Variance modulo 1

Question

The fact that the variance of the sum of independent random variables is the sum of their variances allows one to have a good understanding of how well-concentrated each term $X_i$ in a sum of $n$ independent random variables $S= X_1+X_2+\dots+X_n$ has to be in order for $S$ to be concentraded around the mean. It seems that an analogous fact should be true for the sum taken modulo $1$. More concretely, I believe that the following should be true, but I cannot seem to find a proof or a reference. Help finding either of these would be highly appreciated.

Lemma (tentative). Suppose that $X_1, X_2, ..., X_n$ are (jointly) independent real random variables with $\mathbb{E} X_i = 0$ for each $i \leq n$, and that there exists a real number $\gamma$ such that $$ \mathbb{E} [ \lVert X_1 + X_2 + \dots + X_n - \gamma \rVert_{\mathbb{R}/\mathbb{Z}}^2] \leq \varepsilon $$ for some small $\varepsilon > 0$. Then $$ \mathbb{E} [ \lVert X_1\rVert_{\mathbb{R}/\mathbb{Z}}^2 ] + \mathbb{E} [ \lVert X_2\rVert_{\mathbb{R}/\mathbb{Z}} ^2 ] + \dots + \mathbb{E} [ \lVert X_n\rVert_{\mathbb{R}/\mathbb{Z}}^2 ] \leq C \varepsilon,$$ where $C$ is an absolute constant. Here, $\lVert x\rVert_{\mathbb{R}/\mathbb{Z}} $ denotes the distance of $x$ from the nearest integer.

Rationale:

1. The lemma would be true if $\lVert x\rVert_{\mathbb{R}/\mathbb{Z}} $ were replaced with $\lvert x\rvert_{\mathbb{R}/\mathbb{Z}} $; indeed this is just the additivity of variance.

2. It is not hard to check that the lamma is true if $X_i$ have normal distribution or if $X_i$ have Bernoulli distribution with vales $0$ and $1/2$.

3. On the intuitive level, if we imagine that the distribution of $X_1+\dots+X_n$ is roughly bell-shaped with a single maximum, the only way for $\mathbb{E} [ \lVert X_1 + X_2 + \dots + X_n - \gamma \rVert_{\mathbb{R}/\mathbb{Z}}^2]$ to be small is if $X_1 + X_2 + \dots + X_n$ is very strongly concentrated around that maximum. In particular the ``wrap-around'' issues should not play much role and the above lemma should be roughly equivalent to the version with absolute value in place of the distance from $\mathbb{Z}$.

Answer 1

On the one hand, the proof is very cheap. Let $Z_j=e^{2\pi iX_j}$. $X=\sum_j X_j$, $Z=e^{2\pi i X}$. Note that $\operatorname{Var}_{\mathbb R/\mathbb Z}X\approx 1-|EZ|$ and similarly for $X_j$ and $Z_j$. Now just use the identity $EZ=\prod_j EZ_j$ to conclude.

On the other hand, finding the reference may be a highly non-trivial task, so I leave it to somebody else :-).

P.S. What can be really concluded here is that there exist $\gamma_j$ summing to $\gamma$ with $\sum_jE\|X_j-\gamma_j\|_{\mathbb R/\mathbb Z}^2\le C\varepsilon$. The condition $EX_j=0$ does not allow one to conclude from here that we can take $\gamma_j=0$: take $X_j$ symmetric with values about $\pm \frac 12$ for a counterexample.