I have a set of random variables $X_1,\ldots,X_n$, all Gaussian with mean 0 and variance 1, indepedent. Let $p(x_1,\ldots,x_n)$ be some polynomial that takes products and sums of $x_1,\ldots,x_n$.
What can be said about the concentration of measure of $p(X_1,\ldots,X_n)$ around $E[p(X_1,\ldots,X_n)]$?
If there were only two-order interactions, I think I would look around for concentration of measure for chi-squared random variables, but unfortunately the interaction can be of a higher degree.
Best Answer
Hypercontractivity implies that for a polynomial $P$ of total degree $d$ in Gaussian variables and $q \geq 2$, we have $$ \|P\|_{L^q} \leq (q-1)^{d/2} \|P\|_{L^2} .$$ Applying Markov inequality for the optimal $q$ yields then for $t \geq C_d$ $$ \mathrm{Prob} \left(|P- \mathbf{E} P| \geq t \sqrt{\mathrm{Var}(P)} \right) \leq \exp(-c_d t^{2/d} ) $$ for some constants $C_d,c_d$.
Reference: Corollary 5.49 in Aubrun-Szarek, Alice and Bob meet Banach