  Khnichin's theorem is a surprising and still relatively little known result.
It can be used as a specific criterion for determining whether or not any given
number is irrational. In this paper we apply this theorem as well as the
Gauss--Kuzmin theorem to several thousand high precision (up to more than 53000
significant digits) initial Stieltjes constants $\gamma _{n}$, $n=0,1,...,5000$
in order to confirm that, as is commonly believed, they are irrational numbers
(and even transcendental). We study also the normality of these important
constants.
