Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.

The paper gives a survey of the theory of univariate characteristic functions. These functions were originally introduced as tools in the study of limit theorems but it was later realized that they had an independent mathematical interest. Those parts of the theory which can be found in textbooks are treated only briefly; the main emphasis is placed on more recent developments and areas where active research is still in progress.