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] 3