Homomorphisms from Functional Equations in Probability

From invariance under binomial thinning to unification of the Cauchy and the Gołąb-Schinzel-type equations

“…s 0 > 0: Then (12) gives f (s) = 0 for s ∈ [0, s 0 ). From ( 6) for s = s 0 and t > 0 such that h(s 0 )t < s 0 by (11) we get a = f (s 0 ) which implies that a = 0. Consequently, f is as given in (10).…”

“…For more recent contributions on the Goł ąb-Schinzel equation and its generalizations consult e.g. [7], [8], [9], [4] and [11]. In particular, the latter paper reveals yet another probabilistic (stable laws and random walks) connection of the Goł ąb-Schinzel equation, treated there as a disguised form of the Goldie equation.…”

Sequential regular variation: extensions of Kendall's theorem

“…It then emerges from [BrzM] (cf. [Ost4,§ 9.5] for a more direct approach), and especially [Jab3], that, provided the function H is non-trivial (i.e. its range is not a subset of {0, 1}), then local boundedness of the solution H implies continuity.…”

Stable laws and Beurling kernels

“…The emergence of a particular kind of functional equation, one interpretable as a group homomorphism (see Section 5.3), is linked to the simpler than usual form here of 'probabilistic associativity' (as in [4]) in the incrementation process of the stable random walk; in more general walks, functional equations (and integrated functional equations-see [23]) arise over an associated hypergroup, as with the Kingman-Bessel hypergroup and Bingham-Gegenbauer (ultraspherical) hypergroup (see [4] and [11]). We return to these matters, and connections with the theory of flows, elsewhere- [21].…”