Rough estimation for $L^2$ norm of sum of random variables

Question

Consider three real-valued random variables $X_1,X_2,X_3$ on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$. Further assume $X_1$ and $X_3$ to be independent and let $\mathbb{E}(X_i) = 0$ and $\Vert X_i \Vert_2^2 = \mathbb{E}(X_i^2) < \infty$ for all $i=1,2,3$. Is there a way, to find a constant $C > 0$ not depending on the $X_i$ such that $$ \mathbb{E}((X_1+X_2)^2) \leq C \mathbb{E}((X_1+X_2+X_3)^2)?$$

Answer 1

Fix any $\lambda >0$. Let $X_1,X_2$ be independent where $\operatorname{Var}[X_1]=1$ and $\operatorname{Var}[X_2]=\lambda$, and $X_3=-X_2$. We have $$ \mathbb{E}[(X_1+X_2+X_3)^2] = \|X_1\|_2^2 = 1 $$ while $$ \mathbb{E}[(X_1+X_2)^2] = \|X_1\|_2^2+\|X_2\|_2^2 = 1+\lambda $$ This holds for any $\lambda>0$, and thus rules out any inequality of the form you want, which would imply $\lambda \leq C-1$.