# Anti-concentration of Gaussian quadratic form

Let $X_1,\dots,X_n$ denote i.i.d. standard Gaussian random variables. My question is: Do there exist absolute constants $C,c>0$ such that for every $\epsilon>0$ and positive real numbers $a_1,\dots,a_n>0$, we have the following bound

$\quad \Pr(\sum_{i=1}^n a_i X_i^2 \le \epsilon \sum_{i=1}^n a_i)\le C\epsilon^c\quad ?$

Background: Results of Carbery and Wright (2001) give powerful anti-concentration inequalities for polynomials of Gaussian variables. If we replace the term $\epsilon \sum_{i=1}^n a_i$ by $\epsilon\sqrt{\sum_{i=1}^n a_i^2}$ the bound follows directly from Carbery-Wright with $c=1/2$. Here, however, we need a stronger bound in terms of the $\ell_1$-norm rather than $\ell_2$-norm of the polynomial.

The reason why I believe the claim is still true is that all the variables $X_i$ appear squared so that there are no cancellations. Also, for $n=1$ the claim follows from simple Gaussian anti-concentration bounds with $c=1/2.$

Note: A simple application of Paley-Zygmund gives the bound $\Pr(\sum a_i X_i^2 \ge \epsilon\sum a_i )\ge (1-\epsilon)^2/3.$ This, unfortunately does not imply the statement above.

• Does the implicit constant in $O(\epsilon^c)$ allowed to depend on $n$ and $a_1,\ldots,a_n$? I assume not, but the way it's stated makes it sound like it does to me. Apr 25, 2012 at 0:32
• No, it's supposed to be an absolute constant independent of $a_1,\dots,a_n$ and $n$. I will update the question. Apr 25, 2012 at 0:56
• Anyway, at first glance it looks like it should hold for $c=1/2$. Is that what you expect, or can you rule that case out? (certainly $c > 1/2$ is ruled out by looking at $n=1$). Apr 25, 2012 at 1:07
• Yes, I would expect $c=1/2,$ but I don't want to discourage somebody with an answer that gives $c<1/2.$ Apr 25, 2012 at 1:12
• Using Chernoff's inequality, you can get the result in one line. See my post below. Aug 16, 2020 at 7:55

We can show that $$\mathbb{P}\left(\sum_ia_iX_i^2\le\epsilon\sum_ia_i\right)\le\sqrt{e\epsilon}$$ so that the inequality holds with $c=1/2$ and $C=\sqrt{e}$.

For $\epsilon\ge1$ the right hand side is greater than 1, so the inequality is trivial. I'll prove the case with $\epsilon < 1$ now. Without loss of generality, we can suppose that $\sum_ia_i=1$ (just to simplify the expressions a bit). Then, for any $\lambda\ge0$, \begin{align} \mathbb{P}\left(\sum_ia_iX_i^2\le\epsilon\right)&\le\mathbb{E}\left[e^{\lambda\left(\epsilon-\sum_ia_iX_i^2\right)}\right]\cr &=e^{\lambda\epsilon}\prod_i\mathbb{E}\left[e^{-\lambda a_iX_i^2}\right]\cr &=e^{\lambda\epsilon}\prod_i\left(1+2\lambda a_i\right)^{-1/2}\cr &\le e^{\lambda\epsilon}\left(1+2\lambda\right)^{-1/2}. \end{align} Take $\lambda=(\epsilon^{-1}-1)/2$ to obtain $$\mathbb{P}\left(\sum_ia_iX_i^2\le\epsilon\right)\le e^{(1-\epsilon)/2}\sqrt{\epsilon}.$$

• I don't think this (proof, if not the result) is correct. The first inequality in this proof looks like an incorrect application of Markov's inequality.
– AatG
Nov 14, 2013 at 19:44
• The first inequality is just taking expectations of $1_{\{\sum a_iX_i^2\le\epsilon\}}\le e^{\lambda(\epsilon-\sum a_iX_i^2)}$. Nov 14, 2013 at 21:38
• Ah, yes, carry on, good sir :)
– AatG
Nov 15, 2013 at 22:46

I also would strongly suspect it holds with $c = 1/2$. Here's an argument which, if I haven't made a mistake, gives the upper bound $O(\epsilon^{1/2} \ln^{1/2}(1/\epsilon))$.

Assume without loss of generality that $a_1 \geq a_2 \geq \cdots \geq a_n$ and that $a_1 + \cdots + a_n = 1$. Divide into two cases:

Case 1: $a_1 \geq \frac{1}{100\log(1/\epsilon)}$. In this case it's sufficient to show that we already have $\Pr[a_1 X_1^2 \leq \epsilon] \leq O(\epsilon^{1/2} \ln^{1/2}(1/\epsilon))$. But this holds as you said because of the anticoncentration of a single Gaussian.

