Ibragimov–Iosifescu conjecture for φ-mixing sequences

Template:Short description Template:No footnotes 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 $\varphi$-mixing sequence, for which $𝔼(X_0^2)<\mathrm{\infty }$ and $\mathrm{Var}(S_n)\to +\mathrm{\infty }$. Then $S_n:=\sum _{j=1}^n{X}_j$ is asymptotically normally distributed.

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

Reformulated:

Suppose $X:=(X_k,k\in 𝐙)$ is a strictly stationary sequence of random variables such that $E{X}_0=0,E{X}_0^2<\mathrm{\infty }$ and $E{S}_n^2\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ (that is, such that it has finite second moments and $\mathrm{Var}(X_1+\dots +X_n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$).

Per Ibragimov, under these assumptions, if also $X$ is $\varphi$-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, $\varphi$-mixing sequence of random variables such that $E{X}_1^2<\mathrm{\infty }$ and ${\sigma }_n^2\to \mathrm{\infty }$. Then per Ibragimov $S_n/{\sigma }_n\stackrel{W}{\to }N(0,1)$, and per Iosifescu $S_{[n1]}/{\sigma }_n\stackrel{W}{\to }W$. Also, a related conjecture by Magda Peligrad states that under the same conditions and with ${\varphi }_1<1$, ${\stackrel{\sim }{W}}_n\stackrel{W}{\to }W$.

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.
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