)}{R log log R} = 1.  (3.9)

Consequences:

• If p_{R} = \frac{1}{4}Λ(K, R, B), (B > 1) then for any ε > 0

ξ(R, ∞) ≤ \frac{2(1 + ε)}{B − 1}λ(K, R) log log R a.s.  (3.10)

if R is large enough,

ξ(R, ∞) ≥ \frac{2(1 − ε)}{B − 1}λ(K, R) log_{K+1} R i.o. a.s.  (3.11)

and

lim_{R→∞} P (\frac{B − 1}{2λ(K, R)} ξ(R, ∞) > x) = e^{−x}.  (3.12)

• If p_{R} = \frac{B}{4R^{α}} (0 < α < 1), then

ξ(R, ∞) ≤ \frac{2}{B} (1 + ε)R^{α} log R a.s.  (3.13)

if R is large enough,

ξ(R, ∞) ≥ MR^{α} i.o. a.s.  (3.14)

for any M > 0 and

lim_{R→∞} P (\frac{B ξ(R, ∞)}{2R^{α}} > x) = e^{−x}.  (3.15)

• If p_{R} = \frac{B}{4(log R)^{α}} (α > 0), then

ξ(R, ∞) ≤ O(1)(log R)^{1+α} a.s.  (3.16)

if R is large enough and

ξ(R, ∞) ≥ M(log R)^{α} i.o. a.s.  (3.17)

for any M > 0.

Proof of Theorem 3.2: (3.5) follows from Lemma 3.1. On the other hand, (3.3) also implies that for any M > 0

lim inf_{R→∞} P (ξ(R, ∞) ≥ MD(R, ∞)) > 0

from which it follows that

P (ξ(R, ∞) ≥ MD(R, ∞) i.o.) > 0.

Now to finish our proof we need to apply the zero-one law (in a non-independent setup) exactly in the same way as in Doob [6] page 103, observing that the conditional probability of our tail event given the first n steps of our walk is the same as its unconditional probability, that is for any n = 1, 2, …

P (ξ(R, ∞) ≥ MD(R, ∞) i.o. | X_{1}, X_{2}, ...X_{n}) = P (ξ(R, ∞) ≥ MD(R, ∞) i.o.) .

According to the zero-one law mentioned above, the conditional probability on the left-hand side is either zero or one, and since the unconditional probability on the right-hand side is strictly positive, it should be equal to one, which, in turn, implies (3.6).

Proof of Theorem 3.3: To prove (3.7), we need a few lemmas. Recall the definition of the upcrossing in (3.1) and (3.2). For large values of the local time and upcrossing we have the following invariance principle.

Lemma 3.2. As R → ∞

ξ(R, ∞) − 2ξ(R, ∞, ↑) = O((D(R, ∞) log R)^{1/2+ε} + p_{R}D(R, ∞) log R) a.s.  (3.18)

Proof: Under the condition ξ(R, ∞) = L, ξ(R, ∞, ↑) − 1 has binomial distribution with parameters (L − 1, 1/2 + p_{R}). According to Hoeffding inequality,

P (|ξ(R, ∞, ↑) − 1 − (\frac{1}{2} + p_{R}) (L − 1)| ≥ u(L − 1)^{1/2}) ≤ e^{−Cu^{2}}

with some C > 0, from which as L → ∞,

ξ(R, ∞, ↑) − \frac{L}{2} = O(L^{1/2+ε} + Lp_{R}) a.s.

Putting L = ξ(R, ∞), we get (3.18) from (3.5).

Lemma 3.3. Let

γ_{R} = (\frac{1}{2} + p_{R}) p(R, R + 1, ∞),

and

c_{R} = \frac{γ_{R}}{1 − γ_{R}} .

Then

ζ(R) := \frac{ξ(R, ∞, ↑)}{c_{1} ⋯ c_{R}}, R = 1, 2, …

is a submartingale.

