# Comparison of Rademacher processes

Suppose that $$T$$ is a bounded set in $$\mathbb{R}^n$$ and $$f,g$$ are two nonnegative functions such that $$0\leq f(x)\leq g(x)$$ for all $$x\geq 0$$.

Let $$\epsilon_1,\epsilon_2,\dots,$$ be a Rademacher sequence. Does it hold that $$\mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i f(|t_i|) \right| \leq C\cdot \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i g(|t_i|) \right| \qquad (*)$$ for some absolute constant $$C$$?

It looks intuitively true to me but I cannot find it anywhere in the literature. To compare it with the contraction principle that $$\mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i \alpha_i g(|t_i|) \right| \leq \mathbb{E} \sup_{t\in T} \left| \sum_i \epsilon_i g(|t_i|) \right|, \quad |\alpha_i|\leq 1$$ it seems that the $$\alpha_i$$ in my case would depend on $$t$$ and the proof of the contraction principle above does not seem to go through.

Question: Does (*) hold? Is there a reference in the literature or is there a particularly simple proof?

## 1 Answer

$$\newcommand{\ep}{\varepsilon}$$ Letting $$a_{it}:=f(|t_i|)$$ and $$b_{it}:=g(|t_i|)$$, rewrite your inequality ($$\ast$$) as $$\begin{equation} E\sup_{t\in T}\Big|\sum_{i=1}^N\ep_i a_{it}\Big|\le C\,E\sup_{t\in T}\Big|\sum_{i=1}^N\ep_i b_{it}\Big| \end{equation}$$ for any natural $$N$$ and any set $$T$$, with the condition that $$0\le a_{it}\le b_{it}$$ for all $$i,t$$.

Let now $$T:=2^{[N]}$$, the power set of the set $$[N]:=\{1,\dots,N\}$$. Let $$b_{it}:=1$$ and $$\begin{equation} a_{it}:=1_{i\in t} \end{equation}$$ for all $$i\in[N]$$ and $$t\in T=2^{[N]}$$. Then $$\begin{equation} E\sup_{t\in T}\Big|\sum_{i=1}^N\ep_i a_{it}\Big| \ge E\sum_{i=1}^N\max(0,\ep_i)=N/2, \end{equation}$$ whereas
$$\begin{equation} E\sup_{t\in T}\Big|\sum_{i=1}^N\ep_i b_{it}\Big| =E\Big|\sum_{i=1}^N\ep_i\Big|\le\sqrt{E\Big(\sum_{i=1}^N\ep_i\Big)^2}=\sqrt N. \end{equation}$$

So, your inequality ($$\ast$$) will fail to hold for any $$C>0$$ if $$N>4C^2$$.

• Thanks! I'm aware of the contraction principle and was curious whether something like $(*)$ would hold without the contraction assumption. I constructed an example that $(*)$ fails when $C=1$ (take $T=\{(t,\sqrt{1-t^2}): t\in[0,1]\}$, $f(x)=x$ and $g(x)=1$ on $[0,1]$) and I am more or less convinced that I shouldn't expect $(*)$ to hold. – user58955 Apr 21 '19 at 3:49
• Nice example, thanks! – user58955 Apr 21 '19 at 4:29
• Glad to have been able to help. – Iosif Pinelis Apr 21 '19 at 4:30