Ibragimov–Iosifescu conjecture for φ-mixing sequences explained

Ibragimov–Iosifescu conjecture for φ-mixing sequences in probability theory is the collective name for 2 closely related conjectures by Ildar Ibragimov and .

Conjecture

Let

(X _{n,n\in\Bbb} N) be a strictly stationary \phi -mixing sequence, for which E(X ^{2 )< inf ty} _{0} and \operatorname{Var}(S _{n)\to} +inf ty . Then S _{n:=\sum} ^{nX} ^{j} is asymptotically normally distributed.\phi -mixing coefficients are defined as\phi _{X(n):=\sup(|\mu(B\mid} A) - \mu(B)|, A\in l F ^{m,} B\in l F _{m+n} ,m\in\Bbb N ) ,where l F ^{m} and l 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 { Z}) is a strictly stationary sequence of random variables such thatEX _{0} = 0 , EX ^{2} _{0} < inf ty and ES ^{2} _{n} \to inf ty as n \to inf ty (that is, such that it has finite second moments and \operatorname{Var}(X _{1} + \ldots + X _{n)} \to inf ty as n \to inf ty ).

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 ^{2} _{1} < inf ty and \sigma ^{2} _{n} \to inf ty . 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 . 10.1016/0304-4149(90)90008-G .