Ibragimov–Iosifescu conjecture for φ-mixing sequences

Short description: Collective name for 2 closely-related conjectures in probability theory

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

[math]\displaystyle{ \phi }[/math] -mixing coefficients are defined as [math]\displaystyle{ \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]\displaystyle{ \mathcal F^m }[/math] and [math]\displaystyle{ \mathcal F_{m+n} }[/math] are the [math]\displaystyle{ \sigma }[/math]-algebras generated by the [math]\displaystyle{ X_j, j\leqslant m }[/math] (respectively [math]\displaystyle{ j\geqslant m+n }[/math]), and [math]\displaystyle{ \phi }[/math]-mixing means that [math]\displaystyle{ \phi_X(n)\to 0 }[/math].

Reformulated:

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

Per Ibragimov, under these assumptions, if also [math]\displaystyle{ X }[/math] is [math]\displaystyle{ \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]\displaystyle{ \{X_n\}_n }[/math] be a strictly stationary, centered, [math]\displaystyle{ \phi }[/math]-mixing sequence of random variables such that [math]\displaystyle{ EX^2_1 \lt \infty }[/math] and [math]\displaystyle{ \sigma^2_n \to \infty }[/math]. Then per Ibragimov [math]\displaystyle{ S_n / \sigma_n \overset{W}{\to} N(0, 1) }[/math], and per Iosifescu [math]\displaystyle{ 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]\displaystyle{ \phi_1 \lt 1 }[/math], [math]\displaystyle{ \overset{\sim}{W}_n \overset{W}{\to} W }[/math].

Sources

• 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.