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Recently I came across the following question when I read a paper about Gaussian processes:

Let $g_1,g_2,\ldots,g_N$ be i.i.d. standard Gaussian random variables in $\mathbb{R}$. Then, for any $1\le m\le N$,

$$ \mathbb{E} \max_{|I|=m} \bigg(\sum_{i\in I} g_i^2\bigg)^{1/2} \asymp \sqrt{m\log (eN/m)}, $$ where the maximum is taken over all subsets of $\{1,2,\ldots,N\}$ whose cardinality is $m$. The notation $a \asymp b$ means that $cb\le a\le Cb$ for two absolute constants $c$ and $C$.

My question is: how to prove the bounds stated above? Thanks for any possible ideas or comments!

user1247096
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1 Answers1

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A partial solution at least. For any $I$ with $|I|=m$ let $S_I=\left(\sum_{i\in I} g_i^2\right)^{1/2}$. There are $n = {N\choose m}$ distinct sets $I$, which we can enumerate as $I_1,\ldots,I_n$. Since the $g_i$ are independent, it is well known that all the $S_{I_\ell}$ follow a chi distribution $\chi(m)$.

Denote by $\mu$ the expected value of $\chi(m)$. One useful fact about the chi distribution (not too hard to believe if you look at its density function) is that $\chi(m) - \mu$ is 1-subgaussian, meaning \begin{equation*} \mathbb{E}\left[e^{\lambda (\chi(m) - \mu)}\right] \leq e^{\lambda^2/2} \end{equation*} for all $\lambda$. See here (Proposition 13) for a proof. Now, we can use certain inequalities that are available for subgaussian random variables. For instance, use equation (1) from this answer to get \begin{equation*} \mathbb{E}\left[\max_{1\leq\ell\leq n} (S_{I_\ell} - \mu)\right] \leq \sqrt{2\log n} = \sqrt{2\log {N \choose m}}. \end{equation*} Several bounds and approximations are available for the binomial coefficient. For our case, it is sufficient to use ${N \choose m} \leq \left(\frac{eN}{m}\right)^m$. Hence, from the last expression we get: \begin{equation*} \mathbb{E}\left[\max_{1\leq\ell\leq n} (S_{I_\ell} - \mu)\right] \leq \sqrt{2m\log \frac{eN}{m}}. \end{equation*} Now, take $\mu$ to the right-hand side of the bound and use the fact that $\mu\approx \sqrt{m-1}$ to get the upper bound you need. I am less sure on how to derive the matching lower bound. If the $S_{I_\ell}$ were independent there are some tools available, but they are not.

acrendic
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