Arithmetic properties of multiplicative integer-valued perturbed random walks
Volume 11, Issue 2 (2024), pp. 133–148
Pub. online: 4 January 2024
Type: Research Article
Open Access
Received
4 October 2023
4 October 2023
Revised
12 December 2023
12 December 2023
Accepted
12 December 2023
12 December 2023
Published
4 January 2024
4 January 2024
Abstract
Let (ξ1,η1), (ξ2,η2),… be independent identically distributed N2-valued random vectors with arbitrarily dependent components. The sequence (Θk)k∈N defined by Θk=Πk−1⋅ηk, where Π0=1 and Πk=ξ1⋅⋯⋅ξk for k∈N, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets {Π1,Π2,…,Πk}⊂N and {Θ1,Θ2,…,Θk}⊂N, k∈N, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.
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