TRANSIENT NEAREST NEIGHBOR RANDOM WALK

AND BESSEL PROCESS

Endre Csáki111Research supported by the Hungarian National Foundation for
Scientific Research, Grant No. K 61052 and K 67961.
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences,
Budapest, P.O.B. 127, H-1364, Hungary. E-mail address<EMAIL_ADDRESS>

Antónia Földes222Research supported by a PSC CUNY Grant, No. 69020-0038.
Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd.,
Staten Island, New York 10314, U.S.A. E-mail address<EMAIL_ADDRESS>

Pál Révész1
Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität
Wien, Wiedner Hauptstrasse 8-10/107 A-1040 Vienna, Austria. E-mail address:
<EMAIL_ADDRESS>

Abstract: We prove strong invariance principle between a transient Bessel
process and a certain nearest neighbor (NN) random walk that is constructed
from the former by using stopping times. It is also shown that their local
times are close enough to share the same strong limit theorems. It is shown
furthermore, that if the difference between the distributions of two NN random
walks are small, then the walks themselves can be constructed so that they are
close enough. Finally, some consequences concerning strong limit theorems are
discussed.

AMS 2000 Subject Classification: Primary 60F17; Secondary 60F15, 60J10, 60J55,
60J60.

Keywords: transient random walk, Bessel process, strong invariance principle,
local time, strong theorems.

Running head: NN random walk and Bessel process.

##  1\. Introduction

In this paper we consider a nearest neighbor (NN) random walk, defined as
follows: let $X_{0}=0,\ X_{1},X_{2},\ldots$ be a Markov chain with

$\displaystyle E_{i}$ $\displaystyle:=$ $\displaystyle{\bf P}(X_{n+1}=i+1\mid
X_{n}=i)=1-{\bf P}(X_{n+1}=i-1\mid X_{n}=i)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}1&{\rm if}\quad i=0\\\ 1/2+p_{i}&{\rm
if}\quad i=1,2,\ldots,\end{array}\right.$

where $-1/2\leq p_{i}\leq 1/2,\ i=1,2,\ldots$. In case $0<p_{i}\leq 1/2$ the
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
$p_{i}\sim B/4i$ with $B>0$ as $i\to\infty$. We want to show that in certain
sense, this Markov chain is a discrete analogue of continuous Bessel process
and establish a strong invariance principle between these two processes.

The properties of the discrete 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 [12], or more recent papers by Coolen-Schrijner and Van Doorn [6] and
Dette [9]. In an earlier paper [7] we investigated the local time of this
Markov chain in the transient case.

There is a well-known result in the literature (cf. e.g. Chung [5])
characterizing those sequences $\\{p_{i}\\}$ for which $\\{X_{i}\\}$ is
transient (resp. recurrent).

Theorem A: ([5], page 74) Let $X_{n}$ be a Markov chain with transition
probabilities given in (1.) with $-1/2<p_{i}<1/2$, $i=1,2,\ldots$ Define

$U_{i}:={\frac{1-E_{i}}{E_{i}}}={\frac{1/2-p_{i}}{1/2+p_{i}}}$ (1.4)

Then $X_{n}$ is transient if and only if

$\sum_{k=1}^{\infty}\prod_{i=1}^{k}U_{i}<\infty.$

As a consequence, the Markov chain $(X_{n})$ with $p_{R}\sim
B/4R,\,R\to\infty$ is transient if $B>1$ and recurrent if $B<1$.

The Bessel process of order $\nu$, denoted by $Y_{\nu}(t),\,t\geq 0$ is a
diffusion process on the line with generator

$\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{2\nu+1}{2x}\frac{d}{dx}.$

$d=2\nu+2$ is the dimension of the Bessel process. If $d$ is a positive
integer, then $Y_{\nu}(\cdot)$ is the absolute value of a $d$-dimensional
Brownian motion. The Bessel process $Y_{\nu}(t)$ is transient if and only if
$\nu>0$.

The properties of the Bessel process were extensively studied in the
literature. Cf. Borodin and Salminen [2], Revuz and Yor [19], Knight [14].

Lamperti [15] determined the limiting distribution of $X_{n}$ and also proved
a weak convergence theorem in a more general setting. His result in our case
reads as follows.

Theorem B: ([15]) Let $X_{n}$ be a Markov chain with transition probabilities
given in (1.) with $-1/2<p_{i}<1/2$, $i=1,2,\ldots$ If
$\,\,\lim_{R\to\infty}\,Rp_{R}=B/4>-1/4$, then the following weak convergence
holds:

$\frac{X_{[nt]}}{\sqrt{n}}\Longrightarrow Y_{(B-1)/2}(t)$

in the space D[0,1]. In particular,

$\lim_{n\to\infty}{\bf
P}\left(\frac{X_{n}}{\sqrt{n}}<x\right)=\frac{1}{2^{B/2-1/2}\Gamma(B/2+1/2)}\int_{0}^{x}u^{B}e^{-u^{2}/2}\,du.$

In Theorems A and B values of $p_{i}$ can be negative. In the sequel however
we deal only with the case when $p_{i}$ are non-negative, and the chain is
transient, which will be assumed throughout without mentioning it.

Let

$D(R,\infty):=1+\sum_{j=1}^{\infty}\prod_{i=1}^{j}U_{R+i},$ (1.5)

and define

$p_{R}^{*}:=\frac{\frac{1}{2}+p_{R}}{D(R,\infty)}=1-q^{*}_{R}$ (1.6)

Now let $\xi(R,\infty)$, $R=0,1,2,\ldots$ be the total local time at $R$ of
the Markov chain $\\{X_{n}\\}$, i.e.

$\xi(R,\infty):=\\#\\{n\geq 0:X_{n}=R\\}.$ (1.7)

Theorem C: ([7]) For a transient NN random walk

${\bf P}(\xi(R,\infty)=k)=p_{R}^{*}(q_{R}^{*})^{k-1},\qquad k=1,2,\ldots$
(1.8)

Moreover, $\eta(R,t)$, $R>0$ will denote the local time of the Bessel process,
i.e.

$\eta(R,t):=\lim_{\varepsilon\to
0}\frac{1}{2\varepsilon}\int_{0}^{t}I\\{Y_{\nu}(s)\in(R-\varepsilon,R+\varepsilon)\\}\,ds,\qquad\eta(R,\infty):=\lim_{t\to\infty}\eta(R,t).$

It is well-known that $\eta(R,\infty)$ has exponential distribution (see e.g.
[2]).

