A Multiplicatively Symmetrized Version of the Chung-Diaconis-Graham Random Process

Martin Hildebrand ()
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Martin Hildebrand: University at Albany, State University of New York

Journal of Theoretical Probability, 2022, vol. 35, issue 2, 1216-1225

Abstract: Abstract This paper considers random processes of the form $$X_{n+1}=a_nX_n+b_n\pmod p$$ X n + 1 = a n X n + b n ( mod p ) where p is odd, $$X_0=0$$ X 0 = 0 , $$(a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots $$ ( a 0 , b 0 ) , ( a 1 , b 1 ) , ( a 2 , b 2 ) , … are i.i.d., and $$a_n$$ a n and $$b_n$$ b n are independent with $$P(a_n=2)=P(a_n=(p+1)/2)=1/2$$ P ( a n = 2 ) = P ( a n = ( p + 1 ) / 2 ) = 1 / 2 and $$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$$ P ( b n = 1 ) = P ( b n = 0 ) = P ( b n = - 1 ) = 1 / 3 . This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $$(\log p)^2$$ ( log p ) 2 steps suffice for $$X_n$$ X n to be close to uniformly distributed on the integers mod p for all odd p while order $$(\log p)^2$$ ( log p ) 2 steps are necessary for $$X_n$$ X n to be close to uniformly distributed on the integers mod p.

Keywords: Chung–Diaconis–Graham random process; Random walks; Integers mod p; 60B15; 60G50 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01088-3

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