Gaussian approximation of the characteristic function of Rademacher sum

Question

I am searching an uniform bound for the characteristic function of some Rademacher sum. Specificaly I want to estimate how much the characteristic function is close to a Gaussian.

We have in general the sum $\sum_{i=1}^{+\infty}a_i \epsilon_i$ where the ${\epsilon_i}$ are independent Rademacher random variable and $\sum_{i=1}^{+\infty}a_i^2 <+\infty$.

The characteristic function will be $\phi(t) =\prod_{i=1}^{+\infty}\cos(a_i t)$.

So I want to find a $K$ depending of $\{a_i\}_{i\in \mathbb{N}}$ so that $\sup_{t\in \mathbb{R}}\left|\prod_{i=1}^{+\infty}\cos(a_i t)-\mathrm{e}^{-\frac{||a||_2^2t^2}{2}}\right|<K$.

The goal is to find a criteria determining for which sequence $\{a_i\}_{i\in \mathbb{N}}$ the characteristic function is close to a gaussian. Obviously in the case $a_i=\frac{1}{2^i}$ we have $\phi(t)=\mathrm{sinc}(t)$ which is far to be a Gaussian. I think it have to do with the speed of decay of the $a_i$.

I looked throught the different concentration inequalities such as the Hoeffding's inequality or the Berry–Esseen inequality but most of these inequalities concern the cumulative distribution function.

So I am wondering if there is not a way to translate these inequalities to inequalities concerning the characteristic function.

Answer 1

Too long for a comment, but not a full solution.

Let $X, Y$ be random variables with CDFs $F_X, F_Y$, and characteristic functions $\phi_X, \phi_Y$. It looks like you want an upper bound on $\lVert \phi_X-\phi_Y\rVert_\infty$ in terms of some data relating to $F_X, F_Y$. As I mentioned in the comments, there is a decent amount of study obtaining "reverse" bounds of this type, meaning upper bounds on $\lVert F_X-F_Y\rVert_\infty$ in terms of data related to $\phi_X, \phi_Y$. See for example Proximity of probability distributions in terms of Fourier–Stieltjes transforms by Bobkov.

Anyway, section 19 of that gives something almost of the form you want. Namely

$$\lVert F_X-F_Y\rVert_\infty \geq \frac{1}{2\sqrt{2\pi}}\left|\int_{-\infty}^\infty (\phi_X(t)-\phi_Y(t))\exp(-t^2/2)dt\right|.$$

While this is not an upper bound for $\lVert \phi_X-\phi_Y\rVert_\infty$ in terms of data related to $F_X, F_Y$, it seems worth mentioning still, in case this weaker bound suffices for your application.

Separately, one can follow section 1 (i.e. solely integrate by parts) and take the $\ell_\infty$ norm of the display between equations 1.1 and 1.2 to get that

$$\lVert \phi_X-\phi_Y\rVert_\infty =\lVert t\int_{-\infty}^\infty \exp(ixt)(F_X(x)-F_Y(x))dx\rVert_\infty.$$

where the $\ell_\infty$ norm is for the indicated function of $t$. This also gives some indication of what others have mentioned, namely the issue is that of large $t$. For $t \leq 1$, the above can easily be bounded by $\lVert F_X - F_Y\rVert_\infty$, which might be good enough for you.

In general though, to get non-trivial bounds it suffices to show that the Fourier transform of $F_X(x) - F_Y(x)$ is $o(t^{-1})$ as a function of $t$. I think this will be dependent on the differentiability properties of $F_X - F_Y$ (though I'm no Fourier analyist). Hopefully this reduces the problem to something more tractable though.

Answer 2

Too long for a comment. Have you tried comparing Taylor expansions? I get $$ \prod_{i=1}^n \cos(a_i t) = 1-\left(\frac12\sum_{i=1}^n a_i^2\right)t^2 +\frac{1}{24}\left( \left(\sum_{i=1}^n a_i^2\right)^2+4\sum_{1\le i<j\le n}a_i^2a_j^2 \right) +O(t^6) $$ and, of course, $$ \exp(-||a||^2 t^2/2) = 1-\frac12||a||^2 t^2 +\frac18||a||^4 t^4 +O(t^6). $$

The 2nd-order terms match, so perhaps the condition on the $a_i$ should involve getting the 4th-order terms to match?..

Answer 3

A general comment which admittedly doesn't answer the question: One cannot choose the $a_i$ so that the sum would be Gaussian. For one can break up the sum into two independent pieces, and necessarily each would have to be Gaussian, since the Gaussian cannot have nonGaussian convolution factors. As an aside, neither the Lyapunov or Lindeberg conditions for the Central limit Theorem can be satisfied. To approach normality, some subset of the sum would need to be almost normal, with the remaining part of the sum being small. Of course, your question lies deeper than that.