${\bf P}(\eta(R,\infty)<x)=1-\exp\left(-\frac{\nu}{R}\,x\right).$ (1.9)

For $0<a<b$ let

$\tau:=\tau(a,b)=\min\\{t\geq 0:\,Y_{\nu}(t)\notin(a,b)\\}.$ (1.10)

Then we have (cf. Borodin and Salminen [2], Section 6, 3.0.1 and 3.0.4).

Theorem D: For $0<a<x<b$ we have

${\bf P}_{x}(Y_{\nu}(\tau)=a)=1-{\bf
P}_{x}(Y_{\nu}(\tau)=b)=\frac{x^{-2\nu}-b^{-2\nu}}{a^{-2\nu}-b^{-2\nu}},$
(1.11) ${\bf
E}_{x}e^{-\alpha\tau}=\frac{S_{\nu}(b\sqrt{2\alpha},x\sqrt{2\alpha})+S_{\nu}(x\sqrt{2\alpha},a\sqrt{2\alpha})}{S_{\nu}(b\sqrt{2\alpha},a\sqrt{2\alpha})},$
(1.12)

where

$S_{\nu}(u,v)=(uv)^{-\nu}(I_{\nu}(u)K_{\nu}(v)-K_{\nu}(u)I_{\nu}(v)),$ (1.13)

$I_{\nu}$ and $K_{\nu}$ being the modified Bessel functions of the first and
second kind, resp.

Here and in what follows ${\bf P}_{x}$ and ${\bf E}_{x}$ denote conditional
probability, resp. expectation under $Y_{\nu}(0)=x.$ For simplicity we will
use ${\bf P}_{0}={\bf P},$ and ${\bf E}_{0}={\bf E}.$

Now consider $Y_{\nu}(t),\,t\geq 0$, a Bessel process of order $\nu$,
$Y_{\nu}(0)=0$, and let $X_{n},\,n=0,1,2,\ldots$ be an NN random walk with
$p_{0}=p_{1}=1/2$,

$p_{R}=\frac{(R-1)^{-2\nu}-R^{-2\nu}}{(R-1)^{-2\nu}-(R+1)^{-2\nu}}-\frac{1}{2},\qquad
R=2,3,\ldots$ (1.14)

Our main results are strong invariance principles concerning Bessel process,
NN random walk and their local times.

###### Theorem 1.1.

On a suitable probability space we can construct a Bessel process
$\\{Y_{\nu}(t),\,t\geq 0\\},$ $\nu>0$ and an NN random walk
$\\{X_{n},\,n=0,1,2,\ldots\\}$ with $p_{R}$ as in (1.14) such that for any
$\varepsilon>0$, as $n\to\infty$ we have

$Y_{\nu}(n)-X_{n}=O(n^{1/4+\varepsilon})\qquad{\rm a.s.}$ (1.15)

Our strong invariance principle for local times reads as follows.

###### Theorem 1.2.

Let $Y_{\nu}(t)$ and $X_{n}$ as in Theorem 1.1 and let $\eta$ and $\xi$ their
respective local times. As $R\to\infty$, we have

$\xi(R,\infty)-\eta(R,\infty)=O(R^{1/2}\log R)\quad{\rm a.s.}$ (1.16)

We prove the following strong invariance principle between two NN random
walks.

###### Theorem 1.3.

Let $\\{X_{n}^{(1)}\\}_{n=0}^{\infty}$ and $\\{X_{n}^{(2)}\\}_{n=0}^{\infty}$
be two NN random walk with $p_{j}^{(1)}$ and $p_{j}^{(2)}$, resp. Assume that

$\left|p_{j}^{(1)}-\frac{B}{4j}\right|\leq\frac{C}{j^{\gamma}}$ (1.17)

and

$\left|p_{j}^{(2)}-\frac{B}{4j}\right|\leq\frac{C}{j^{\gamma}}$ (1.18)

$j=1,2,\ldots$ with $B>1$, $1<\gamma\leq 2$ and some non-negative constant
$C$. Then on a suitable probability space one can construct
$\\{X_{n}^{(1)}\\}$ and $\\{X_{n}^{(2)}\\}$ such that as $n\to\infty$

$|X_{n}^{(1)}-X_{n}^{(2)}|=O((X_{n}^{(1)}+X_{n}^{(2)})^{2-\gamma})=O((n\log\log
n)^{1-\gamma/2})\quad{\rm a.s.}$

The organization of the paper is as follows. In Section 2 we will present some
well-known facts and prove some preliminary results. Sections 3-5 contain the
proofs of Theorems 1.1-1.3, respectively. In Section 6 we prove strong
theorems (most of them are integral tests) which easily follow from Theorems
1.1 and 1.2 and the corresponding results for Bessel process. In Section 7,
using our Theorem 1.3 in both directions, we prove an integral test for the
local time of the NN-walk, and a strong theorem for the speed of escape of the
Bessel process.

##  2\. Preliminaries

###### Lemma 2.1.

Let $Y_{\nu}(\cdot)$ be a Bessel process starting from $x=R$ and let $\tau$ be
the stopping time defined by (1.10) with $a=R-1$ and $b=R+1$. Let $p_{R}$ be
defined by (1.14). Then as $R\to\infty$

$p_{R}=\frac{2\nu+1}{4R}+O\left(\frac{1}{R^{2}}\right),$ (2.1) ${\bf
E}_{R}(\tau)=1+O\left(\frac{1}{R}\right),$ (2.2) $Var_{R}(\tau)=O(1).$ (2.3)

Proof: For $\nu=1/2$, i.e. for $d=3$-dimensional Bessel process, in case
$x=R$, $a=R-1$, $b=R+1$ we have

${\bf E}_{R}(e^{\lambda\tau})=\frac{1}{\cos(\sqrt{2\lambda})}$

which does not depend on $R$. We prove that this holds asymptotically in
general, when $\nu>0$.

