Ignatov's theorem

In probability and mathematical statistics, Ignatov's theorem is a basic result on the distribution of record values of a stochastic process.

Statement
Let X1, X2, ... be an infinite sequence of independent and identically distributed random variables. The initial rank of the nth term of this sequence is the value r such that Xi ≥ Xn for exactly r values of i less than or equal to n. Let denote the stochastic process consisting of the terms Xi having initial rank k; that is, Yk,j is the jth term of the stochastic process that achieves initial rank k. The sequence Yk is called the sequence of kth partial records. Ignatov's theorem states that the sequences Y1, Y2, Y3, ... are independent and identically distributed.

Note
The theorem is named after Tzvetan Ignatov (1942-2024) a Bulgarian professor in probability and mathematical statistics at Sofia University. Due to it and his general contributions to mathematics, Prof. Ignatov was granted a Doctor Honoris Causa degree in 2013 from Sofia University. The recognition is given on extremely rare occasions and only to scholars with internationally landmark results.