Preprint:
On Dynamic Bit Processes
D. Khoshnevisan, D. A. Levin, and P. J.
Méndez-Hernández
Abstract.
Let Xk(t):=(X1(t),…,Xk(t)) denote a k-vector of
i.i.d. random variables, each taking the values 1 or 0 with
respective probabilities p∈(0,1) and 1-p. As a process
indexed by t≥0, Xk is constructed---following
Benjamini, Häggström, Peres, and Steif (2003)---so that
it is strong Markov with invariant measure
((1-p)δ0+pδ1)k.
We derive sharp estimates for the
probability that ``X1(t)+…+Xk(t)=
k-λ for some t∈F,'' where F⊆[0,1] is nonrandom and compact. We do this
in two very different settings: (i) Where λ is a constant; and
(ii) Where λ=k/2, k is even, and p=q=1/2. We prove that the
probability is described by the Kolmogorov capacitance of F for
case (i) and Howroyd's (1/2)-dimensional box-dimension profiles
for case (ii). We also present sample-path consequences, and a
connection to capacities that answers a question of Benjamini et al (2003).
Keywords.
Dynamical sequences, ε-capacity, box-dimension profiles.
AMS Classification (2000)
60J25, 60J05, 60Fxx, 28A78, 28C20.
Support. The research of D. Kh.
was supported in part by a grant from
the National Science Foundation (DMS-0706728).
Pre/E-Prints. This paper is available in
Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A.
davar@math.utah.edu |
David Asher Levin
Department of Mathematics
University of Oregon
Eugene, OR 97403--1221, U.S.A.
dlevin@uoregon.edu |
Pedro J. Méndez-Hernández
Escuela de Matemática
Universidad de Costa Rica
San Pedro de Montes de Oca, Costa Rica
pedro.mendez@ucr.ac.cr |
Last Update: June 13, 2009
© 2009 - Davar Khoshnevisan, David Asher Levin, and
Pedro J. Méndez-Hernández