Using the identity (cf. [2], page 449 and [21], page 78)

$\displaystyle K_{\nu}(x)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{\frac{\pi}{2\sin(\nu\pi)}(I_{-\nu}(x)-I_{\nu}(x))}\quad{\rm
if}\,\,\nu\,\,{\rm is\,\,not\,\,an\,\,integer}\\\ &\\\
\lim_{\mu\to\nu}K_{\mu}(x)\quad{\rm if\,\,}\nu\,\,{\rm
is\,\,an\,\,integer}\end{array}\right.$

and the series expansion

$I_{\nu}(x)=\sum_{k=0}^{\infty}\frac{(x/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)},$

one can see that the coefficient of $-\alpha$ in the Taylor series expansion
of the Laplace transform (1.12) is

${\bf
E}_{x}(\tau)=\frac{1}{2(\nu+1)}\frac{(b^{2}-x^{2})a^{-2\nu}+(x^{2}-a^{2})b^{-2\nu}-(b^{2}-a^{2})x^{-2\nu}}{a^{-2\nu}-b^{-2\nu}}$

from which by putting $x=R$, $a=R-1$, $b=R+1$, we obtain

${\bf
E}_{R}(\tau)=\frac{1}{2(\nu+1)}\frac{(2R+1)(R-1)^{-2\nu}+(2R-1)(R+1)^{-2\nu}-4R^{1-2\nu}}{(R-1)^{-2\nu}-(R+1)^{-2\nu}}$

giving (2.2) after some calculations.

(2.3) can also be obtained similarly, but it seems quite complicated. A
simpler argument is to use moment generating function and expansion of the
Bessel functions for imaginary arguments near infinity. Put $\alpha=-\lambda$
into (1.12) to obtain

${\bf
E}_{x}(e^{\lambda\tau})=\frac{S_{\nu}(ib\sqrt{2\lambda},ix\sqrt{2\lambda})+S_{\nu}(ix\sqrt{2\lambda},ia\sqrt{2\lambda})}{S_{\nu}(ib\sqrt{2\lambda},ia\sqrt{2\lambda})},$
(2.7)

where $i=\sqrt{-1}$. We use the following asymptotic expansions (cf. Erdélyi
et al. [11], page 86, or Watson [21], pages 202, 219)

$I_{\nu}(z)=(2\pi z)^{-1/2}\left(e^{z}+ie^{-z+i\nu\pi}+O(|z|^{-1})\right),$
$K_{\nu}(z)=\left(\frac{\pi}{2z}\right)^{1/2}\left(e^{-z}+O(|z|^{-1})\right).$

Hence one obtains for $\lambda>0$ fixed, and $x<b,\,$

$S_{\nu}(ib\sqrt{2\lambda},ix\sqrt{2\lambda})=(-2\lambda
bx)^{-\nu}(I_{\nu}(ib\sqrt{2\lambda})K_{\nu}(ix\sqrt{2\lambda})-I_{\nu}(ix\sqrt{2\lambda})K_{\nu}(ib\sqrt{2\lambda}))$
$=\frac{1}{2}(-2\lambda
bx)^{-\nu-1/2}\left(e^{i(b-x)\sqrt{2\lambda}}-e^{-i(b-x)\sqrt{2\lambda}}+O\left(\frac{1}{x}\right)\right),\quad
x\to\infty.$

One can obtain asymptotic expansions similarly for
$S_{\nu}(ix\sqrt{2\lambda},ia\sqrt{2\lambda})$,
$S_{\nu}(ib\sqrt{2\lambda},ia\sqrt{2\lambda})$. Putting these into (2.7), with
$x=R$, $a=R-1$, $b=R+1$, we get as $R\to\infty$

${\bf
E}_{R}(e^{\lambda\tau})=\frac{(R^{2}+R)^{-\nu-1/2}+(R^{2}-R)^{-\nu-1/2}}{(R^{2}-1)^{-\nu-1/2}}\,\,\frac{e^{i\sqrt{2\lambda}}-e^{-i\sqrt{2\lambda}}+O\left(\frac{1}{R}\right)}{e^{2i\sqrt{2\lambda}}-e^{-2i\sqrt{2\lambda}}+O\left(\frac{1}{R}\right)}$
$=\frac{1}{\cos(\sqrt{2\lambda})}+O\left(\frac{1}{R}\right).$

Hence putting $\lambda=1$, there exists a constant $C$ such that ${\bf
E}_{R}(e^{\tau})\leq C$ for all $R=1,2,\ldots$ By Markov’s inequality we have

${\bf P}_{R}(\tau>t)={\bf P}_{R}(e^{\tau}>e^{t})\leq Ce^{-t},$

from which ${\bf E}_{R}(\tau^{2})\leq 2C$, implying (2.3). $\Box$

Here and throughout $C,C_{1},C_{2},\ldots$ denotes unimportant positive
(possibly random) constants whose values may change from line to line.

Recall the definition of the upper and lower classes for a stochastic process
$Z(t),\,t\geq 0$ defined on a probability space $(\Omega,{\cal F},P)$ (cf.
Révész [18], p. 33).

The function $a_{1}(t)$ belongs to the upper-upper class of $Z(t)$
($a_{1}(t)\in{\rm UUC}(Z(t)$) if for almost all $\omega\in\Omega$ there exists
a $t_{0}(\omega)>0$ such that $Z(t)<a_{1}(t)$ if $t>t_{0}(\omega).$

The function $a_{2}(t)$ belongs to the upper-lower class of $Z(t)$
($a_{1}(t)\in{\rm ULC}(Z(t)$) if for almost all $\omega\in\Omega$ there exists
a sequence of positive numbers
$0<t_{1}=t_{1}(\omega)<t_{2}=t_{2}(\omega)<\ldots$ with
$\lim_{i\to\infty}t_{i}=\infty$ such that $Z(t_{i})\geq a_{2}(t_{i})$,
$(i=1,2,\ldots).$

The function $a_{3}(t)$ belongs to the lower-upper class  of $Z(t)$
($a_{3}(t)\in{\rm LUC}(Z(t)$) if for almost all $\omega\in\Omega$ there exists
a sequence of positive numbers
$0<t_{1}=t_{1}(\omega)<t_{2}=t_{2}(\omega)<\ldots$ with
$\lim_{i\to\infty}t_{i}=\infty$ such that $Z(t_{i})\leq a_{3}(t_{i})$,
$(i=1,2,\ldots).$

The function $a_{4}(t)$ belongs to the lower-lower class of $Z(t)$
($a_{4}(t)\in{\rm LLC}(Z(t)$) if for almost all $\omega\in\Omega$ there exists
a $t_{0}(\omega)>0$ such that $Z(t)>a_{4}(t)$ if $t>t_{0}(\omega).$

The following lower class results are due to Dvoretzky and Erdős [10] for
integer $d=2\nu+2$. In the general case when $\nu>0$, the proof is similar
(cf. also Knight [14] and Chaumont and Pardo [4] in the case of positive self-
similar Markov processes).