Case 2: $a_1 \leq \frac{1}{100\log(1/\epsilon)}$. In this case, note that $\sigma^2 := \sum_i a_i^2 \leq a_1 \sum_i a_i \leq \frac{1}{100\ln(1/\epsilon)}$. Now the random variables $X_i^2$ are nice enough that one should be able to apply a Chernoff Bound to them. (I think I could provide a source for this if necessary.) Since $Y := \sum a_i X_i^2$ has mean $1$ and standard deviation $\sqrt{2}\sigma$ (I think, maybe the constant $\sqrt{2}$ is wrong), we should have a statement like $\Pr[|Y - 1| \geq t \cdot \sqrt{2}\sigma] \leq \exp(-t^2/c)$, where $c$ is a quite modestly small universal constant. So $\Pr[Y \leq \epsilon] \leq \Pr[|Y-1| \geq 1/2] \leq \exp(-\frac{1}{8c\sigma^2}) \leq \exp(-\frac{12.5}{c} \ln(1/\epsilon)) = \epsilon^{12.5/c}$, which is smaller than $\epsilon^{1/2}$ if $c$ is not too large, and even if $c$ is too large, one can make $100$ larger.

You can directly use Chernoff's inequality, to get $$LHS \le e^{-\Lambda^*(\epsilon)}$$, where $$\Lambda^*(\epsilon) := (\epsilon-1-\log\epsilon)/2$$ is the Fenchel-Legendre transform of the log-MGF $$\Gamma$$ of $$X_1^2$$ (this has been computed in this SE post). Simplifying then gives

$$LHS \le e^{-\Lambda^*(\epsilon)} = e^{-(\epsilon-1-\log\epsilon)/2} = e^{(1-\epsilon)/2}\sqrt{\epsilon} \le \sqrt{e\epsilon}.$$

It looks like all the previous answers (including mine) are quite off by a huge margin, if the vector $$a=(a_1,\ldots,a_n)$$ is somewhat dense in the sense that $$\|a\|_1/\|a\|_\infty$$ is substantially larger than $$1$$. For example, if $$a_i=1/n$$ for all $$i$$, then we would expect to have the upper-bound to be of orde $$\epsilon^{\Omega(n)}$$, and not $$\epsilon^{1/2}$$. See Fact 2.1 of this blogpost.

In this post, I will provide an upper-bound which has somewhat optimal dependence on $$a$$.

## Main tool: concentration function under affine transformation

Given a random vector $$Y$$ taking values in $$\mathbb R^m$$, its concentration function is defined by setting, for any $$\epsilon \ge 0$$,

$$\mathcal L(Y,\epsilon) := \sup_{y \in \mathbb R^m} \mathcal L_y(Y,\epsilon),\text{ where }\mathcal L_y(Y, \epsilon) := \mathbb P(\|Y-y\|_2 \le \epsilon).$$

Now, let $$A$$ be a deterministic $$m \times n$$ matrix and let $$X=(X_1,\ldots,X_n)$$ be a random vector taking values in $$\mathbb R^n$$, with independent components verifying $$\max_{1 \le i \le n}\mathcal L(X_i,\epsilon) \le \ell(\epsilon).$$ It is well-known (e.g see Theorem 1.5 of this paper by M. Rudelson) that for any $$\delta \in (0,1)$$, there exists a positive constant $$C_\delta$$ such that $$\mathcal L(AX,\epsilon\|A\|_{HS}) \le (C_\delta \ell(\epsilon))^{(1-\delta)r(A)},$$ where $$r(A) := \|A\|_{HS}^2/\|A\|_{op}^2 \ge 1$$. In particular, if the $$X_i$$'s have standard normal distribution, then we may take $$\ell(\epsilon)=\epsilon$$, and get for any $$\epsilon \ge 0$$ and $$\delta \in (0,1)$$, $$\mathcal L(AX,\epsilon\|A\|_{HS}) \le (C_\delta \epsilon)^{(1-\delta)r(A)}.$$

## Application to OP's question

In particular, taking $$A$$ to be the $$n \times n$$ diagonal matrix $$\mbox{diag}(\sqrt a)$$ with $$a = (a_1,\ldots,a_n)$$, we have

$$\begin{split} \mathbb P(\sum_{i=1}^n a_i X_i^2 \le \epsilon \sum_{i=1}^n a_i) &= \mathbb P(\|AX\|_2 \le \sqrt\epsilon\|A\|_{HS})\\ & = \mathcal L_0(AX, \sqrt\epsilon\|A\|_{HS})\\ & \le \mathcal L(AX,\sqrt\epsilon\|A\|_{HS}) \\ &\le (C_\delta \sqrt \epsilon)^{(1-\delta)r(A)}\\ & = (C_\delta \sqrt{\epsilon})^{(1-\delta)s(a)}, \end{split}$$ where $$s(a) := \|a\|_1 / \|a\|_\infty \in [1,n]$$ measures the sparsity of $$a$$. As a sanity check, if $$a_i=c > 0$$ for all $$i \in [n]$$, then $$s(a) = \|a\|_1 / \|a\|_\infty = n$$, and our upper-bound above recovers the correct order in $$\epsilon$$, namely $$\epsilon^{\Omega(n)}$$.