Ibragimov–Iosifescu conjecture for φ-mixing sequences

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

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

Reformulated:

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

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

• 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.