Theorem E: Let $\nu>0$ and let $b(t)$ be a non-increasing, non-negative
function.

  * •

$t^{1/2}b(t)\in{\rm LLC}(Y_{\nu}(t))$ if and only if
$\displaystyle{\int_{1}^{\infty}(b(2^{t}))^{2\nu}\,dt<\infty.}$

It follows e.g. that in case $\nu>0$, for any $\varepsilon>0$ we have

$Y_{\nu}(t)\geq t^{1/2-\varepsilon}$ (2.8)

almost surely for all sufficiently large $t$.

In fact, from our invariance principle it will follow that the integral test
in Theorem E holds also for our Markov chain $(X_{n})$. In the proof however
we need an analogue of (2.8) for $X_{n}$.

One can easily calculate the exact distribution of $\xi(R,\infty),$ the total
local time of $X_{n}$ of Theorem 1.1 according to Theorem C.
Lemma A: If $p_{R}$ is given by (1.14), then $\xi(R,\infty)$ has geometric
distribution (1.8) with

$p_{R}^{*}=\frac{\frac{1}{2}+p_{R}}{D(R,\infty)}=\frac{(\frac{1}{2}+p_{R})((R+1)^{2\nu}-R^{2\nu})}{(R+1)^{2\nu}}=\frac{\nu}{R}+O\left(\frac{1}{R}\right).$
(2.9)

###### Lemma 2.2.

For any $\delta>0$ we have

$X_{n}\geq n^{1/2-\delta}$

almost surely for all large enough $n$.

Proof: From Lemma A it is easy to conclude that almost surely for some
$R_{0}>0$

$\xi(R,\infty)\leq CR\log R$

if $R\geq R_{0}$, with some random positive constant $C$. Hence the time
$\displaystyle{\sum_{R=1}^{S}}\xi(R,\infty)$ which the particle spent up to
$\infty$ in $[1,S]$ is less than

$\sum_{R=1}^{R_{0}-1}\xi(R,\infty)+C\sum_{R=R_{0}}^{S}R\log R\leq
C_{1}S^{2+\delta}$

with some (random) $C_{1}>0$. Consequently, after $C_{1}S^{2+\delta}$ steps
the particle will be farther away from the origin than $S$. Let

$n=[C_{1}S^{2+\delta}],$

then

$S\geq\left(\frac{n}{C_{1}}\right)^{1/(2+\delta)}$

and hence

$X_{n}\geq\left(\frac{n}{C_{1}}\right)^{1/(2+\delta)}\geq n^{1/2-\delta}$

for $n$ large enough. This proves the Lemma. $\Box$

##  3\. Proof of Theorem 1.1

Define the sequences $(\tau_{n})$, $t_{0}=0,$
$t_{n}:=\tau_{1}+\ldots+\tau_{n}$ as follows:

$\displaystyle\tau_{1}$ $\displaystyle:=$ $\displaystyle\min\\{t:\ t>0,\
Y_{\nu}(t)=1\\},$ $\displaystyle\tau_{2}$ $\displaystyle:=$
$\displaystyle\min\\{t:\ t>0,\ Y_{\nu}(t+t_{1})=2\\},$ $\displaystyle\tau_{n}$
$\displaystyle:=$ $\displaystyle\min\\{t:\ t>0,\
|Y_{\nu}(t+t_{n-1})-Y_{\nu}(t_{n-1})|=1\\}\quad{\rm for}\quad n=3,4,\ldots$

Let $X_{n}=Y_{\nu}(t_{n})$. Then (cf. (1.14)) it is an NN random walk with
$p_{0}=p_{1}=1/2$,

$p_{R}=\frac{(R-1)^{-2\nu}-R^{-2\nu}}{(R-1)^{-2\nu}-(R+1)^{-2\nu}}-\frac{1}{2},\qquad
R=2,3,\ldots$

Let ${\cal F}_{n}$ be the $\sigma$-algebra generated by $(\tau_{k},\
Y_{\nu}(\tau_{k}))_{k=1}^{n}$ and consider

$M_{n}:=\sum_{i=1}^{n}(\tau_{i}-{\bf E}(\tau_{i}\mid{\cal F}_{i-1})).$

Then the sequence $(M_{n})_{n\geq 1}$ is a martingale with respect to $({\cal
F}_{n})_{n\geq 1}$. It follows from (2.2) of Lemma 2.1 that for $i=2,3,\ldots$
we have

${\bf E}(\tau_{i}\mid{\cal F}_{i-1})={\bf E}(\tau_{i}\mid
Y_{\nu}(t_{i-1}))=1+O\left(\frac{1}{Y_{\nu}(t_{i-1})}\right).$

Hence

$|t_{n}-n|\leq|M_{n}|+|\tau_{1}-1|+C_{1}\sum_{i=2}^{n}\frac{1}{Y_{\nu}(t_{i-1})}=|M_{n}|+|\tau_{1}-1|+C_{1}\sum_{i=2}^{n}\frac{1}{X_{i-1}}$

with some (random) constant $C_{1}$. By (2.3) of Lemma 2.1 we have ${\bf
E}M_{n}^{2}\leq Cn$. Let $\varepsilon>0$ be arbitrary and define
$n_{k}=[k^{1/\varepsilon}]$. From the martingale inequality we get

${\bf P}\left(\max_{n_{k-1}\leq n\leq n_{k}}|M_{n}|\geq
C_{1}n_{k-1}^{1/2+\varepsilon}\right)\leq\frac{C_{2}}{n_{k}^{2\varepsilon}},$

hence we obtain by Borel-Cantelli lemma

$\max_{n_{k-1}\leq n\leq n_{k}}|M_{n}|\leq C_{1}n_{k-1}^{1/2+\varepsilon}$

almost surely for large $k$. Hence we also have

$|M_{n}|=O(n^{1/2+\varepsilon})\qquad{\rm a.s.}$

By Lemma 2.2

$\sum_{i=2}^{n}\frac{1}{X_{i-1}}=O(n^{1/2+\varepsilon})\qquad{\rm a.s.},$

consequently

$|t_{n}-n|=O(n^{1/2+\varepsilon})\qquad{\rm a.s.}$ (3.1)

