Let $(X_n)_n$ be a sequence of i.i.d random variables following the uniform distribution $\mathcal{U}([0;1]).$ Let $f:\mathbb{R}^3 \rightarrow\mathbb{R},$ be a continuous function.
Prove that $$W_n=\frac{1}{\sqrt{n}}(\sum_{k=1}^nf(X_k,X_{k+1},X_{k+2})-n\int_{[0;1]^3}f(x,y,w)dxdydw)$$ converges in distribution.
The exercise seems to be an application of the central limit theorem, since $Y_n=f(X_n,X_{n+1},X_{n+2})-E[f(X_1,X_2,X_3)]$ have the same distribution, but the problem is that $(Y_n)_n$ are not independent, I tried to write $$W_n=\frac{1}{\sqrt{n}}\sum_{k=1}^{\left \lfloor{\frac{n}{3}}\right \rfloor}Y_{3k}+\frac{1}{\sqrt{n}}\sum_{k=0}^{\left \lfloor{\frac{n-1}{3}}\right \rfloor}Y_{3k+1}+\frac{1}{\sqrt{n}}\sum_{k=0}^{\left \lfloor{\frac{n-2}{3}}\right \rfloor}Y_{3k+2},$$ This way, $(Y_{3n})_n,(Y_{3n+1})_n,(Y_{3n+2})_n$ are independent, so we can apply the central limit theorem, but in the expression of $W_n$ the considered random variables aren't independent so that we can say that the sum converges in distribution.
Is there a way to solve the problem? And more generally, the result remains true if $(X_n)_n$ is i.i.d without knowing the related distribution?
This exercise is inspired from problem 10, page 156, from Theory of Probability and Random Processes. Maybe there is a way using Markov chains and ergodicity, but this is not in the program, since in the mentioned book, Markov chains and related results are explained, later, in other chapters.
Let $Y_i:=f(X_i,X_{i+1},X_{i+2})$ and $\widetilde{Y}_i:=Y_i-\mathsf{E}Y_i$. It is clear that $Y_i$ and $Y_{i+j}$ are independent for $j\ge 3$. Split the sum of $\widetilde{Y}_i$'s into $K$ blocks of size $p$ with gaps of size $2$ between each pair of consecutive blocks, i.e., write \begin{align} \sum_{i=1}^{n} \widetilde{Y}_i&=(\widetilde{Y}_1+\cdots+\widetilde{Y}_p)+(\widetilde{Y}_{p+1}+\widetilde{Y}_{p+2})+(\widetilde{Y}_{p+3}+\cdots+\widetilde{Y}_{2p+2})+\cdots\\ &=\sum_{j=1}^K B_{n,j}+\sum_{j=1}^K S_{n,j}+R_n, \end{align} where $B_{n,j}:=(\widetilde{Y}_{(j-1)p+2j-1}+\cdots+\widetilde{Y}_{jp+2(j-1)})$, $S_{n,j}=\widetilde{Y}_{jp+2j-1}+\widetilde{Y}_{jp+2j}$, and $R_n$ is the remainder term.
Choose $K=\ln(n)$ and $p=n/\ln(n)$. Since $\{B_{n,j}\}_{j=1}^K$ is a triangular array of centered i.i.d. random variables, we can apply the central limit theorem (for triangular arrays). The approximation error consists of at most $3\ln(n)$ terms ($K$ terms of size $2$ and at most $\ln(n)$ terms in $R_n$) and is asymptotically negligible.