View source for Ibragimov–Iosifescu conjecture for φ-mixing sequences

{{Short description|Collective name for 2 closely-related conjectures in probability theory}}
{{No footnotes|date=October 2022}}
'''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]].

== Conjecture ==
Let <math>(X_n,n\in\Bbb N)</math> be a strictly stationary <math>\phi</math>-mixing sequence, for which <math>\mathbb E(X_0^2)<\infty</math> and <math>\operatorname{Var}(S_n)\to +\infty</math>. Then <math>S_n:=\sum_{j=1}^nX_j</math> is asymptotically normally distributed.

<math>\phi</math>
-mixing coefficients are defined as
<math>\phi_X(n):=\sup(|\mu(B\mid A)-\mu(B)|, A\in\mathcal F^m, B\in \mathcal F_{m+n},m\in\Bbb N )</math>,
where <math>\mathcal F^m</math> and <math>\mathcal F_{m+n}</math> are the <math>\sigma</math>-algebras generated by the <math>X_j, j\leqslant m</math> (respectively <math>j\geqslant m+n</math>), and <math>\phi</math>-mixing means that <math>\phi_X(n)\to 0</math>.

Reformulated:

Suppose <math>X:=(X_k, k \in {\mathbf Z})</math> is a strictly stationary sequence of random variables such that
<math>EX_0 = 0, \ EX_0^2 < \infty</math> and <math>ES_n^2 \to \infty</math> as <math>n \to \infty</math> (that is, such that it has finite second moments and <math>\operatorname{Var}(X_1 + \ldots + X_n) \to \infty</math> as <math>n \to \infty</math>).

Per Ibragimov, under these assumptions, if also <math>X</math> is <math>\phi</math>-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 <math>\{X_n\}_n</math> be a strictly stationary, centered, <math>\phi</math>-mixing sequence of random variables such that <math>EX^2_1 < \infty</math> and <math>\sigma^2_n \to \infty</math>. Then per Ibragimov <math>S_n / \sigma_n \overset{W}{\to} N(0, 1)</math>, and per Iosifescu <math>S_{[n1]} / \sigma_n \overset{W}{\to} W</math>. Also, a related conjecture by [[Magda Peligrad]] states that under the same conditions and with <math>\phi_1 < 1</math>, <math>\overset{\sim}{W}_n \overset{W}{\to} W</math>.

== Sources ==
*I.A. Ibragimov and [[Yuri Linnik|Yu.V. Linnik]], Independent and Stationary Sequences of Random Variables, [[:nl:Noordhoff Uitgevers|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. [[:ro:Editura Academiei Române|Publishing House of the Romanian Academy]], [[Bucharest]], 1977.
*{{cite journal |last=Peligrad |first=Magda |date=August 1990 |title=On Ibragimov–Iosifescu conjecture for φ-mixing sequences |url= |journal=Stochastic Processes and their Applications |volume=35 |issue=2 |pages=293-308 |doi=10.1016/0304-4149(90)90008-G |access-date=|doi-access= }}

[[Category:Conjectures]]
[[Category:Probability theory]]
{{DEFAULTSORT:Ibragimov-Iosifescu conjecture for φ-mixing sequences}}