Proof: Let T_{R} be the first hitting time of R by {X_{n}}, e.g. T_{R} = min{n : X_{n} = R}. Then we have

P_{R}(ξ(R, T_{R−1}, ↑) = j, T_{R−1} < ∞) = (\frac{1}{2} − p_{R}) γ_{R}^{j}, j = 0, 1, …,  (3.19)

P_{R}(ξ(R, ∞, ↑) = j, T_{R−1} = ∞) = (\frac{1}{2} + p_{R} − γ_{R}) γ_{R}^{j−1}, j = 1, 2, …  (3.20)

Observe that

ξ(R, ∞, ↑) = ∑_{m=1}^{ξ(R−1,∞,↑)−1} ξ_{m} + ξ̃,

where ξ_{m}, m = 1, 2... has distribution (3.19) and ξ̃ has distribution (3.20). Then

E(e^{λξ(R,∞,↑)}, ξ(R − 1, ∞, ↑) = i) = (E(e^{λξ_{1}}))^{i−1}E(e^{λ_{ξ}})  (3.21)

= \frac{(\frac{1}{2} + p_{R} − γ_{R}) e^{λ} (\frac{1}{2} − p_{R})^{i−1}}{(1 − γ_{R}e^{λ})^{i}} ,

hence

E(e^{λξ(R,∞,↑)} | ξ(R − 1, ∞, ↑) = i) = e^{λ} (\frac{1 − γ_{R}}{1 − γ_{R}e^{λ}} )^{i} ,

from which

E(ξ(R, ∞, ↑) | ξ(R − 1, ∞, ↑)) = c_{R}ξ(R − 1, ∞, ↑) + 1,  (3.22)

which easily implies the lemma. □

Now we prove the upper bound, i.e.

lim sup_{R→∞} \frac{ξ(R, ∞, ↑)}{D(R, ∞) log log R} ≤ 1 a.s.,  (3.23)

which also implies (3.7) by Lemma 3.2.

With an easy calculation we get from (3.21) that

E(e^{λξ(R,∞,↑)}) = \frac{e^{λ}}{D(R, ∞) − e^{λ}(D(R, ∞) − 1)}.  (3.24)

Using that ζ(R) is submartingale, from (3.24) we have with R_{k} = [exp(k/ log k)], C_{k} = c_{1}c_{2} … c_{R_{k}},

u_{k} = (1 + ε)D(R_{k}, ∞) log log R_{k},

P ( max_{R_{k}≤R<R_{k+1}} ζ(R) ≥ \frac{u_{k+1}}{C_{k+1}} )

≤ exp(−λu_{k+1}/C_{k+1})E(exp(λζ(R_{k+1})))

= \frac{exp(λ/C_{k+1})(1 − u_{k+1})}{D(R_{k+1}, ∞) − exp(λ/C_{k+1})(D(R_{k+1}, ∞) − 1)}.

It can be seen that the optimal choice for λ is given by

exp(λ/C_{k+1}) = \frac{(u_{k+1} − 1)D(R_{k+1}, ∞)}{u_{k+1}(D(R_{k+1}, ∞) − 1)},

and we get finally

P ( max_{R_{k}≤R<R_{k+1}} ζ(R) ≥ \frac{u_{k+1}}{C_{k+1}} ) = \frac{O(1) log log R_{k+1}}{(log R_{k+1})^{1+ε}} .

Hence by Borel-Cantelli lemma for large k and R_{k} ≤ R < R_{k+1} we have

ζ(R) ≤ \frac{(1 + ε)D(R, ∞) log log R}{c_{1} ⋯ c_{R}c_{R+1} ⋯ c_{R_{k+1}}},

i.e.

ξ(R, ∞, ↑) ≤ \frac{(1 + ε)D(R, ∞) log log R}{c_{R+1} ⋯ c_{R_{k+1}}}.

If p_{R} = Λ(K, R, B)/4, then as R → ∞ (cf. (2.9))

D(R, ∞) ∼ \frac{λ(K, R)}{B − 1}

and

c_{R} ∼ \frac{1 + 2p_{R} − 1/D(R, ∞)}{1 − 2p_{R} + 1/D(R, ∞)} ∼ exp(4p_{R} − 2/D(R, ∞)) ∼ exp (Λ(K, R, B) − \frac{2(B − 1)}{λ(K, R)}) .

