- Question:
Let $Z=f(X_1,...,X_n)$ be a function of $n$ independent random variables $X_1,...,X_n$. Let $X_i'$ be an independent copy of $X_i$. Use martingale decomposition to prove $$ \text{var}[Z]\le \frac{1}{2}\sum_{i=1}^n E[(f(X_1,...,X_{i-1},X_i',X_{i+1},...,X_n)-f(X_1,...,X_n))^2] $$
- Other information:
- Our professor gives a hint: first prove for an arbitrary random variable $Y$, if $Y'$is an independent copy, then $\text{var}[Y]=\frac{1}{2}E[(Y-Y')^2]$. This is easy to do, just a direct calculation.
- By martingale decomposition, I think our prof refers to the following operation: let X be an arbitrary random variable $X\in L^2(\Omega,\mathcal{F},P)$. $\mathcal{F}_0=\{\emptyset,\Omega\}\subset \mathcal{F}_1\subset ... \subset \mathcal{F}_n=\mathcal{F}$ is a filtration. Then let $X_i=E[X|\mathcal{F}_i]$, we have the following decompostion: $$ X = E[X] + \sum_{i=1}^n X_i - X_{i-1} $$ From this, by the orthogonality of martingale increments, we will get: $$ \text{var}[X]=\sum_{i=1}^n E[(X_i-X_{i-1})^2]= \sum_{i=1}^n E[\text{var}[X_i|\mathcal{F_i}]] $$ Our Prof says $\text{var}[X_i|\mathcal{F_i}]$ can be controlled by coupling techs, but I do not know how to deal with it, and I cannot find some materials about this method.
Could someone help me with it? Thanks a lot.