Assume that $ ( X , \mathcal A , \mu ) $ is a probability space, $ T : X \to X $ is an ergodic transformation, and $ f \in L ^ 2 ( \mu ) $ is such that $ \int f \ \mathrm d \mu = 0 $ and $ \frac { S _ m ( f ) } { \| S _ m ( f ) \| } $ converges in distribution to $ \mathcal N ( 0 , 1 ) $. Here, $ \mathcal N ( 0 , 1 ) $ is the standard normal distribution, $ \| . \| $ is the $ L ^ 2 ( \mu ) $ norm and $ S _ m ( f ) = \sum _ { i = 0 } ^ { m - 1 } f \circ T ^ m $. Prove that for any $ \phi \in L ^ 2 ( \mu ) $, $ \frac { S _ m ( f + \phi \circ T - \phi ) } { \| S _ m ( f + \phi \circ T - \phi ) \| } $ also converges in distribution to $ \mathcal N ( 0 , 1 ) $.
This problem comes from studying the Central Limit Theorem (CLT) for dynamical systems. In "On the Central Limit Theorem for Dynamical Systems" by Burton and Denker, the authors claim without proof that if the function $ f \in L ^ 2 ( \mu ) $ satisfies CLT, then for any functions of the form $ \phi \circ T - \phi $ for some ergodic $ T $, $ f + \phi \circ T - \phi $ also satisfies CLT. Consequently, the functions satisfying CLT are dense in the subspace of the functions in $ L ^ 2 ( \mu ) $ with zero integral, because functions of the form $ \phi \circ T - \phi $ are dense in that subspace.
I can observe that $ S _ m ( f + \phi \circ T - \phi ) = S _ m ( f ) + \phi \circ T ^ m - \phi $. As $ T $ is ergodic, the fraction $ \frac { S _ m ( f ) + \phi \circ T ^ m - \phi } m $ converges almost sure to $ 0 $, by Birkhoff ergodic theorem. But I can't replace the $ m $ in the denominator by $ \| S _ m ( f ) + \phi \circ T ^ m - \phi \| $.
I'm familiar with the proof of CLT in case of an i.i.d.; but in this case $ ( f \circ T ^ m ) _ m $ is not an i.i.d. sequence. I can't figure out how to approach this problem.
Burton, Robert; Denker, Manfred, On the central limit theorem for dynamical systems, Trans. Am. Math. Soc. 302, 715-726 (1987). ZBL0628.60030.
I can see the result if we assume that $\lVert S_m(f) \rVert\to\infty$. In this case, $$ \lVert S_m(f) \rVert-2\lVert \phi \rVert\leqslant \| S _ m ( f ) + \phi \circ T ^ m - \phi \|\leq \lVert S_m(f) \rVert+2\lVert \phi \rVert $$ hence $$\tag{1} \lim_{m\to\infty}\frac{ \| S _ m ( f ) + \phi \circ T ^ m - \phi \|}{\lVert S_m(f) \rVert}=1 $$ and $$ \tag{2} \left(\phi\circ T^m-\phi\right)/\lVert S_m(f) \rVert\to 0\mbox{ in probability} $$ hence the central limit theorem for $f+\phi\circ T-\phi$ follows from (1), (2) and standard properties of convergence in distribution of random variables.
However, it might be an other story if we do not assume that $\lVert S_m(f) \rVert\to\infty$. Indeed, let $\Omega$ be a product space $\mathbb R^{\mathbb Z}$, $T$ the shift and $\mu$ the product measure of $\mu_i$, the standard Gaussian measure on $\mathbb R$. Let $f\colon (x_i)_{i\in\mathbb Z}\mapsto x_0-x_1$, then $S_m(f)/\lVert S_n(f)\rVert$ is standard normal. However, adding a co-boundary of the form $\phi\circ T-\phi$ where $\phi\colon (x_i)_{i\in\mathbb Z}\mapsto x_0x_1-x_1x_2$ can spoil the central limit theorem.
Nevertheless, it is not what is claimed in the paper: they just say that when, the dynamical system is ergodic, the co-boundaries are dense in the subspace of mean-zero functions, giving an other argument for the proof of the fact that the set of the functions which satisfy the central limit theorem is dense in $\mathbb L^2$.
In other words, if $CLT$ is the set of square integrable functions such satisfying the central limit theorem in the paper, pick a centered $f_0$ in the set $CLT$ such that $\lVert S_n(f_0)\rVert\to\infty$ (the existence is given by Theorem 1. Then by the argument given at the beginning of the answer, $f_0+\phi\circ T-\phi$ belongs to $CLT$ for each square integrable function $\phi$ hence the closure of $CLT$ in $\mathbb L^2$ contains the closure of the set of functions of the form $f_0+\phi\circ T-\phi$, which is $\mathbb L^2$.