Characterization problems in mathematical statistics are statements in which the description
of possible distributions of random variables follows from properties of some
functions in these variables. One of the famous examples of a characterization problem
is the classical Kac–Bernstein theorem ([65], [13]). This theorem characterizes
a Gaussian distribution by the independence of the sum 1 C 2 and of the difference
1 2 of independent random variables j . Taking into account that the characteristic
function of the random variable j with distribution j is the expectation
fj .y/ D Oj .y/ D EŒeij y, it is easily verified that the Kac–Bernstein theorem is
equivalent to the statement that, in the class of normalized continuous positive definite
functions, all solutions to the Kac–Bernstein functional equation
f1.u C v/f2.u v/ D f1.u/f1.v/f2.u/f2.v/; u; v 2 R;
are of the form fj .y/ D expfy2 C ibj yg, where  0, and bj 2 R.