Functional equations and distribution functions

Title: Functional equations and distribution functions
Creator: Borwein, Jonathan M.; Girgensohn, Roland
Relation: Results in Mathematics Vol. 26, Issue 3-4, p. 229-237
Publisher Link: http://dx.doi.org/10.1007/BF03323043
Publisher: Birkhaeuser Verlag
Resource Type: journal article
Date: 1994
Description: We consider the functional equation f(t)={1∖over b}{∖mathop∖sum∧{b-1}∖limits_{∖nu=0}}f∖Bigg({t-∖beta_{∖nu}∖over a}∖Bigg)∖∖∖{∖rm for∖ all}∖ t∖in {∖rm R}, where 0 < a < 1, b in N{1} and −1 = β₀ ≤ β₁ ≤ … ≤ β_b-1 =1 are given parameters, f R → R is the unknown. We show that there is a unique bounded function f which solves (F) and satisfies f(t) = 0 for t <~-1/(1 − a), f(t) = 1 for t > 1/(1 − a). This solution can be interpreted as the distribution function of a certain random series. It is known to be either singular or absolutely continuous, but the problem for which parameters it is absolutely continuous is largely open. We collect some previously established partial answers and generalize them. We also point out an interesting connection to the so-called Schilling equation.
Subject: distribution functions; Bernoulli distributions; singular functions; Schilling equation; Vieta’s product
Identifier: http://hdl.handle.net/1959.13/1042564
Identifier: uon:14081
Identifier: ISSN:1422-6383
Language: eng
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