If K = 1, then

Λ(1, R, B) − \frac{2(B − 1)}{λ(1, R)} ∼ \frac{2 − B}{R} , B ≠ 2

and

Λ(1, R, 2) − \frac{2}{λ(1, R)} = \frac{o(1)}{R},

and if K > 1, then

Λ(K, R, B) − \frac{2(B − 1)}{λ(K, R)} ∼ \frac{1}{R}.

Hence for large k and R_{k} ≤ R ≤ R_{k+1} we have

c_{R+1} ⋯ c_{R_{k+1}} ∼ exp (C log \frac{R_{k+1}}{R} )

with some constant C if K = 1, B ≠ 2 or K > 1 and C = o(1) if K = 1, B = 2. In view of lim_{k→∞} R_{k+1}/R_{k} = 1, for any ε > 0, one can choose k large enough such that

c_{R+1} ⋯ c_{R_{k+1}} ≥ 1 − ε, i.e.

ξ(R, ∞, ↑) ≤ \frac{(1 + ε)D(R, ∞) log log R}{1 − ε}.

Since ε > 0 is arbitrary, (3.23) follows.

To prove the lower bound (3.8), consider an increasing sequence of sites R_{k} to be determined later. Let

τ_{k} = min{n : X_{n} = R_{k}},

the time of the first visit at the site R_{k}, and define

Z(k) := ξ(R_{k}, τ_{k+1}).

Observe that {Z(k), k = 1, 2...} are independent. Following the proof of Lemma 3.1 we can conclude that

P(Z(k) ≥ L) = (1 + o_{R_{k}}(1))×
× [(1 − \frac{1}{2}(1 − p(R_{k}, R_{k} + 1, R_{k+1}))) (1 + O((1 − p(R_{k}, R_{k} + 1, R_{k+1}))p_{R_{k}}))]^{L−1} .(3.25)

Based on (2.8) it is easy to calculate that

D(R_{k}, R_{k+1}) = (1 + o_{R_{k}}(1)) \frac{λ(K − 1, R_{k})}{B − 1} log_{K−1} R_{k} (1 − (\frac{log_{K−1} R_{k}}{log_{K−1} R_{k+1}} )) =
= (1 + o_{R_{k}}(1)) \frac{λ(K, R_{k})}{B − 1} (1 − (\frac{log_{K−1} R_{k}}{log_{K−1} R_{k+1}} )) .  (3.26)

Define the sequence R_{k} by

log_{K} R_{k} := k log Q

with some Q > 1 (we intentionally forget about the technicalities arising from the fact that the sites should be integers). It is easy to see that with this choice of R_{k}

\frac{log_{K−1} R_{k}}{log_{K−1} R_{k+1}} = \frac{1}{Q}.

Let

L(k) = 2\frac{λ(K, R_{k})}{B − 1} \frac{Q − 1}{Q} log_{K+1} R_{k}.

From (2.4) we get that

P(Z(k) ≥ L(k)) ∼ exp(− log_{K+1} R_{k}) = \frac{1}{log_{K} R_{k}} = \frac{1}{k log Q}.

Applying Borel-Cantelli lemma and then letting Q → ∞, we get (3.8). □

Our next issue was to investigate how small could be the local time of our process. More precisely we wanted to know whether it is true that in the transient case there are always infinitely many sites with local time equal to 1. In fact we managed to prove in some sense much more, and in some sense much less. Namely, we prove the following two theorems. Define for N ≥ 2

f(N, R) = f(N, R, ε) = \frac{1}{log 2} (∑_{j=2}^{N} log_{j} R + ε log_{N} R)

and

g(N, R) = f(N, R, 0).

Theorem 3.4. Let p_{R} = \frac{Λ(1, R, B)}{4} with B > 1 and let N ≥ 2. Then

• with probability 1 there exist infinitely many R for which

ξ(R + j, ∞) = 1

for each j = 0, 1, 2, …, [g(N, R)].