It is well-known (cf. [2], p. 69) that $Y_{\nu}(t)$ satisfies the stochastic
differential equation

$dY_{\nu}(t)=dW(t)+\frac{2\nu+1}{2Y_{\nu}(t)}dt,$ (3.2)

where $W(t)$ is a standard Wiener process. Hence

$X_{n}-Y_{\nu}(n)=Y_{\nu}(t_{n})-Y_{\nu}(n)=W(t_{n})-W(n)+\int_{t_{n}}^{n}\frac{2\nu+1}{2Y_{\nu}(s)}\,ds,$

consequently,

$|X_{n}-Y_{\nu}(n)|\leq|W(t_{n})-W(n)|+\frac{(2\nu+1)|t_{n}-n|}{2}\max_{\min(n,t_{n})\leq
t\leq\max(n,t_{n})}\frac{1}{Y_{\nu}(t)}.$

Now by (3.1) and (2.8) the last term is $O(n^{2\varepsilon})$ almost surely
and since for the increments of the Wiener process (cf. [8], page 30)

$|W(t_{n})-W(n)|=O(n^{1/4+\varepsilon})\qquad{\rm a.s.}$

as $n\to\infty$, we have (1.15) of Theorem 1.1. $\Box$

##  4\. Proof of Theorem 1.2

For $R>0$ integer define

$\displaystyle\kappa_{1}$ $\displaystyle:=$ $\displaystyle\min\\{t\geq
0:\,Y_{\nu}(t)=R\\},$ $\displaystyle\delta_{1}$ $\displaystyle:=$
$\displaystyle\min\\{t\geq\kappa_{1}:\,Y_{\nu}(t)\notin(R-1,R+1)\\},$
$\displaystyle\kappa_{i}$ $\displaystyle:=$
$\displaystyle\min\\{t\geq\delta_{i-1}:\,Y_{\nu}(t)=R\\},$
$\displaystyle\delta_{i}$ $\displaystyle:=$
$\displaystyle\min\\{t\geq\kappa_{i}:\,Y_{\nu}(t)\notin(R-1,R+1)\\},$
$\displaystyle\kappa^{*}$ $\displaystyle:=$ $\displaystyle\max\\{t\geq
0\,:Y_{\nu}(t)=R\\},$

$i=2,3,\ldots$

Consider the local times at $R$ of the Bessel process during excursions around
$R$, i.e. let

$\zeta_{i}:=\eta(R,\delta_{i})-\eta(R,\kappa_{i}),\quad i=1,2,\ldots,$
$\tilde{\zeta}:=\eta(R,\infty)-\eta(R,\kappa^{*}).$

We have

$\eta(R,\infty)=\sum_{i=1}^{\xi(R,\infty)-1}\zeta_{i}+\tilde{\zeta}.$

###### Lemma 4.1.

${\bf
E}\left(e^{\lambda\eta(R,\infty)}\right)=\frac{p^{*}_{R}\,\varphi(\lambda)}{1-q_{R}^{*}\,\varphi(\lambda)},$
(4.1)

where

$p_{R}^{*}=\frac{A_{R}}{A_{R}+B_{R}}\,\frac{(R+1)^{2\nu}-R^{2\nu}}{(R+1)^{2\nu}},\quad
q_{R}^{*}=1-p_{R}^{*},$ (4.2)

$\varphi(\lambda)=\frac{\nu(A_{R}+B_{R})}{\nu(A_{R}+B_{R})-\lambda
R^{2\nu+1}A_{R}B_{R}},$ (4.3)

and

$A_{R}=(R-1)^{-2\nu}-R^{-2\nu},\qquad B_{R}=R^{-2\nu}-(R+1)^{-2\nu}.$ (4.4)

Proof: By ([2], p. 395, 3.3.2) $\zeta_{i}$ are i.i.d. random variables having
exponential distribution with moment generating function $\varphi(\lambda)$
given in (4.3). Moreover, it is obvious that $\tilde{\zeta}$ is independent
from $\sum_{i=1}^{\xi(R,\infty)-1}\zeta_{i}.$ Furthermore, $\tilde{\zeta}$ is
the local time of $R$ under the condition that starting from $R,$ $Y_{\nu}(t)$
will reach $R+1$ before $R-1.$ Hence its distribution can be calculated from
formula 3.3.5(b) of [2], and its moment generating function happens to be
equal to $\varphi(\lambda)$ of (4.3). $\Box$

We can see

$\theta:={\bf E}(\zeta_{i})={\bf
E}(\tilde{\zeta})=\frac{\nu(A_{R}+B_{R})}{R^{2\nu+1}A_{R}B_{R}}=1+O\left(\frac{1}{R}\right),\quad
R\to\infty.$

${\bf P}(|\eta(R,\infty)-\xi(R,\infty)|\geq u)={\bf
P}\left(\left|\sum_{i=1}^{\xi(R,\infty)-1}(\zeta_{i}-\theta)+\tilde{\zeta}-\theta\right|\geq
u\right)$ $\leq{\bf P}(\xi(R.\infty)>N)+{\bf P}\left(\max_{k\leq
N}\left|\sum_{i=1}^{k}(\zeta_{i}-\theta)\right|\geq u\right)$

$\leq(q^{*}_{R})^{N}+e^{-\lambda
u}\left(\left(\frac{e^{\lambda\theta}}{1+\lambda\theta}\right)^{N}+\left(\frac{e^{-\lambda\theta}}{1-\lambda\theta}\right)^{N}\right).$

In the above calculation we used the common moment generating function (4.3)
of $\zeta_{i}$ and $\tilde{\zeta}$, the exact distribution of $\xi(R,\infty)$
(see (1.8)) and the exponential Kolmogorov inequality. Estimating the above
expression with standard methods and selecting

