Let $X\_1,\ldots, X\_n$ be independent Rademacher random variables (i.e. $\mathbb{P}(X\_i=\pm 1)=1/2$). Consider the random polynomial $$P\_{n}(t)=c+X\_{1}t+X\_2t^2+\cdots+X\_{n}t^n.$$ 


Is it well known how to get good upper bounds on probabilities of type
$$\mathbb{P}(|\max\_{t\in [0,x]}|P\_{n}(t)|-\mathbb{E}\max\_{t\in [0,x]}|P\_{n}(t)||>y)?$$



Concentration of maxima of a random polynomial with Rademacher coefficients
Theorem 5.3.2 in [Talagrand's book](https://link.springer.com/book/10.1007/978-3-642-54075-2) states the following (using here somewhat different notations): 



> 
> Let $X\_1,X\_2,\dots$ be independent Rademacher random variables. Let $U$ be a subset of the closed ball $B(u\_0,s)$ in $\ell^2$ centered at some $u\_0\in\ell^2$ and of some radius $s>0$. Let 
>  $$S:=\sup\_{u\in U}\Big(c(u)+\sum\_{i\ge1}u\_iX\_i\Big),$$
>  where $c\colon U\to\mathbb R$. Then 
>  $$P(|S-ES|\ge y)\le C\exp\Big(-\frac{y^2}{Cs^2}\Big) $$
>  for some universal real constant $C>0$ and all real $y>0$. 
> 
> 
> If $ES$ in the latter display is replaced by the median of $S$, then $C$ may be replaced by $4$.
> 
> 
> 


Take now any real 
$$s\ge\inf\_{u\in\mathbb R^n}\max\_{t\in[0,x]}\sqrt{\sum\_{j=1}^n(u\_j-t^j)^2}.$$
Then, by the above statement, the probability in question is bounded from above by $ C\exp\Big(-\dfrac{y^2}{Cs^2}\Big)$. In particular, here one may take $s=\sqrt{\sum\_{j=1}^n x^{2j}}$. 


