A note on absorption probabilities in one-dimensional random walk via complex-valued martingales
Dennis Gilliland,
Shlomo Levental and
Yimin Xiao
Statistics & Probability Letters, 2007, vol. 77, issue 11, 1098-1105
Abstract:
Let {Xn,n[greater-or-equal, slanted]1} be a sequence of i.i.d. random variables taking values in a finite set of integers, and let Sn=Sn-1+Xn for n[greater-or-equal, slanted]1 and S0=0 be a random walk on , the set of integers. By using the zeros, together with their multiplicities, of the rational function , we characterize the space U of all complex-valued martingales of the form {g(Sn),n[greater-or-equal, slanted]0} for some function . As an application we calculate the absorption probabilities of the random walk {Sn,n[greater-or-equal, slanted]0} by applying the optional stopping theorem simultaneously to a basis of the martingale space U. The advantage of our method over the classical approach via the Markov chain techniques (cf. Kemeny and Snell [1960. Finite Markov Chains. Van Nostrand, Princeton, NJ.]) is in the size of the matrix that is needed to be inverted. It is much smaller by our method. Some examples are presented.
Keywords: Random; walks; Martingales; Optional; stopping; theorem; Absorption; probabilities (search for similar items in EconPapers)
Date: 2007
References: View complete reference list from CitEc
Citations:
Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0167-7152(07)00048-X
Full text for ScienceDirect subscribers only
Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.
Export reference: BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text
Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:77:y:2007:i:11:p:1098-1105
Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01
Access Statistics for this article
Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul
More articles in Statistics & Probability Letters from Elsevier
Bibliographic data for series maintained by Catherine Liu ().