Concentration Square Increments of a MDS I have a martingale difference sequence $\{ X_t \}$ where each $X_t$ is subGaussian.
Are there concentration inequalities for
$$
\sum_{t=1}^T X^2_t - E \left( \sum_{t=1}^T X^2_t \right)
$$
 A: It seems that in inequalities for $\sum_{t=1}^T X^2_t - E \left( \sum_{t=1}^T X^2_t \right)$, the fact that $\{ X_t \}$ is a martingale differences sequences cannot be really exploited. For example, let $\left(\varepsilon_t\right)_{t\geqslant 1}$ be a i.i.d. sequence of random variable taking the values $+1$ and $-1$ with probability $1/2$ and let $\left(Y_t\right)_{t\geqslant 1}$ be any sequence of sub-Gaussian random variables. Then defining $X_t:=\varepsilon_tY_t$ and $\mathcal F_t$ the $\sigma$-algebra generated by $\varepsilon_i$, $1\leqslant i\leqslant t$ and $Y_k$, $k\geqslant 1$, the sequence $\left(X_t,\mathcal F_t\right)_{t\geqslant 1}$ is a martingale differences sequence and 
$$\sum_{t=1}^T X^2_t - E \left( \sum_{t=1}^T X^2_t \right)=\sum_{t=1}^T Y^2_t - E \left( \sum_{t=1}^T Y^2_t \right).$$
Without specifying any independence assumption on $\left(Y_t\right)_{t\geqslant 1}$, nothing can be said. As extreme examples, one could assume $\left(Y_t\right)_{t\geqslant 1}$ i.i.d., or $Y_t=Y$ for all $t$.
