The number of favorite points of a simple random walk

Transcription

The number of favorite points of a simple random walk
```The number of favorite points of a simple
random walk
Richard F. Bass
June 8, 2013
Abstract
If Vn is the set of favorite points of a simple random walk at time
n, can the cardinality of Vn be equal to 3 infinitely often?
In [3] Erdös and Révész introduced the concept of the favorite points of
a simple random walk. Independently and around the same time Bass and
Griffin [1] studied the same notion, which they called the most visited sites
of the random walk.
Here are the definitions. Let X1 , X2 , . . . be an independent identically
distributed P
sequence of random variables with P(Xi = 1) = P(Xi = −1) = 12 .
Then Sn = ni=1 Xi is a simple random walk. Define
Nnk
=
n
X
1(Sj =k) ,
j=1
the number of times the site k has been visited up to time n, and let
Nn∗ = max Nnk .
k∈Z
Let
Vn = {k ∈ Z : Nnk = Nn∗ },
the collection of favorite points or most visited sites of the random walk at
time n.
1
The main result of [3] is that there exists a function f (n) that is not in
the upper class for Sn but is in the upper class for Vn+ = max{k : k ∈ Vn }.
This answered a question posed in [1].
The main result of [1] is that Vn− = min{|k| : k ∈ Vn } is transient, and
that moreover
Vn−
= ∞,
a.s.
(1)
lim inf √
n→∞
n/(log n)γ
if γ > 11. In particular, with probability one, 0 is not in Vn infinitely often,
which answered a question in [3].
A question posed by Erdös and Révész in 1984 in [3] concerns the cardinality of Vn . Let #A denote the cardinality of a set A. It is easy to see
that
P(#Vn ≥ 2 i.o.) = 1.
The question Erdös and Révész asked 29 years ago is:
Open problem: Determine
P(#Vn ≥ 3 i.o.).
(2)
Tóth [6] proved that
P(#Vn ≥ 4 i.o.) = 0.
This is where things stand today. See [5] for more on some of the questions
posed in [3].
Once could ask the analogous question for Brownian motion. Let Lxt
be a jointly continuous version of the local times for Brownian motion, let
L∗t = supx∈R Lxt and let
Ut = {x ∈ R : Lxt = L∗t }.
Eisenbaum [2] and Leuridan [4] independently proved that with probability
one, supt # Ut = 2.
2
References
[1] R.F. Bass and P.S. Griffin, The most visited site of Brownian motion
and simple random walk. Z. Wahrsch. Verw. Gebiete 70 (1985) 417-436.
[2] N. Eisenbaum, Temps locaux, excursions et lieu le plus visité par un
mouvement brownien lineaire. Thèse de doctorat, Université de Paris 7,
1989.
[3] P. Erdös and P. Révész, On the favourite points of a random walk. Mathematical Structure-Computational Mathematics-Mathematical Modelling 2, 152–157. Bulgarian Academy of Sciences, Sofia, 1984.
[4] C. Leuridan, Problèmes lié aux temps locaux du mouvement brownien:
estimations de normes H p , théorèmes de Ray-Knight sur le tore, point
le plus visité. Thése de doctorat, Université Joseph Fourier, Grenoble,
1994.
[5] Z. Shi and B. Tóth, Favorite sites of simple random walk. Period. Math.
Hungar. 41 (2000) 237–249.
[6] B. Tóth, No more than three favorite sites for simple random walk. Ann.
Probab. 29 (2001) 484–503.
Richard F. Bass
Department of Mathematics
University of Connecticut
Storrs, CT 06269-3009, USA
[email protected]
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