It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces.
The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.
denote the collection of all probability measures defined on
The following statements are equivalent: Since Prokhorov's theorem expresses tightness in terms of compactness, the Arzelà–Ascoli theorem is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the modulus of continuity or an appropriate analogue—see tightness in classical Wiener space and tightness in Skorokhod space.
There are several deep and non-trivial extensions to Prokhorov's theorem.