TY - JOUR

T1 - Remarks on the speeds of a class of random walks on the integers

AU - Boudabra, Maher

AU - Markowsky, Greg

N1 - Funding Information:
We would like to thank an anonymous referee for helpful comments.
Publisher Copyright:
© 2021 Elsevier B.V.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/8

Y1 - 2021/8

N2 - In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance [2] and [1]. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of recurrent random walks of having speed zero, and show by example that this property need not be shared by deterministic sequences. However, if we consider the inter-arrival times (times at which the sequence is equal to 0) then we find a sufficient deterministic condition for a sequence to have zero speed, and show that this can be used to derive several probabilistic results.

AB - In recent years, there has been an interest in deriving certain important probabilistic results as consequences of deterministic ones; see for instance [2] and [1]. In this work, we continue on this path by deducing a well known equivalence between the speed of random walks on the integers and the growth of the size of their ranges. This result is an immediate consequence of the Kesten-Spitzer-Whitman theorem, and by appearances is probabilistic in nature, but we will show that it follows easily from an elementary deterministic result. We also investigate the common property of recurrent random walks of having speed zero, and show by example that this property need not be shared by deterministic sequences. However, if we consider the inter-arrival times (times at which the sequence is equal to 0) then we find a sufficient deterministic condition for a sequence to have zero speed, and show that this can be used to derive several probabilistic results.

KW - Random walk

KW - Speed of random walks

UR - http://www.scopus.com/inward/record.url?scp=85105590532&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2021.112436

DO - 10.1016/j.disc.2021.112436

M3 - Article

AN - SCOPUS:85105590532

VL - 344

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 8

M1 - 112436

ER -