Bounded stationary CLT under summable strong mixing

Put ${\gamma }_r=\mathrm{Cov}(Z_0,Z_r)$. By [F1]–[F2], $\mathrm{\mid }{\gamma }_r\mathrm{\mid }\le 4B^2{\alpha }_Z(r)$ for $r\ge 1$, so $D_Z<\mathrm{\infty }$. Expanding the square of a sum indexed by a finite subset $I$ gives its diagonal terms $\mathrm{\mid }I\mathrm{\mid }{\gamma }_0$ and off-diagonal terms $2{\gamma }_{j-i}$ for pairs $i<j$ in $I$. For each fixed positive lag there are at most $\mathrm{\mid }I\mathrm{\mid }$ such pairs. Thus the asserted bound $\mathbb{E}({\sum }_{i\in I}{Z}_i{)}^2\le D_Z\mathrm{\mid }I\mathrm{\mid }$ holds, also for $I=\mathrm{\varnothing }$. For a consecutive block, the same expansion gives $\mathrm{Var}(T_n)\mathrm{/}n={\gamma }_0+2{\sum }_{r=1}^{n-1}(1-r\mathrm{/}n){\gamma }_r$. The summable majorant $2\mathrm{\mid }{\gamma }_r\mathrm{\mid }$ and [F4] imply convergence to the stated ${\sigma }_Z^2$, which is nonnegative as a limit of variances.

Extend the coefficient convention by ${\alpha }_Z(0)=1$. It is nonincreasing: increasing the separation restricts the future sigma-algebra in [F1]. For ordered indices $i\le j\le k\le l$, write $a=j-i$ and $c=l-k$. If $a>0$, centering of $Z_i$ and [F2] applied to $Z_i$ and $Z_j{Z}_k{Z}_l$ give $\mathrm{\mid }\mathbb{E}{Z}_i{Z}_j{Z}_k{Z}_l\mathrm{\mid }\le 4B^4{\alpha }_Z(a)$; if $a=0$, the trivial bound $B^4$ suffices. Splitting off the centered last factor gives the same bound with $c$. Therefore $\mathrm{\mid }\mathbb{E}{Z}_i{Z}_j{Z}_k{Z}_l\mathrm{\mid }\le 4B^4{\alpha }_Z(\max (a,c))$. There are at most $24$ orderings, at most $p$ choices each for the first index and the middle gap, and $2r+1$ pairs $(a,c)$ with maximum $r$. Hence $\mathbb{E}{T}_p^4\le 96B^4{p}^2{\sum }_{r=0}^p(2r+1){\alpha }_Z(r)=o(p^3)\mathrm{.}$ To justify the last limit, for each fixed $R$, $p^{-1}{\sum }_{r\le R}r{\alpha }_Z(r)\to 0$, whereas $p^{-1}{\sum }_{R<r\le p}r{\alpha }_Z(r)\le {\sum }_{r>R}{\alpha }_Z(r)\to 0$ as $R\to \mathrm{\infty }$; the unweighted part divided by $p$ also tends to zero. Repeated indices are included by the convention at zero.

Set ${\epsilon }_p=\mathbb{E}{T}_p^4\mathrm{/}{p}^3$ and ${\eta }_n={\sup }_{p\ge \lfloor \sqrt{n}\rfloor }{\epsilon }_p$. By step 1.2, ${\eta }_n\to 0$. For sufficiently large $n$, define $h_n=\min (n^{1\mathrm{/}8},{\eta }_n^{-1\mathrm{/}4})$, interpreting the second entry as $+\mathrm{\infty }$ if ${\eta }_n=0$, and ignore the finitely many earlier $n$. Then $h_n\to \mathrm{\infty }$, $h_n=o(n^{1\mathrm{/}4})$, and $h_n^2{\eta }_n\to 0$. Put $p_n=\lceil \sqrt{n}{h}_n\rceil$, $q_n=\lfloor \sqrt{n}\mathrm{/}{h}_n\rfloor$, and $k_n=\lfloor n\mathrm{/}(p_n+q_n)\rfloor$. Thus $p_n,q_n\to \mathrm{\infty }$, $p_n\mathrm{/}n\to 0$, $q_n\mathrm{/}{p}_n\to 0$, and $k_n{p}_n\mathrm{/}n\to 1$. Retain $k_n$ consecutive blocks of length $p_n$, separated by discarded blocks of length $q_n$, and discard the final remainder. Their discarded index set $I_n$ has cardinality at most $k_n{q}_n+p_n+q_n=o(n)$. Step 1.1 gives $\mathrm{\parallel }{n}^{-1\mathrm{/}2}{\sum }_{i\in I_n}{Z}_i{\mathrm{\parallel }}_2^2\le D_Z\mathrm{\mid }{I}_n\mathrm{\mid }\mathrm{/}n\to 0$, controlling all gaps together without asserting their independence.