• with probability 1 for any ε > 0 and R large enough there exists an S

R ≤ S ≤ f(N, R, ε)

such that

ξ(S, ∞) > 1.

Let

f^{∗}(R, ϵ) = \frac{(1 + ε)(1 − α) log R}{log 2} and g^{∗}(R) = f^{∗}(R, 0).

Theorem 3.5. Let p_{R} = \frac{B}{4R^{α}} (0 < α < 1). Then • with probability 1 there exists infinitely many R for which

ξ(R + j, ∞) = 1

for each j = 0, 1, 2, …, g^{∗}(R).

• with probability 1 for each R large enough and ε > 0 there exists an S,

R ≤ S ≤ f^{∗}(R, ε)

such that

ξ(S, ∞) > 1.

Furthermore, we conjecture that for p_{i} ≥ B/(4i), where B > 1, with probability 1 there are always infinitely many sites with local time 1. On the other hand, recently James et al.

[10] proved that for p_{i} ∼ Λ(2, i, B) with B > 1 with probability 1 there are only finitely many cutpoints, hence finitely many points with local time 1. We note that it can be seen with a similar argument that this is the case for p_{i} ∼ Λ(K, i, B) for all K ≥ 2 as well.

Proof of Theorem 3.4: At first we prove the second statement. Recall the notation of λ(N, R) and observe that

R2^{g(N,R)} = λ(N, R) and R2^{f(N,R)} = λ(N − 1, R)(log_{N−1} R)^{1+ϵ}.  (3.27)

Now the proof of the second statement is a trivial consequence of

Lemma 3.4. For every N ≥ 2 integer as R → ∞

P { ⋂_{j=1}^{f(N,R)}{ξ(R + j, ∞) = 1}} =

= ∏_{j=1}^{f(N,R)} (\frac{1}{2} + \frac{B}{4(R + j)}) (1 − p(R + f(N, R), R + f(N, R) + 1, ∞)) =
= (1 + o_{R}(1))\frac{1}{2^{f(N,R)}} \frac{B − 1}{R} = (1 + o_{R}(1))\frac{B − 1}{λ(N − 1, R)(log_{N−1} R)^{1+ε}} .

Proof: Obvious by (2.10). □

The proof of the first statement of the theorem is based on the following

Lemma 3.5. For every N ≥ 2 integer as R → ∞

P \{ ⋂_{j=1}^{g(N,R)}\{ξ(R + j, ∞) = 1\}\} = \frac{O(1)}{λ(N, R)},  (3.28)

P := P(N, R, S) :=  (3.29)
= P { ⋂_{j=1}^{g(N,R)}{ξ(R + j, ∞) = 1} ∩ ⋂_{j=1}^{g(N,S)}{ξ(S + j, ∞) = 1}} ≤

≤ \{ \begin{matrix}\frac{(1 + o_{R}(1))(B − 1)^{2}}{λ(N, R)λ(N, S − R)} & if S ≥ R + g(N, R), \\ \frac{O(1)2^{R}}{2^{S+g(S,N)}} \frac{B − 1}{S + g(N, S)} & if R < S < R + g(N, R). \end{matrix}

Proof: (3.28) follows from Lemma 3.1 and (3.27). In case R < S < R + g(N, R) we have

P = ∏_{i=R}^{S+g(N,S)} (\frac{1}{2} + \frac{B}{4i}) (1 − p(S + g(N, S), S + g(N, S) + 1, ∞)) ≤
≤ O(1)\frac{1}{2^{S+g(N,S)−R}} \frac{B − 1}{S + g(N, S)}.

In case S > R + g(N, R) we have

P = (1 + o_{R}(1))\frac{1}{2^{g(N,R)}} (1 − p(R, R + 1, S))\frac{1}{2^{g(N,S)}} (1 − p(S, S + 1, ∞)) =
= (1 + o_{R}(1))\frac{1}{2^{g(N,R)} 2^{g(N,S)}} \frac{B − 1}{R^{B}} \frac{1}{R^{1−B} − S^{1−B}} \frac{B − 1}{S} =
= (1 + o_{R}(1))\frac{1}{2^{g(N,R)} 2^{g(N,S)}} \frac{(B − 1)^{2}}{R} \frac{S^{B−2}}{S^{B−1} − R^{B−1}} ≤
≤ (1 + o_{R}(1))\frac{1}{2^{g(N,R)} 2^{g(N,S−R)}} \frac{(B − 1)^{2}}{R} \frac{1}{(S − R)} ≤
≤ (1 + o_{R}(1))\frac{(B − 1)^{2}}{λ(N, R)λ(N, S − R)}.

