Homomorphisms from Functional Equations in Probability

Adam J. Ostaszewski ()
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Adam J. Ostaszewski: London School of Economics

Chapter Chapter 9 in Developments in Functional Equations and Related Topics, 2017, pp 171-213 from Springer

Abstract: Abstract We showcase the significance to probability theory of homomorphisms and their simplifying rôle by reference to the Goldie functional equation (GFE), an equation at the heart of regular variation theory (RV) encoding asymptotic flows, but with an apparent lack of symmetry. Like the Gołąb–Schinzel equation (GS), of which it is a disguised equivalent, it and its Pexiderized form can be transmuted into homomorphy under a ‘generalized circle product’ due to Popa, conformally with the Pompeiu equation. This not only forges a specific direct connection to Beurling’s Tauberian Theorem, but also generally both helps simplify classical RV-analysis, lending it a flow-type intuition as a guide, and elevates it to unfamiliar contexts. This is illustrated by a new approach to the one-dimensional random walks with stable laws. We review some new literature, offer some new insights and, in Sections 9.4 and 9.5, some new contributions; possible generalizations are indicated in Section 9.6.

Keywords: Random walks; Stable laws; Goldie equation; Gołąb–Schinzel equation; Regular variation; Circle groups; Hypergroups; Primary 26A03; Secondary 26E07 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-61732-9_9

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