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Orthogonality of Measures Induced by Random Walks with Scenery

Published online by Cambridge University Press:  12 September 2008

C. Douglas Howard
C. Douglas Howard
Affiliation:
Department of Applied Mathematics and Physics, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA Email: [email protected]
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Abstract

Thinking of a deterministic function s: ℤ → ℕ as ‘scenery’ on the integers, a simple random walk on ℤ generates a random record of scenery ‘observed’ along the walk. We address this question: If t:ℤ → ℕ is another scenery on the integers and we are handed a random scenery record obtained from either s or t, under what circumstances can the source be distinguished? We allow ourselves to use information about s and t together with information contained in the scenery record. It has been conjectured that it is sufficient for t to be neither a translate of s nor a translate of the reflection of s. We show that this condition is sufficient to ensure distinguishability if s−1(δ) is finite and non-empty for some δ ∈ℕ.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 3 , September 1996 , pp. 247 - 256
Copyright
Copyright © Cambridge University Press 1996

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References

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