Ibragimov–Iosifescu conjecture for φ-mixing sequences in probability theory is the collective name for 2 closely related conjectures by Ildar Ibragimov and ro:Marius Iosifescu.

Let be a strictly stationary -mixing sequence, for which and . Then is asymptotically normally distributed.

-mixing coefficients are defined as , where and are the -algebras generated by the (respectively ), and -mixing means that .

Reformulated:

Suppose is a strictly stationary sequence of random variables such that and as (that is, such that it has finite second moments and as ).

Per Ibragimov, under these assumptions, if also is -mixing, then a central limit theorem holds. Per a closely related conjecture by Iosifescu, under the same hypothesis, a weak invariance principle holds. Both conjectures together formulated in similar terms:

Let be a strictly stationary, centered, -mixing sequence of random variables such that and . Then per Ibragimov , and per Iosifescu . Also, a related conjecture by Magda Peligrad states that under the same conditions and with , .

• I.A. Ibragimov and Yu.V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen, 1971, p. 393, problem 3.
• M. Iosifescu, Limit theorems for ϕ-mixing sequences, a survey. In: Proceedings of the Fifth Conference on Probability Theory, Brașov, 1974, pp. 51-57. Publishing House of the Romanian Academy, Bucharest, 1977.
• Peligrad, Magda (August 1990). "On Ibragimov–Iosifescu conjecture for φ-mixing sequences". Stochastic Processes and their Applications. 35 (2): 293–308. doi:10.1016/0304-4149(90)90008-G.