Ibragimov's block-independence lemma: $\|E[\prod_j f_j]-\prod_j E[f_j]\|\le 16(m-1)\alpha(n)\prod_j M_j$

Let $Y=(Y_i)_{i\ge0}$ be a measurable process on a probability space $(\Omega,\mathcal F,P)$ with strong mixing coefficients $\alpha(n)$. Consider $m$ blocks of time indices

separated by gaps of length at least $n$, that is $b_j+n\le a_{j+1}$ for every $j$, and let $f_j$ be a complex random variable measurable with respect to $\sigma(Y_i: a_j\le i\le b_j)$ with $\|f_j\|_\infty\le M_j$. Then the expectation of the product factorizes up to an error of $16\alpha(n)$ per junction:

This is Ibragimov's block-independence lemma, the combinatorial heart of every characteristic-function proof of a central limit theorem for mixing sequences. Taking $f_j=\exp(\mathrm{i}t\,S_j)$, where $S_j$ is the sum of the observations in the $j$-th block (so $M_j=1$), it says that the characteristic function of a sum of well-separated big blocks differs from that of independent blocks by at most $16(m-1)\alpha(n)$. In the Bernstein big-block/small-block scheme the number of blocks $m$ grows with the sample size while the gap $n$ grows fast enough that $m\,\alpha(n)\to0$, which is precisely how a mixing-rate hypothesis such as $\sum_n\alpha(n)^{\delta/(2+\delta)}<\infty$ or $\alpha(n)=O(a^n)$ enters the proof.

The bound follows from the two-block case, the covariance inequality $|E[UV]-E[U]E[V]|\le16\|U\|_\infty\|V\|_\infty\alpha(n)$ for bounded complex variables measurable with respect to a past and a future separated by $n$, applied once at each of the $m-1$ junctions.

Formalization Note The block schedule is given by two sequences $a,b:\mathbb N\to\mathbb N$ and the hypotheses constrain only the indices actually used, $j<m$; blocks are allowed to be arbitrary index intervals rather than consecutive runs. For $m=0$ and $m=1$ both sides are $0$, so the statement is trivially true there and the constant $m-1$ is a truncated natural subtraction. Boundedness is stated pointwise.