I have the following statistics/probability question:
Let $A$ and $B$ be two finite sets of real numbers. Let $n \geq 1$. Let $T_1,...,T_n$ be random variables with values in $A$ (not necessarily independent). Let $Q$ be a random variable with values in $B$. Let $\epsilon_1, \epsilon_2,...,\epsilon_n$ be a sequence of independent random variables which are identically normally distributed, with mean 0 and variance $\sigma>0$. Assume that for all $1 \leq i \leq n$, $\epsilon_i$ is independent of $T_i$ and of $Q$ (note that $\epsilon_i$ may be correlated with $T_{i-1}$ for instance). Let \begin{equation*} X_i=T_i+Q+\epsilon_i \end{equation*} I would like to construct a random variable $Q_n$ that is $(X_1,X_2,...,X_n)$-measurable, and such that for some constant $C$ and $d>0$ that depend only on $A$, $B$ and $\sigma$, we have \begin{equation*} \left\|Q_n-Q\right\|_{L^2} \leq C n^{-d}. \end{equation*} In words, knowing $X_1,X_2,...,X_n$, I would like to approximate $Q$ in an efficient way.
Many thanks!