Ibragimov–Iosifescu conjecture for φ-mixing sequences

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 $(X_{n},n\in {\mathbb {N}})$ be a strictly stationary $\phi$-mixing sequence, for which $\mathbb {E} (X_{0}^{2})<\infty$ and $\operatorname {Var} (S_{n})\to +\infty$. Then $S_{n}:=\sum _{j=1}^{n}X_{j}$ is asymptotically normally distributed.

$\phi$ -mixing coefficients are defined as $\phi _{X}(n):=\sup(|\mu (B\mid A)-\mu (B)|,A\in {\mathcal {F}}^{m},B\in {\mathcal {F}}_{m+n},m\in {\mathbb {N}})$, where ${\mathcal {F}}^{m}$ and ${\mathcal {F}}_{m+n}$ are the $\sigma$-algebras generated by the $X_{j},j\leqslant m$ (respectively $j\geqslant m+n$), and $\phi$-mixing means that $\phi _{X}(n)\to 0$.

Reformulated:

Suppose $X:=(X_{k},k\in {\mathbf {Z} })$ is a strictly stationary sequence of random variables such that $EX_{0}=0,\ EX_{0}^{2}<\infty$ and $ES_{n}^{2}\to \infty$ as $n\to \infty$ (that is, such that it has finite second moments and $\operatorname {Var} (X_{1}+\ldots +X_{n})\to \infty$ as $n\to \infty$).

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

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.