Summability and monotonicity imply $r{\alpha }_Z(r)\to 0$, since $\lfloor r\mathrm{/}2\rfloor {\alpha }_Z(r)\le {\sum }_{s=\lfloor r\mathrm{/}2\rfloor }^r{\alpha }_Z(s)$. For large $n$, $n\mathrm{/}{p}_n\le \sqrt{n}\mathrm{/}{h}_n\le 2q_n$, so $k_n{\alpha }_Z(q_n)\le 2q_n{\alpha }_Z(q_n)\to 0$. Let $B_{n,j}$ denote the sum in retained block $j$. At a fixed real frequency $t$, put $U_j=e^{it{B}_{n,j}\mathrm{/}\sqrt{n}}$. Both each factor and every partial product have modulus at most one. The partial product through block $j$ is measurable before the gap preceding block $j+1$, so [F2] bounds the discrepancy in factoring its expectation from $U_{j+1}$ by $16{\alpha }_Z(q_n)$. Telescoping these factorizations gives $\mid \mathbb{E}{e}^{it{\sum }_{j=1}^{k_n}{B}_{n,j}\mathrm{/}\sqrt{n}}-{\prod }_{j=1}^{k_n}\mathbb{E}{e}^{it{B}_{n,j}\mathrm{/}\sqrt{n}}\mid \le 16(k_n-1){\alpha }_Z(q_n)⟶0.$ This compares characteristic functions only and requires no independent copies or coupling construction.

Write $W_{n,j}=B_{n,j}\mathrm{/}\sqrt{n}$. These variables are centered and each has the law of $T_{p_n}\mathrm{/}\sqrt{n}$ by stationarity. Steps 1.1 and 2.1 imply ${\sum }_{j=1}^{k_n}\mathbb{E}{W}_{n,j}^2=(k_n{p}_n\mathrm{/}n)\mathrm{Var}(T_{p_n})\mathrm{/}{p}_n\to {\sigma }_Z^2$. Also ${\sum }_{j=1}^{k_n}\mathbb{E}{W}_{n,j}^4=\frac{k_n\mathbb{E}{T}_{p_n}^4}{n^2}\le \frac{p_n^2}{n}{\epsilon }_{p_n}⟶0,$ because $p_n^2\mathrm{/}n=O(h_n^2)$ and ${\epsilon }_{p_n}\le {\eta }_n$. For every $\delta >0$, the Lindeberg sum of these block marginals is at most ${\delta }^{-2}{\sum }_j\mathbb{E}{W}_{n,j}^4\to 0$. Applying [F5] first to $\mathrm{\mid }W\mathrm{\mid },W^2$ and then the finite-sum Cauchy–Schwarz inequality yields ${\sum }_j\mathbb{E}\mathrm{\mid }{W}_{n,j}{\mathrm{\mid }}^3\le ({\sum }_j\mathbb{E}{W}_{n,j}^2{)}^{1\mathrm{/}2}({\sum }_j\mathbb{E}{W}_{n,j}^4{)}^{1\mathrm{/}2}\to 0$. Finally ${\max }_j(\mathbb{E}{W}_{n,j}^2{)}^2\le {\sum }_j\mathbb{E}{W}_{n,j}^4\to 0$. No uniform-integrability hypothesis or deterministic block bound tending to zero has been assumed.

For fixed $t$, [F6] gives $\mathbb{E}{e}^{it{W}_{n,j}}=1-a_{n,j}+r_{n,j}$, where $a_{n,j}=t^2\mathbb{E}{W}_{n,j}^2\mathrm{/}2\ge 0$ and ${\sum }_j\mathrm{\mid }{r}_{n,j}\mathrm{\mid }\le \mathrm{\mid }t{\mathrm{\mid }}^3{\sum }_j\mathbb{E}\mathrm{\mid }{W}_{n,j}{\mathrm{\mid }}^3\mathrm{/}3\to 0$. Step 3.2 gives ${\sum }_j{a}_{n,j}\to t^2{\sigma }_Z^2\mathrm{/}2$ and ${\sum }_j{a}_{n,j}^2\le ({\max }_j{a}_{n,j}){\sum }_j{a}_{n,j}\to 0$. The elementary real Taylor bound $\mathrm{\mid }1-a-e^{-a}\mathrm{\mid }\le a^2\mathrm{/}2$ for $a\ge 0$ follows by integrating the second derivative $e^{-a}\le 1$ twice. Since both $\mathrm{\mid }\mathbb{E}{e}^{it{W}_{n,j}}\mathrm{\mid }$ and $e^{-a_{n,j}}$ are at most one, the product telescoping inequality now gives $\mid {\prod }_j\mathbb{E}{e}^{it{W}_{n,j}}-e^{-{\sum }_j{a}_{n,j}}\mid \le {\sum }_j\mathrm{\mid }{r}_{n,j}\mathrm{\mid }+\frac{1}{2}{\sum }_j{a}_{n,j}^2⟶0.$ Thus the product characteristic function tends to $e^{-t^2{\sigma }_Z^2\mathrm{/}2}$.

The discarded normalized sum in step 2.1 tends to zero in $L^2$ and hence in $L^1$ by [F5]. The elementary bound $\mathrm{\mid }{e}^{ix}-e^{iy}\mathrm{\mid }\le \mathrm{\mid }x-y\mathrm{\mid }$, obtained by integrating the derivative of $e^{iu}$ along the real interval, makes its effect on each characteristic function tend to zero. Combine steps 3.1 and 4.1: $\mathbb{E}{e}^{it{T}_n\mathrm{/}\sqrt{n}}\to e^{-t^2{\sigma }_Z^2\mathrm{/}2}$ for every real $t$. Under [A1], [F7] identifies this with the characteristic function of $N(0,{\sigma }_Z^2)$, and [F8] gives the asserted weak convergence. This remains valid for ${\sigma }_Z^2=0$; all finite-moment and discarded-set assertions were already proved in step 1.1.