$N=CR\log R,\quad u=CR^{1/2}\log R,\quad\lambda=\frac{u}{\theta^{2}N}$

we conclude that

${\bf P}(|\eta(R,\infty)-\xi(R,\infty)|\geq CR^{1/2}\log R)\leq
C_{1}\exp{\left(-\frac{C\log R}{2\theta}\right)}.$

With a big enough $C$ the right hand side of the above inequality is summable
in $R,$ hence Theorem 1.2 follows by the Borel-Cantelli lemma. $\Box$

##  5\. Proof of Theorem 1.3

Let $p_{j}^{(1)}$ and $p_{j}^{(2)}$ as in Theorem 1.3. Define the two-
dimensional Markov chain $(X_{n}^{(1)},X_{n}^{(2)})$ as follows. If
$p_{j}^{(1)}\geq p_{k}^{(2)}$, then let

$\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j+1,k+1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}+p_{k}^{(2)}$ $\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j+1,k-1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle p_{j}^{(1)}-p_{k}^{(2)}$ $\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j-1,k-1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}-p_{j}^{(1)}.$

If, however $p_{j}^{(1)}\leq p_{k}^{(2)}$, then let

$\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j+1,k+1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}+p_{j}^{(1)}$ $\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j-1,k+1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle p_{k}^{(2)}-p_{j}^{(1)}$ $\displaystyle{\bf
P}\left((X_{n+1}^{(1)},X_{n+1}^{(2)})=(j-1,k-1)\mid(X_{n}^{(1)},X_{n}^{(2)})=(j,k)\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}-p_{k}^{(2)}.$

Then it can be easily seen that $X_{n}^{(1)}$ and $X_{n}^{(2)}$ are two NN
random walks as desired. Consider the following 4 cases.

  * •

(i) $p_{j}^{(1)}\leq p_{k}^{(2)}$, $j\leq k$,

  * •

(ii) $p_{j}^{(1)}\leq p_{k}^{(2)}$, $j\geq k$,

  * •

(iii) $p_{j}^{(1)}\geq p_{k}^{(2)}$, $j\leq k$,

  * •

(iv) $p_{j}^{(1)}\geq p_{k}^{(2)}$, $j\geq k$.

In case (i) from (1.17) and (1.18) we obtain

$\frac{B}{4j}-\frac{C}{j^{\gamma}}\leq\frac{B}{4k}+\frac{C}{k^{\gamma}}\leq\frac{B}{4k}+\frac{C}{kj^{\gamma-1}},$

implying

$k-j\leq\frac{2Cj^{2-\gamma}}{B/4-Cj^{1-\gamma}}=O(j^{2-\gamma})$

if $j\to\infty$. So in this case if $X_{n}^{(1)}=j$ and $X_{n}^{(2)}=k$, then
we have

$0\leq X_{n}^{(2)}-X_{n}^{(1)}=O((X_{n}^{(1)})^{2-\gamma})$

if $n\to\infty$.

In case (ii) either $X_{n+1}^{(1)}-X_{n+1}^{(2)}=X_{n}^{(1)}-X_{n}^{(2)}$, or
$X_{n+1}^{(1)}-X_{n+1}^{(2)}=X_{n}^{(1)}-X_{n}^{(2)}-2$, so that we have

$-2\leq X_{n+1}^{(1)}-X_{n+1}^{(2)}\leq X_{n}^{(1)}-X_{n}^{(2)}.$

Similar procedure shows that in case (iii)

$-2\leq X_{n+1}^{(2)}-X_{n+1}^{(1)}\leq X_{n}^{(2)}-X_{n}^{(1)}$

and in case (iv)

$0\leq X_{n}^{(1)}-X_{n}^{(2)}=O((X_{n}^{(2)})^{2-\gamma}).$

Hence Theorem 1.3 follows from the law of the iterated logarithm for
$X_{n}^{(i)}$ (cf. [3]). $\Box$

##  6\. Strong theorems

As usual, applying Theorem 1.1 and Theorem 1.3, we can give limit results
valid for one of the processes to the other process involved.

In this section we denote $Y_{\nu}(t)$ by $Y(t)$ and define the following
related processes.

$M(t):=\max_{0\leq s\leq t}Y(s),\qquad Q_{n}:=\max_{1\leq k\leq n}X_{k}.$

The future infimums are defined as

$I(t):=\inf_{s\geq t}Y(s),\qquad J_{n}:=\inf_{k\geq n}X_{k}.$

Escape processes are defined by

$A(t):=\sup\\{s:\,Y(s)\leq t\\},\qquad G_{n}:=\sup\\{k:\,X_{k}\leq n\\}.$

Laws of the iterated logarithm are known for Bessel processes (cf. [2]) and NN
random walks (cf. [3]) as well. Upper class results for Bessel process read as
follows (cf. Orey and Pruitt [16] for integral $d$, and Pardo [17] for the
case of positive self-similar Markov processes).

Theorem F: Let $a(t)$ be a non-decreasing non-negative continuous function.
Then for $\nu\geq 0$

$\displaystyle{t^{1/2}a(t)\in{\rm UUC}(Y(t))\qquad
if\,\,and\,\,only\,\,if\qquad\int_{1}^{\infty}\frac{(a(x))^{2\nu+2}}{x}e^{-a^{2}(x)/2}\,dx<\infty.}$

Now Theorems 1.1, 1.3 and Theorems E and F together imply the following
result.

###### Theorem 6.1.

Let $\\{X_{n}\\}$ be an NN random walk with $p_{R}$ satisfying

$p_{R}=\frac{B}{4R}+O\left(\frac{1}{R^{1+\delta}}\right),\quad R\to\infty$

with $B>1$ and for some $\delta>0$. Let furthermore $a(t)$ be a non-decreasing
non-negative function. Then

$\displaystyle{n^{1/2}a(n)\in{\rm UUC}(X_{n})\qquad
if\,\,and\,\,only\,\,if\qquad\sum_{k=1}^{\infty}\frac{(a(k))^{B+1}}{k}e^{-a^{2}(k)/2}<\infty.}$

If $b(t)$ is a non-increasing non-negative function, then

$\displaystyle{n^{1/2}b(n)\in{\rm LLC}(X_{n})\qquad
if\,\,and\,\,only\,\,if\qquad\sum_{k=1}^{\infty}(b(2^{k}))^{B-1}<\infty.}$

Next we prove the following invariance principles for the processes defined
above.

