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 X_i ≥ X_n for exactly r values of i less than or equal to n. Let Y_k = (Y_k,1, Y_k,2, Y_k,3, ...) denote the stochastic process consisting of the terms X_i having initial rank k; that is, Y_k,j is the jth term of the stochastic process that achieves initial rank k. The sequence Y_k 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.

References

• Ilan Adler and Sheldon M. Ross, "Distribution of the Time of the First k-Record", Probability in the Engineering and Informational Sciences, Volume 11, Issue 3, July 1997, pp. 273–278
• Ron Engelen, Paul Tommassen and Wim Vervaat, "Ignatov's Theorem: A New and Short Proof", Journal of Applied Probability, Vol. 25, A Celebration of Applied Probability (1988), pp. 229–236
• Ignatov, Z., "Ein von der Variationsreihe erzeugter Poissonscher Punktprozess", Annuaire Univ. Sofia Fac. Math. Mech. 71, 1977, pp. 79–94
• Ignatov, Z., "Point processes generated by order statistics and their applications". In: P. Bartfai and J. Tomko, eds., Point Processes and Queueing Problems, Keszthely (Hungary). Coll. Mat. Soc. 5. Janos Bolyai 24, 1978, pp. 109–116
• Ross, S. M., "Introduction to probability models", Section 3.6.5, Academic press, 12th edition, 2019
• Samuels, S., "All at once proof of Ignatov's theorem", Contemp. Math. 125, 1992, pp. 231–237
• Yi-Ching Yao, "On Independence of k-Record Processes: Ignatov's Theorem Revisited", The Annals of Applied Probability, Vol. 7, No. 3 (Aug., 1997), pp. 815–821
• Doctor Honoris Causa degree, 2013, in English
• Doctor Honoris Causa degree, 2013, in Bulgarian
• In memoriam, 2024, in Bulgarian