Hence we have the second statement of the lemma. □ Now we turn to the proof of the first statement of the theorem. Let

A(R) = ⋂_{j=1}^{g(N,R)}{ξ(R + j, ∞) = 1}.

Then by (3.28)

∑_{R=1}^{T} P(A(R)) = O(1) log_{N} T  (3.30)

and

∑_{R=1}^{T} ∑_{S=R+1}^{T} P(A(R)A(S)) =

= ∑_{R=1}^{T} ∑_{S=R+1}^{R+g(N,R)} P(A(R)A(S)) + ∑_{R=1}^{T} ∑_{S=R+g(N,R)+1}^{T} P(A(R)A(S)) =: I + II.

By (3.29) we have

I ≤ O(1) ∑_{R=1}^{T} ∑_{S=R+1}^{R+g(N,R)} \frac{2^{R}}{2^{S+g(N,S)}} \frac{1}{S + g(N, S)} ≤  (3.31)
≤ O(1) ∑_{R=1}^{T} \frac{1}{(R + g(N, R))2^{g(N,R)}} ∑_{j=1}^{g(N,R)} \frac{1}{2^{j}} ≤
≤ O(1) ∑_{R=1}^{T} \frac{1}{R2^{g(N,R)}} = O(1) ∑_{R=1}^{T} \frac{1}{λ(N, R)} ≤ O(1)(log_{N} T)

and

II ≤ O(1) ∑_{R=1}^{T} ∑_{S=R+g(N,R)+1}^{T} \frac{1}{λ(N, R)} \frac{1}{λ(N, S − R)} ≤  (3.32)
≤ O(1)(log_{N} T)^{2}.

By (3.28) and (3.29)

∑_{R=1}^{T} ∑_{S=R+1}^{T} P(A(R)A(S)) ≤ O(1)(log_{N} T)^{2}.  (3.33)

(3.30), (3.33) and the Kochen-Stone Borel–Cantelli lemma (see e.g. Spitzer [18], page 317) imply the first statement with positive probability. Now to finish our proof we need to apply the zero-one law (again in a non-independent set up) as in the proof of Theorem 3.2, observing that for any n = 1, 2, …

P(A(R) i.o. | X_{1}, X_{2}, ...X_{n}) = P(A(R) i.o.).

□

Proof of Theorem 3.5: The proof goes along the same line as the proof of Theorem 3.4. The only point which needs a little different approach is the proof of the counterpart of Lemma 3.5. Namely, in the proof of this lemma we need an upper bound for 1−p(R, R+1, S), which is equivalent of getting a lower bound for D(R, S). Observe that in Lemma 2.4 we have an asymptotic formula for D(R, ∞). Now to get a lower bound for D(R, S) we need a less precise calculation (the statement of Theorem 3.5 does not depend on B, which was important in Lemma 2.4). It is enough to observe that

U_{i} ≥ C exp (−\frac{B^{∗}}{i^{α}} )

with an appropriate choice of C and B^{∗} > B. After this observation, with some tedious calculation, somewhat similar to Lemma 2.4, we get that

D(R, S) ≥ CR^{α} (1 − (\frac{S}{R})^{α} exp (C_{1}(R^{1−α} − S^{1−α}))) .  (3.34)

It is easy to see

D(R, S) ≥ C_{2}R^{α}

if S ≥ R + R^{α}/ log R. On the other hand, if R < S < R + R^{α}/ log R then it can be seen that

D(R, S) > C_{3}(S − R)

and this is enough to carry through the argument in Lemma 3.5. We omit the details. □

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