###### Theorem 6.2.

Let $Y(t)$ and $X_{n}$ as in Theorem 1.1. Then for any $\varepsilon>0$ we have

$|M(n)-Q_{n}|=O(n^{1/4+\varepsilon})\quad{\rm a.s.}$ (6.1)

and

$|I(n)-J_{n}|=O(n^{1/4+\varepsilon})\quad{\rm a.s.}$ (6.2)

Proof: Define $\tilde{s},s^{*},\tilde{k},k^{*}$ by

$Y(\tilde{s})=M(n),\quad Y(s^{*})=I(n),\quad X_{\tilde{k}}=Q_{n},\quad
X_{k^{*}}=J_{n}.$

Then as $n\to\infty$, we have almost surely

$Q_{n}-M(n)=X_{\tilde{k}}-Y(\tilde{s})\leq
X_{\tilde{k}}-Y(\tilde{k})=O(n^{1/4+\varepsilon})$

and

$M(n)-Q_{n}=Y(\tilde{s})-X_{\tilde{k}}=Y(\tilde{s})-Y([\tilde{s}])-(X_{[\tilde{s}]}-Y([\tilde{s}]))+X_{[\tilde{s}]}-X_{\tilde{k}}$
$\leq
Y(\tilde{s})-Y([\tilde{s}])-(X_{[\tilde{s}]}-Y([\tilde{s}])=Y(\tilde{s})-Y([\tilde{s}])+O(n^{1/4+\varepsilon})$

By (3.2) and recalling the results on the increments of the Wiener process
(see [8] page 30) we get

$Y(\tilde{s})-Y([\tilde{s}])=W(\tilde{s})-W([\tilde{s}])+\int_{[\tilde{s}]}^{\tilde{s}}\frac{2\nu+1}{2Y(s)}\,ds$
$\leq\sup_{0\leq t\leq n}\sup_{0\leq s\leq
1}|W(t+s)-W(t)|+\frac{2\nu+1}{2}\max_{[\tilde{s}]\leq
t\leq\tilde{s}}\frac{1}{Y(t)}=O(\log n)\quad{\rm a.s.},$

since $Y(t)$ in the interval $([\tilde{s}],\tilde{s})$ is bounded away from
zero. Hence (6.1) follows.

To show (6.2), note that $n\leq s^{*}\leq n^{1+\alpha}$ and $n\leq k^{*}\leq
n^{1+\alpha}$ for any $\alpha>0$ almost surely for all large $n$. Then as
$n\to\infty$

$I(n)-J_{n}\leq
Y(k^{*})-X_{k^{*}}=O((k^{*})^{1/4+\varepsilon})=O(n^{(1+\alpha)(1/4+\varepsilon)})\quad{\rm
a.s.}$

On the other hand,

$J_{n}-I(n)\leq
X_{k^{*}}-Y([s^{*}])+Y([s^{*}])-Y(s^{*})=O(n^{(1+\alpha)(1/4+\varepsilon)})+Y([s^{*}])-Y(s^{*}).$

By (3.2), taking into account that when applying this formula the integral
contribution is negative, and recalling again the results on the increments of
the Wiener process, we get

$Y([s^{*}])-Y(s^{*})\leq W([s^{*}])-W(s^{*})\leq\sup_{0\leq t\leq
n^{1+\alpha}}\sup_{0\leq s\leq 1}|W(t+s)-W(t)|=O(\log n)\quad{\rm a.s.}$

as $n\to\infty$. Hence

$|I(n)-J_{n}|=O(n^{(1+\alpha)(1/4+\varepsilon)})\quad{\rm a.s.}$

Since $\alpha>0$ and $\varepsilon>0$ are arbitrary, (6.2) follows. This
completes the proof of Theorem 6.2. $\Box$

###### Theorem 6.3.

Let $X_{n}^{(1)}$ and $X_{n}^{(2)}$ as in Theorem 1.3 and let $Q_{n}^{(1)}$
and $Q_{n}^{(2)}$ be the corresponding maximums, while let $J_{n}^{(1)}$ and
$J_{n}^{(2)}$ be the corresponding future infimum processes. Then for any
$\varepsilon>0$, as $n\to\infty$ we have

$|Q_{n}^{(1)}-Q_{n}^{(2)}|=O(n^{1-\gamma/2+\varepsilon})\quad{\rm a.s.}$ (6.3)

and

$|J_{n}^{(1)}-J_{n}^{(2)}|=O(n^{1-\gamma/2+\varepsilon})\quad{\rm a.s.}$ (6.4)

Proof: Define $\tilde{k}_{i},k_{i}^{*},\,i=1,2$ by

$X^{(i)}_{\tilde{k}_{i}}=Q_{n}^{(i)},\quad X_{k_{i}^{*}}^{(i)}=J_{n}^{(i)}.$

Then

$|Q_{n}^{(1)}-Q_{n}^{(2)}|\leq\max(X_{\tilde{k}_{1}}^{(1)}-X_{\tilde{k}_{1}}^{(2)},X_{\tilde{k}_{2}}^{(1)}-X_{\tilde{k}_{2}}^{(2)})=O((n\log\log
n)^{1-\gamma/2})\quad{\rm a.s.},$

proving (6.3).

Moreover, for any $\alpha>0$, $n\leq k_{i}^{*}\leq n^{1+\alpha}$ almost surely
for large $n$, hence we have

$|J_{n}^{(1)}-J_{n}^{(2)}|\leq\max(X_{k_{1}^{*}}^{(1)}-X_{k_{1}^{*}}^{(2)},X_{k_{2}^{*}}^{(1)}-X_{k_{2}^{*}}^{(2)})=O((n\log\log
n)^{(1+\alpha)(1-\gamma/2)})\quad{\rm a.s.}$

Since $\alpha$ is arbitrary, (6.4) follows.

This completes the proof of Theorem 6.3. $\Box$

Khoshnevisan et al. [13] (for $I(t)$ and $A(t)$), Adelman and Shi [1], and Shi
[20] (for $Y(t)-I(t)$) proved the following upper and lower class results.

Theorem G: Let $\varphi(t)$ be a non-increasing, and $\psi(t)$ be a non-
decreasing function, both non-negative. Then for $\nu>0$

  * •

$\displaystyle{t^{1/2}\psi(t)\in{\rm UUC}(I(t))}$ if and only if
$\displaystyle{\int_{1}^{\infty}\frac{(\psi(x))^{2\nu}}{x}e^{-\psi^{2}(x)/2}\,dx<\infty,}$

  * •

$\displaystyle{t^{2}\varphi(t)\in{\rm LLC}(A(t))}$ if and only if
$\displaystyle{\int_{1}^{\infty}\frac{1}{x\varphi^{\nu}(x)}e^{-1/2\varphi(x)}\,dx<\infty.}$

  * •

$\displaystyle{t^{1/2}\psi(t)\in{\rm UUC}(Y(t)-I(t))}$ if and only if
$\displaystyle{\int_{1}^{\infty}\frac{1}{x\psi^{2\nu-2}(x)}e^{-\psi^{2}(x)/2}\,dx<\infty,}$

Theorem H: Let $\rho(t)>0$ be such that $(\log\rho(t))/\log t$ is non-
decreasing. Then

  * •

$\displaystyle{1/\rho(t)\in{\rm LLC}(M(t)-I(t))}$ if and only if
$\displaystyle{\int_{1}^{\infty}\frac{dx}{x\log\rho(x)}<\infty.}$

Taking into account that $J_{n}$ and $G_{n}$ are inverses of each other,
immediate consequences of Theorems F, G, H, Theorems 6.2 and 6.3 are the
following upper and lower class results.

###### Theorem 6.4.

Let $X_{n}$ be as in Theorem 6.1 and let $\varphi(t)$ be a non-increasing and
$\psi(t)$ be a non-decresing function, both non-negative. Then

  * •

$\displaystyle{n^{1/2}\psi(n)\in{\rm UUC}(J_{n})}$ if and only if
$\displaystyle{\sum_{k=1}^{\infty}\frac{(\psi(k))^{B-1}}{k}e^{-\psi^{2}(k)/2}<\infty,}$

  * •

$\displaystyle{n^{2}\varphi(n)\in{\rm LLC}(G_{n})}$ if and only if
$\displaystyle{\sum_{k=1}^{\infty}\frac{1}{k\varphi^{(B-1)/2}(k)}e^{-1/2\varphi(k)}<\infty.}$

  * •

$\displaystyle{n^{1/2}\psi(n)\in{\rm UUC}(X_{n}-J_{n})}$ if and only if
$\displaystyle{\sum_{k=1}^{\infty}\frac{1}{k\psi^{B-3}(k)}e^{-\psi^{2}(k)/2}<\infty,}$

###### Theorem 6.5.

Let $\rho(t)>0$ be such that $(\log\rho(t))/\log t$ is non-decreasing.

  * •

$\displaystyle{1/\rho(n)\in{\rm LLC}(Q_{n}-J_{n})}$ if and only if
$\displaystyle{\sum_{k=2}^{\infty}\frac{1}{k\log\rho(k)}<\infty.}$

##  7\. Local time

We will need the following result from Yor [22], page 52.

Theorem J: For the local time of a Bessel process of order $\nu$ we have

$\eta(R,\infty)\stackrel{{\scriptstyle{\cal
D}}}{{=}}(2\nu)^{-1}R^{1-2\nu}Y_{0}^{2}(R^{\,2\nu}),$

where $Y_{0}$ is a two-dimensional Bessel process and
$\stackrel{{\scriptstyle{\cal D}}}{{=}}$ means equality in distribution.

Hence applying Theorem F for $\nu=0$, we get

Theorem K: If $f(x)$ is non-decreasing, non-negative function, then

  * •

$\displaystyle{Rf(R)\in{\rm UUC}(\eta(R,\infty))}$ if and only if
$\displaystyle{\int_{1}^{\infty}\frac{f(x)}{x}e^{-\nu f(x)}\,dx<\infty}.$

From this and Theorem 1.2 we get the following result.

###### Theorem 7.1.

If $f(x)$ is non-decreasing, non-negative function, then

  * •

$\displaystyle{Rf(R)\in{\rm UUC}(\xi(R,\infty))}$ if and only if
$\displaystyle{\sum_{k=1}^{\infty}\frac{f(k)}{k}e^{-\nu f(k)}<\infty.}$

In [7] we proved the following result.

Theorem L: Let
$\displaystyle{p_{R}=\frac{B}{4R}+O\left(\frac{1}{R^{\gamma}}\right)}$ with
$B>1$, and $\gamma>1$. Then with probability $1$ there exist infinitely many
$R$ for which

$\xi(R+j,\infty)=1$

for each $j=0,1,2,\ldots,[\log\log R/\log 2]$. Moreover, with probability $1$
for each $R$ large enough and $\varepsilon>0$ there exists an

$R\leq S\leq{\frac{(1+\varepsilon)\log\log R}{\log 2}}$

such that

$\xi(S,\infty)>1.$

Remark 1: In fact in [7] we proved this result in the case when $p_{R}=B/{4R}$
but the same proof works also in the case of Theorem L.

This theorem applies e.g. for the case when $p_{R}$ is given by (1.14), which
in turn, gives the following result for the Bessel process.

Let

  * (i)

$\kappa(R):=\inf\\{t:\ Y_{\nu}(t)=R\\}$,

  * (ii)

$\kappa^{*}(R):=\sup\\{t:\ Y_{\nu}(t)=R\\}$,

  * (iii)

$\Psi(R)$ be the largest integer for which the event

$A(R)=\bigcap_{j=-1}^{\Psi(R)}\\{\kappa^{*}(R+j)<\kappa(R+j+1)\\}$

occurs.

$A(R)$ means that $Y_{\nu}(t)$ moves from $R$ to $R+1$ before returning to
$R-1$, it goes from $R+1$ to $R+2$ before returning to $R$, $\ldots$ and also
from $R+\Psi(R)$ to $R+\Psi(R)+1$ and it never returns to $R+\Psi(R)-1$. We
say that the process $Y_{\nu}(t)$ escapes through $(R,R+\Psi(R))$ with large
velocity.

###### Theorem 7.2.

$\limsup_{R\to\infty}{\frac{\Psi(R)}{\log\log R}}={\frac{1}{\log 2}}\quad{\rm
a.s.}$

Remark 2: The statement of Theorem 7.2 (for integral $d=2\nu+2$) was
formulated in [18], p. 291 as a Conjecture.

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